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Products of manifolds with fibered corners

Manifolds with fibered corners arise as resolutions of stratified spaces, in many body compactifications of vector spaces, moduli spaces, and other settings. We define a category of fibered corners manifolds which has products and transverse fiber products, generalizing both the resolutions of products of stratified spaces and many body products, which are special cases. The product in the fibered corners category is a resolution of the cartesian product by blow-up which we call the 'ordered product'. This ordered product is a natural product for wedge (aka incomplete edge) metrics and quasi-fibered boundary metrics, a class which includes QAC and QALE metrics.


Introduction
There is a large body of work in geometric analysis in which non-compact and/or singular spaces are compactified and/or resolved to manifolds with corners, on which detailed analysis, especially of asymptotic expansions at boundary faces, gives precise results not readily available by other means. In many of these problems, the manifolds admit a natural fibered corners structure in which each boundary hypersurface is the total space of a fiber bundle comparable in a precise sense to its neighboring bundles at the corners.
Fibered corners structures arise in particular in two settings: the first setting is the resolution of (smoothly) stratified spaces [5,3,4,1], which are often equipped with wedge (or 'iterated incomplete edge') metrics, Riemannian metrics degenerating conically along the strata in an iterated fashion. The second setting is typified in simplest form by many body spaces, which are vector spaces which have been radially compactified and subsequently lown up along the boundaries of a family of linear subspaces [31,21,6]. Euclidean metrics on the original vector spaces become quasi-asymptotically conic (QAC) metrics on the associated many body spaces, and this asymptotic geometry generalizes to the manifold setting in the form of QAC manifolds [12] and even more generally in the form of 'quasi-fibered boundary' (QFB, aka Φ) manifolds [10], extending the scattering and fibered boundary structures of [27] and [26] on manifolds with boundary, respectively, as well as the quasi-asymptotically locally euclidean (QALE) metrics introduced by Joyce [17,19,8]. Important examples of QFB manifolds include the compactifications of the hyperKähler moduli spaces of SU(2) monopoles of arbitrary charge [13]. Roughly speaking, spaces arising in the first setting are modelled (iteratively and in parameterized fashion) on the small ends of cones, while those arising in the second setting are modelled on the large ends.
While a fair amount has been written about manifolds with fibered corners, this article establishes some aspects of an associated category, in particular the existence of products and fiber products in this category. The results presented here will have applications to the construction of various pseudodifferential calculi associated to manifolds with fibered corners. In particular, these results are used in an essential way in [24] and [23], the authors' work on elliptic pseudodifferential operator analysis and L 2 -cohomology of QFB manifolds. In fact, it will be shown in a forthcoming work that a wide variety of existing pseudodifferential caculi adapted to various geometric settings can be constructed within the fibered corners category, and that doing so simplifies and clarifies many of their features.
In more detail, a fibered corners structure 1 on a compact (not necessarily connected) manifold with corners X consists of a locally trivial fiber bundle structure φ H : H → B H (hereafter simply called a fibration) on each boundary hypersurface H (again not necessarily connected), the base B H and typical fiber F H of which are also manifolds with corners (and in fact fibered corners), satisfying a comparability condition wherever two hypersurfaces intersect (see Definition 4.1), which among other things determines a partial order on the set M 1 (X) of boundary hypersurfaces, in which H < H ′ whenever H ∩ H ′ = ∅ and H has a strictly coarser fibration. As a matter of notation, we write H ∼ H ′ when H and H ′ are comparable, meaning H = H ′ , H < H ′ , or H ′ < H.
A key observation, seemingly overlooked in the literature, is that this structure and associated partial order extends to the larger set M 1,0 (X) = M 1 (X) ∪ {X} of boundary hypersurfaces along with the interior, provided the interior itself is equipped with a fibration φ X : X → B X similarly comparable to the φ H . The two most common situations arise by taking one of the two trivial fibration structures on X: • If B X = X and φ X = 1, making X ∈ M 1,0 (X) maximal in the order, we say X is interior maximal, denoted by X = X max , which turns out to be natural in the setting of stratified spaces. • If B X = * is a single point and φ X is the constant map, making X ∈ M 1,0 (X) minimal in the order, we say X is interior minimal, denoted by X = X min , which turns out to be natural in the setting of many body spaces and QFB manifolds. We treat the general case of an arbitrary fibration on X below, obtaining results for interior maximal and interior minimal spaces by specialization.
Even with interior fibrations specified, the cartesian product X × Y is not generally a manifold with fibered corners in any natural way, since the product fibrations are not generally comparable where they meet. Instead, the product in the fibered corners category is the ordered product, defined for interior maximal manifolds with fibered corners by the iterated blow-up of all codimension 2 corners of the form H × G, where H ∈ M 1 (X) and G ∈ M 1 (Y ), taken in any order consistent with the partial order on the product M 1 (X) × M 1 (Y ); in other words H ′ × G ′ must be blown-up prior to H × G whenever H ′ < H and G ′ < G, and any order satisfying this condition leads the same space up to diffeomorphism.
On the other hand, for interior minimal manifolds with fibered corners, the ordered product is defined by the opposite order blow-up where ← − M 1 (X) denotes the partially ordered set M 1 (X) with order reversed; in other words H ′ × G ′ must be blown up prior to H × G whenever H ′ > H and G ′ > G. In general, these are not diffeomorphic, i.e., X max ∼ × Y max ∼ = X min ∼ × Y min , whenever one of the factors has a corner of codimension at least 2. Both products are in fact special cases of the general ordered product where M ≷ 1 (X) = {H ∈ M 1 (X) : H ≷ X}. Our main results about the ordered product are summarized as follows.
Theorem (Theorems 4.13, 4.8, 6.6, 6.6). For manifolds X and Y with fibered corners: where ∼ ⋆ denotes the join product (see below). (c) For interior minimal spaces, the boundary fibrations of X min ∼ × Y min have the form (1. 5) In contrast to the ordered product, there is in fact a remarkably non-trivial diffeomorphism X ∼ ⋆ max Y ∼ = X ∼ ⋆ min Y between the minimal and maximal join products which is proved in §7, from which it follows that the associated stratified space to (1.5) is also the topological join of the stratified spaces of the factors. A general version of the join which appears in the boundary fibrations for a general ordered product (1.1) is somewhat more complicated and depends on the place of X in the order on M 1,0 (F H ) and M 1,0 (B H ), viewed as ordered subsets of M 1,0 (X). • The boundary fibrations (1.3) in the interior minimal case (and more generally whenever X and Y are not both maximal) depend on equivalence classes of boundary defining functions on X and Y , where two defining functions are equivalent if their ratio is basic over each boundary hypersurface (see Definition 4.4). Such a choice of equivalence class of boundary defining functions is therefore part of the data for X and Y as objects in the general fibered corners category (unless all spaces are taken to be interior maximal). This equivalence relation also arises naturally in the consideration of QFB metrics (see §6). • The morphisms referenced in part (d) are taken to be those interior b-maps f : X → Y which are simple, meaning all boundary exponents are 0 or 1, -b-normal, in particular boundary hypersurfaces map either to boundary hypersurfaces of the target or to the interior, but never to faces of higher codimension, so determining a map f ♯ : M 1,0 (X) → M 1,0 (Y ), ordered, meaning f ♯ : M 1,0 (X) → M 1,0 (Y ) is a map of ordered sets, -fibered, meaning f restricts over each H ∈ M 1,0 (X) to a map of fiber bundles f H for some f H consistent with the equivalence classes of boundary defining functions on X and Y (see Definition 4.4 for a precise statement), except in the case that all spaces are interior maximal. In contrast to the usual category of manifolds with corners and b-maps 2 , the fibered corners category has all transverse fiber products: Theorem (Theorems 3.12, 3.13 and Proposition 4.14). If f : X → Z and g : Y → Z are fibered corners morphisms which are b-transverse, meaning that is a manifold with fibered corners, embedded as a p-submanifold in the ordered product and satisfying the universal property of the fiber product in the category of fibered corners. The boundary fibrations on X ∼ × Z Y have the form except for the case that H ∈ M 1 (X) and G ∈ M 1 (Y ) are proper boundary hypersurfaces with f ♯ (H) = g ♯ (G) = Z ∈ M 0 (Z), in which case the boundary fibrations have a more complicated form degeneration of each stratum onto the next. There is a well-known equivalence between a smoothly stratified space X on one hand, and a manifold with fibered corners X on the other, wherein X is obtained by collapsing the fibers of the boundary fibrations of X, while X is recovered from X by iteratively resolving the strata by a kind of blow-up which resolves cones to cylinders [3,1].
Theorem (Theorem 5.1). If X and Y are manifolds with fibered corners associated to smoothly stratified spaces X = i s i and Y = j s ′ j , then the interior maximal ordered product X max ∼ × Y max is the manifold with fibered corners associated to the product X ×Y = i,j s i ×s ′ j of stratified spaces.
The second is the product of many body spaces, which are compactifications of finite dimensional real vector spaces V obtained by iteratively blowing up the radial compactification V along the boundaries of a finite set S V of linear subspaces. Such a space is naturally an interior minimal manifold with fibered corners.
Theorem An overview of the paper is as follows. We first recall some notation and background on manifolds with corners in §2, including some of the theory of generalized blow-up from [22] (though all blowups here are ordinary ones), which is used throughout to simplify the proofs of our results. As it happens, a number of properties of the ordered product do not actually depend on the fibration structures of the boundary hypersurfaces of the factors, but rather only on the induced order on their hypersurfaces. For this reason, and since it may have more general applications, we first develop much of the theory in a larger category of ordered corners in §3 in which manifolds are equipped only with a suitable orderings of their hypersurfaces and morphisms are consistent with these. The properties of products in this ordered corners cateogry are proved in §3.1 and the properties of fiber products are proved in §3. 3. In §3.2 we give a detailed analysis of the product structure of boundary hypersurfaces of X ∼ × Y in preparation for the fibered corners category; a key role is played by so-called compressed projection maps which in a certain sense extend the boundary fibrations to fibrations involving a cone over the fiber or base, depending on context. We then review fibered corners structures in §4 and obtain the results stated as parts (a)-(d) of the main product theorem above as well as the fiber product theorem in §4.1. The connection to stratified spaces and many body spaces is discussed in sections §5.1 and §5.2, respectively. In §6 we introduce the geometric structures encoded by the wedge and Φ tangent bundles, proving the product result for Φ structures in §6.1 and for wedge structures in §6.2. In fact we give two proofs each of the product results for wedge and QFB metrics, one based on a direct analysis of the respective tangent bundles, and another alternate proof based on examination of the form of the lifted product metrics. In §7, we prove the diffeomorphism between the minimal and maximal join products, which apart from its conceptual relevance in this paper, gives an unusual example of a diffeomorphism of blow-ups of manifolds with corners which is non-trivial on the interior. Finally, Appendix A contains a short exposition of some technical results concerning tubular neighborhood structures with nice properties and an Ehresmann lemma in the fibered corners category.

Background
We briefly recall some of the important notions from the theory of manifolds with corners used below, though we assume the reader is already somewhat familiar; for more complete references and/or more liesurely introductions, see [15,16,29,28].
A manifold with corners is a Hausdorff space X locally diffeomorphic 3 to open sets in R n + , where R + = [0, ∞). Every point has a well defined codimension, given by the number of vanishing R + factors in any chart, and the closure of a (connected) set of points with a fixed codimension is a boundary face, with the set of boundary faces of codimension k denoted by M k (X); in particular M 1 (X) is the set of boundary hypersurfaces. We write M(X) = k M k (X) for the set of all boundary faces. The interior, X • , of X is the set of points with codimension 0, and the depth of X is the maximum codimension occurring on X, or equivalently the maximum number of boundary hypersurfaces that have a nonempty mutual intersection. We assume boundary hypersurfaces (and therefore all boundary faces) are embedded; in particular each boundary face is again a manifold with corners. A function ρ H : X → R + is boundary defining for H ∈ M 1 (X) if ρ −1 H (0) = H and dρ H = 0 on H; the ratio ρ H /ρ ′ H of two boundary defining functions for H is a strictly positive smooth function.
Note that we do not require that a manifold with corners is connected, nor do we strictly require that boundary faces are connected, despite how they have just been defined! Indeed, it will be useful below to allow certain unions of disjoint boundary hypersurfaces to be considered as a single hypersurface, or what is called in [5] a collective boundary hypersurface. This identification of collective boundary hypersurfaces (i.e., which components are to be considered to belong to 'the same' hypersurface) constitues additional data on the manifold with corners, and in the event that such data has been specified, we abuse notation by using M 1 (X) to refer to the set of collective boundary hypersurfaces; in the absence of such a specification M 1 (X) consists by default of connected components only.
The b-vector fields V b (X) ⊂ V(X) = C ∞ (X; T X) are by definition the Lie subalgebra of smooth vector fields tangent to all boundary faces. These form a locally free sheaf of constant rank, defining the b-tangent vector bundle spanned in local coordinates (x 1 , . . . , x k , y 1 , . . . , y l ) ∈ R k + × R l by x i ∂ x i and ∂ y j . Over x i = 0, the b-tangent vector x i ∂ x i = ρ H i ∂ ρ H i is actually independent of the choice of boundary defining coordinate x i . The inclusion V b (X) ⊂ V(X) induces a natural bundle map b T X → T X which is an isomorphism over the interior; at p ∈ X, the kernel defines the b-normal space b N p F , where F is the unique boundary face whose interior contains p. Taken together, these form a vector subbundle, b N F , of b T X over the interior of F which extends by continuity to the whole of F . In fact b N F → F is trivialized globally by the frame ρ H i ∂ ρ H i , where ρ H i are any boundary defining functions for the hypersurfaces H 1 , . . . , H k whose intersection has F as a connected component.
An interior b-map (hereafter simply a b-map) f : X → Y is a smooth map such that, for any sets {ρ H : H ∈ M 1 (X)} and {ρ G : G ∈ M 1 (Y )} of boundary defining functions for X and Y , where a G > 0 is smooth and strictly positive and the boundary exponents e(G, H) ∈ Z + are nonnegative integers. We will say that a b-map f is rigid with respect to fixed sets of boundary defining functions on X and Y if each a G in (2.1) is identically 1; while this is not a standard notion it turns out to be useful below.
Returning to standard concepts, a b-map f is said to be simple if each e(G, H) ∈ {0, 1}. Under f , every boundary face F ∈ M k (X) is mapped into a unique smallest face G ∈ M l (Y ) (determined by condition that f (F • ) ⊂ G • ), an assignment we denote by In general the differential of a b-map extends by continuity from the interior to a bundle map 3) over f , and this b-differential restricts over F ∈ M(X) to a bundle map where e(G, H) ∈ Z + are the boundary exponents of f .
3) is a surjective (resp. injective) on fibers, and is said to be b-normal provided each map (2.4) is surjective on fibers; note that the latter condition holds if and only if it holds on all hypersurfaces F ∈ M 1 (X). In particular a b-normal map cannot map a boundary hypersurface into any face of codimension 2 or more, so (2.2) restricts to a map a notation which is frequently employed below. An equivalent condition to b-normality is that for each H ∈ M 1 (X), there is at most one G ∈ M 1 (Y ) for which the boundary exponent e(G, H) is nonzero.
A b-normal map can be rigidified in the sense that, for any set of boundary defining functions on Y , there is a set of boundary defining functions on X with respect to which f is rigid; indeed, starting with an arbitrary set {ρ H : H ∈ M 1 (X)}, this is achieved by replacing and g(•, •) forms a right inverse for e(•, •) viewed as a |f ♯ (M 1 (X))|×|M 1 (X)| matrix. In particular, a b-normal map can be put locally into the normal coordinate form: where (x, y) ∈ R k ′ + × R l ′ are coordinates centered at f (p) ∈ Y and (x, y) ∈ R k + × R l are coordinates centered at p ∈ X, and where x = x e is shorthand for Remark. We will occasionally employ the exponential notation . , x k ) and e = [e(j, i)] is an n × k matrix. In this convention (x e ) g = x ge , where ge is the usual matrix product of g and e.
A b-normal b-submersion is said to be a b-fibration, and such a map can always be written in the local normal form f : (x, y) → (x e , pr l ′ y), (2.7) where pr l ′ : (y 1 , . . . , y l ) → (y 1 , . . . , y l ′ ) denotes the projection onto the first l ′ < l coordinates. The restriction of a b-fibration to any boundary face F ∈ M(X) is again a b-fibration f : F → f ♯ (F ) onto its image in Y . A simple b-fibration with the property that each boundary hypersurface G of Y has a unique hypersurface H ∈ f −1 ♯ (G) in X which maps to it is in fact a fiber bundle of manifolds with corners; indeed in this case it follows from the local normal form (2.7) (in which e is a projection matrix up to permutation) that f is a surjective submersion of compact manifolds with corners, and the standard argument for manifolds then shows that it is a fiber bundle.
A subset P ⊂ X is a p-submanifold if it covered by coordinate charts (x, y) in which it is locally defined by the vanishing of codim(P ) of the coordinates; in particular P intersects every hypersurface H ∈ M 1 (X) transversally within the smallest face F ∈ M(X) such that P ⊂ F . Examples include boundary faces themselves.
The blow-up of a p-submanifold P ⊂ X is the space where the front face of the blow-up, S + P , consists of the inward pointing spherical normal bundle of P , the fiber at p ∈ P of which consists of normal vectors to P at p which have unit length with respect to any norm and point inward, with smooth structure generated by polar coordinates normal to P . This is equipped with a canonical b-map β : [X; P ] → X (which is not generally b-normal), given by the bundle projection S + P → P at the front face and the identity elsewhere. If S ⊂ X is another set, the lift of S to [X; P ] is defined by β −1 (S) if S ⊂ P , and the closure of S \ P in [X; P ] otherwise, and provided S lifts to a p-submanifold in [X; P ], the iterated blow-up is well-defined. In particular, every boundary face lifts to a p-submanifold under a blow-up, and the iterated blow-up of any number of boundary faces is always well-defined. Three additional topics from manifolds with corners will be used below. As these are either new or not well-known, we devote a subsection to each for ease of reference.

Normal Models.
For H ∈ M 1 (X) we define ν : N + H → H to be the inward pointing normal (not b-normal!) bundle to H, which will be referred to as the normal model of H in X. That is to say, N H = T X| H /T H, which is a non-canonically trivial line bundle with well-defined inward pointing subspace N + H, taken to include the zero section, with an action by (0, ∞) fixing the zero section H 0 which we identify with H itself. It is convenient to extend this to the case H = X as well, in which case N + X is the unique 0-dimensional bundle X 0 → X, identifed with X itself, with trivial (0, ∞) action. If f : X → Y is a b-normal b-map, then the differential descends to a well-defined (0, ∞)-equivariant map In this latter case df = ν * (f | H ) factors as the composition of the bundle projection ν : This consideration applies in particular to smooth functions, with f ∈ C ∞ (X; R) regarded as a b-map f : X → R with f ♯ (H) = R ∈ M 0 (R), lifting to df = ν * f : N + H → R. In contrast, boundary defining functions on X, regarded as b-maps ρ H ′ : It follows from this observation that (2.8) is a b-normal b-map serving as an infinitesimal model for f itself near H ⊂ X, and we frequently use such models below.

2.2.
Generalized blow-up. We will make use of some of the theory of generalized blow-up developed in [22], and we now recall the relevant details here, keeping the treatment as self-contained as possible. We begin with the observation that within each b-normal space b N p F , F ∈ M(X), there is a freely generated monoid (meaning a commutative semigroup with identity) which is well-defined by the independence of the b-normal vectors ρ i ∂ ρ i of the choices ρ i = ρ H i of boundary defining functions, and which is likewise independent of p ∈ F , so we just denote this monoid by b M F from now on and denote the generator associated to the boundary hypersurface Whenever F ⊂ G, the normal monoid b M G (which is generated by fewer of the same h i ) includes as a face of b M F , which, up to reordering, is just the inclusion Since the coefficients of (2.4) are nonnegative integers, the b-differential of a b-map f : which are consistent with the inclusions (2.9). Note that if f is both simple and b-normal then in (2.10) each generator of b M F maps to a generator of b M f ♯ (F ), and conversely: the condition that generators map to generators in (2.10) implies that f is simple and b-normal.
In the particular case that f = β : X = [X; F 1 , . . . , F n ] → X is the blow-down map of an iterated blow-up of boundary faces {F i } ⊂ M(X) of X, the images under (2.10) of the monoids from X form a complete decomposition of each monoid b M G of the target, in the sense that b M G is a union of such image monoids b df * ( b M G), G ∈ M( X), meeting along boundary faces. Such decompositions constitute what is called a refinement in [22,20].
For example, if F = H 1 ∩ · · · ∩ H l , with b M F = Z + h 1 , . . . , h l , then for any G ⊆ F , the refinement of b M G induced by the blow-up β : [X; F ] → X consists of the l submonoids of b M G in which precisely one of the generators {h 1 , . . . , h l } is replaced by h 1 + · · · + h l ; this corresponds to the geometric process of 'barycentric subdivision' of b M F , considered as a face of b M G. Examples are illustrated in Figure 1.
As summarized below, the monoids of the refinement actually characterize the iterated blow-up [X; F 1 , . . . , F n ] completely up to diffeomorphism, and an arbitrary refinement determines what is called a generalized blow-up 5 , meaning a manifold with corners X and a suitable blow-down map β : X → X, the images of the induced monoid homomorphisms (2.10) of which constitute the given refinement. In particular, the boundary hypersurfaces of X and their incidence relations (meaning their intersection properties) can be simply read off from the monoids of the refinement: hypersurfaces are in bijection with generators (i.e., 1 dimensional monoids) and intersect if the associated generators together generate a 2 dimensional monoid in the refinement. The main results we use below are the following, which assert the characterization of an iterated boundary blow-up by its monoid refinement, as well as a criterion for lifting b-maps under such a blow-up.   meaning maps intertwining addition which send 0 to 0 5 While the refinements considered here arise from ordinary iterated boundary blow-up, the general theory in [22] accounts for inhomogeneous blow-ups, as well as refinements which do not arise from any classical iterated boundary blow-up, hence the term 'generalized blow-up'. The paper [20] extends the notion of refinement and generalized blow-up to the category [18] of manifolds with generalized corners. 6 Or more generally, a generalized blow-up. the property that f = β • f : is an iterated blow-up of boundary faces of Y and and is an iterated blow-up of the same boundary faces in a different order (i.e., for a permutation σ of {1, . . . , n}), then as long as the two associated refinements of the monoids of Y agree, there is a canonical diffeomorphism Y ∼ = Y ′ extending the identity in the interior.
Though we refer to [22] for the full proof of these results, we give a sketch here for the convenience of the unfamiliar reader. First, the space Y may be constructed directly from the monoid refinement of Y as follows. Locally, if (x, y) ∈ R l + × R n−l are coordinates on Y centered at an interior point of a face G ∈ M l (Y ), then for each monoid of dimension l in the refinement of b M G, let ν = [ν (i, j)] ∈ Mat(l × l, Z + ) denote the matrix of the inclusion map with respect to the generators (which is unimodular and in particular invertible) and construct a coordinate patch (t, y) ∈ R l + × R n−l for Y with local blow-down map β : (t, y) → (x, y) = (t ν , y). (2.11) In particular t = x ν −1 gives 'blow-up coordinates' on Y as rational combinations of the x. For adjacent monoids in the refinement, associated to matrices ν and ν ′ respectively, the transition maps are seen to be diffeomorphisms on open subsets where some of the variables (associated to those generators which are not common to both monoids) are strictly positive, and glue together naturally with respect to coordinate transitions on Y itself to determine Y and the blow down map β : Y → Y whose boundary exponents are the ν (i, j).
For the lifting result, suppose f is given locally by ) with a i > 0, so that the matrix µ = [µ (i, j)] ∈ Mat(l × k, Z + ) of boundary exponents represents the associated monoid given locally by {x ′ = 0}). The condition that the latter factors through some monoid in the refinement means that µ = ν µ for some ν as above, and then gives the local coordinate expression for the unique lift f : X → Y ; it is straightforward to check that this is functorial and in particular behaves well with respect to changes of coordinates.

Rational combinations of boundary defining functions.
A rational combination of boundary defining functions {ρ i } on a space X associated to boundary hypersurfaces H 1 , . . . , H n is defined to be product of the form (2.13) It will be of interest to know where in X such a combination determines a smooth function locally, and more importantly, when σ lifts globally to a smooth function under a blow-up β : X → X. These properties may be determined by considering the logarithmic differential Moreover, on such a neighborhood σ (resp. σ −1 ) vanishes on a boundary hypersurface H j ∩ U if and only if b dσ(h j ) > 0 (resp. b dσ(h j ) < 0).
Proof. The first statement is a consequence of σ having a local coordinate expression of the form σ(x, y) = a(x, y) j x a j j where each a j has a determinate sign and a is a smooth positive function. The second statement corresponds to the fact that the local expression for σ has a term x a j j with a j > 0 (resp. a j < 0).
If β : X = [X; F 1 , . . . , F n ] → X is an iterated bounday blow-up, the behavior of the pullback β * σ of σ to X can be analyzed using the fact that b d(β * σ)(m) = b dσ(β * m) for m ∈ b M F , F ∈ M( X). Corollary 2.3. A rational combination σ lifts to a smooth b-map β * σ : X → [0, ∞] if and only if b dσ takes a determinate sign on each monoid in the refinement associated to X, and in this case where h is the generator in the refinement associated to the hypersurface H ∈ M 1 ( X).
Note that boundary defining functions on a blow-up Y of Y are given locally by rational combinations of boundary defining functions from Y , as follows from writing t = x ν −1 in (2.11). Moreover, if a map f : X → Y is locally rigid and satisfies the hypotheses of Theorem 2.1, then its lift f : X → Y is rigid with respect to the rational boundary defining functions on Y as follows from (2.12).
It is evident from Corollary 2.3 that a set of the form {σ = 1} ⊂ X (or more generally {σ = c} for c ∈ (0, ∞)) lifts to an interior p-submanifold of a blow-up X whenever σ lifts to be smooth on X, and by iteration, a set of the form {σ 1 = · · · = σ r = 1} lifts to an interior p-submanifold of X if each of the σ i lifts to be smooth; equivalently each b dσ i is determinate on each monoid associated to the blow-up X.
On the other hand, {σ 1 = · · · = σ r = 1} may lift to a p-submanifold of X even when none of the individual rational combinations σ i lift to be smooth, as in the following result (a specialization of [22,Prop. 10.3]) that will be used below. Note that the property that b dσ has a determinate sign on a monoid M = Z + g 1 , . . . , g l is equivalent to the property that the intersection of M with the subspace ker b dσ is a face (possibly {0}) of M , meaning a submonoid generated by a (possibly empty) subset of {g 1 , . . . , g l }. Indeed, this is the case if and only if the remaining generators of M all lie entirely within one or the other of the half-spaces determined by b dσ > 0 or b dσ < 0. . Let σ i be rational combinations of boundary defining functions on X and c i ∈ (0, ∞) for 1 ≤ i ≤ r. Then the set {σ 1 = c 1 , . . . , σ r = c r } ⊂ X lifts to an interior p-submanifold of the iterated blow-up X = [X; F 1 , . . . , F n ] provided that each monoid M in the refinement associated to X intersects the subspace r i=1 ker b dσ i along a face of M . Proof. The key observation is that the equations {log σ i = log c i : 1 ≤ i ≤ r} may be replaced by equivalent Z-linear combinations. Thus for any given monoid M = Z + g 1 , . . . , g l in the refinement associated to X, the hypothesis that M meets r i=1 ker b dσ i along a face of M means that the log σ i may be replaced by Z-linear combinations log σ i = j b ij log σ j with the property that b d σ i = j b ij b dσ j is determinate on M , and then it follows from Corollary 2.3 that the set lifts to a p-submanifold in any local coordinates on X associated to M .
As an example, in Figure 1, neither of the rational combinations ρ 1 /ρ 2 or ρ 1 /ρ 3 lifts to be smooth in any of the depicted blow-ups, yet the set {ρ 1 /ρ 2 = ρ 1 /ρ 3 = 1} lifts to a p-submanifold in the blow-ups depicted in (b) and (d), since ker b d(

Ordered corners
Definition 3.1. The category of manifolds with ordered corners is as follows: • An object is a manifold with corners X equipped with a partial order on the set of its boundary hypersurfaces along with its interior-considered for our purposes as an 'honorary boundary hypersurface'-with the property that every pair of faces in M 1,0 (X) which meet are comparable with respect to the order; equivalently, incomparable faces must be disjoint. In particular every element must be comparable to X ∈ M 0 (X).
• A morphism is a simple, b-normal, interior b-map f : X → Y which is ordered, meaning that the induced map is order preserving. Remark.
• As noted in §2, we allow for the possibility that M 1 (X) consists of collective boundary hypersurfaces, that is, unions of disjoint components which are identified as a single element of M 1 (X). This also applies to X ∈ M 0 (X) in case X is disconnected; in fact we always require M 0 (X) to consist of a single element. • We do not require that a pair of faces be comparable if and only if they are non-disjoint; it is sometimes convenient to consider orders in which possibly disjoint faces are deemed comparable, such as a total order; nevertheless, for any admissible order on M 1,0 (X) there is a minimal suborder with the property that elements are comparable if and only if they are non-disjoint.
We will often be interested in the case that X ∈ M 0 (X) is either maximal or minimal, in which case we say X is respectively interior maximal or interior minimal. We denote by X max or X min the manifold with ordered corners in which the interior is made maximal or minimal, respectively. We introduce the notation with respect to which M 1,0 (X) decomposes as a disjoint union Note that X ∈ M 0 (X) forms a basepoint for the ordered set M 1,0 (X), in that it is comparable to every other element and is preserved by the map f ♯ : M 1,0 (X) → M 1,0 (Y ) associated to a morphism f : X → Y (as f is interior); thus the association X → M 1,0 (X) is a functor from the category of manifolds with ordered corners to the category of pointed ordered sets. Note also that ∪ {Y }, with notation as above.
3.1. Products. Products of manifolds with ordered corners can be motivated by considering the functor X → M 1,0 (X). The product of M 1,0 (X) and M 1,0 (Y ) as ordered sets is the cartesian product M 1,0 (X) × M 1,0 (Y ) equipped with the product order, depicted schematically as follows: Notation Convention 3.2. Here and below we employ the following diagrammatic notation convention for products of ordered sets: an arrow of the form A × B → A × B ′ between products with a single factor in common (and B ∩ B ′ = ∅) means that (a, b) < (a, b ′ ) for each a ∈ A, b ∈ B, and b ′ ∈ B ′ . Conversely, an arrow of the form A × B ′ → A ′ × B with no common factor on both sides means that However, the product in the category of pointed ordered sets is the subset of pairs which are comparable to the base point (X, Y ) ∈ M 0 (X) × M 0 (Y ): Corollary 3.5. If X = X min and Y = Y max are interior minimal and maximal, respectively, then the cartesian product X × Y ≡ X ∼ × Y is already the product of X and Y in the category of manifolds with ordered corners.
Proof of Theorem 3.4. While it is possible to prove these results by local coordinate computations, the sheer number of coordinate charts on X ∼ × Y makes this tedious. Instead, we employ the generalized blow-up machinery discussed in Section 2.2 to identify the monoid refinement of X ∼ × Y as a generalized blow-up of X × Y , from which its ordered corners structure and universal property are easily derived. We will see that the refinement consists of monoids which are freely generated by sums of generators h + g forming maximally ordered chains in M 1,0 (X) × M 1,0 (Y ), under the identification of hypersurfaces H with their associated monoid generators h ∈ b M H and with X and Y identified with 0.
Thus, consider an arbitrary corner in X × Y . By relabeling if necessary, we may assume this has the form of hypersurfaces of X and Y , respectively, which, since we work locally near the corner, we may assume constitute all of the hypersurfaces of X and Y . The monoid associated to this corner of X × Y has the form where we denote the generators associated to the hypersurfaces by lower case letters. Identifing . . , h k and g 0 := 0 ∈ Z + g −l ′ , . . . , g l with the honorary hypersurfaces X and Y determines an identification between the ordered set M 1,0 (X) × M 1,0 (Y ) and the elements . . , g l equipped with the product order on summands (where we fix the order of addition). In other words, we regard h i and g j as h i + 0 = h i + g 0 and 0 + g j = h 0 + g j , respectively and equip the sums with the product order induced by Assuming temporarily that X and Y are minimal in M 1,0 (X) and The first blow-up of H k × G l subdivides (3.5) into the pair of monoids In particular, the incomparable pair {h k , g l } = {h k + 0, 0 + g l } is replaced by either of the comparable pairs {h k + g l , h k + 0} or {h k + g l , 0 + g l }, with the result that the sum h k + g l , which is comparable to all other generators h i = h i + 0, i < k and g j = 0 + g j , j < l. Proceding through the rest of the iterated blow-up of in any order consistent with the product order) has the effect of subsequently subdividing each remaining monoid with incomparable generators into two new monoids by replacing the highest indexed incomparable pair {h i , g j } with one of the comparable pairs {h i + g j , h i + 0} or {h i + g j , 0 + g j }, at the end of which process (3.5) is replaced by the collection of monoids generated by totally ordered chains among Note that had we instead blown up in the opposite order M > 1 (X) × M > 1 (Y ), we would have started with the blow-up of H 1 × G 1 , in which {h 1 + 0, 0 + g 1 } is replaced by either {h 1 + g 1 , h 1 + 0} or {h 1 + g 1 , 0 + g 1 }; however, the sum h 1 + g 1 is then incomparable to the remaining generators h i + 0, i > 0 and 0 + g j , j > 0 in the resulting monoids, an incomparability which is not resolved by any subsequent blow-up. In particular, the iterated blow

or indeed the iterated blow-up in any other order besides
) fails to obtain an ordered corners structure consistent with (3.3).
Likewise, assume temporarily that X and Y are maximal in M 1,0 (X) and M 1,0 (Y ), so k = l = 0, and consider the iterated blow- has the effect of iteratively subdividing each remaining monoid with incomparable generators h i and g j , −k ′ ≤ i ≤ −1 and −l ′ ≤ j ≤ −1 into two new monoids, by replacing its lowest indexed incomparable pair {h i , g j } with one of the comparable pairs {h i + g j , h i + 0} or {h i + g j , 0 + g j }. At the end of this process (3.5) is replaced by the collection of monoids generated by maximal totally ordered chains among the generators Again, had we begun instead by blowing up In the general case, since the monoids generated by totally ordered chains decompose as products of factors with generators indexed by i, j < 0 on the one hand and factors with generators indexed by i, j > 0 on the other, it follows from Theorem 2.1 that an equivalent blow-up is obtained by combining the above procedures in either order; in other words [X ×Y ; , as both are associated to the same refinement of monoids.
Since X ∼ × Y has boundary hypersurfaces associated to the one dimensional monoids Z + h i + g j in the refinement, with hypersurfaces meeting if and only if their associated monoids generate a two dimensional monoid, it follows immediately that X ∼ × Y admits an ordered corners structure with M 1,0 (X ∼ × Y ) order isomorphic to the pointed product To prove the universal property, suppose that an arbitrary corner in Z has the local form Given ordered morphisms f : Z → X and g : Z → Y , with respect to which this corner of Z maps into the corner is determined on generators by a map of pointed ordered sets which is encoded by an assignment e i → h α(i) for an increasing sequence α(−m ′ ) ≤ · · · ≤ α(m) with α(0) = 0. Similarly, the homomorphism Z + e −m ′ , . . . , e −1 , e 1 , . . . , e m → Z + g −l ′ , . . . , g −1 , g 1 , . . . g l is determined by an assignment e i → g β(i) for an increasing sequence β(−m ′ ) ≤ · · · ≤ β(m) with β(0) = 0. It follows that the product homomorphism and hence its image is the submonoid generated by the totally ordered chain , which is precisely a monoid in the refinement discussed above (or a face thereof). As a consequence of Theorem 2.1, the map Z → X × Y factors uniquely through the blow-up X ∼ × Y , and moreover the map Z → X ∼ × Y is simple and b-normal since generators are mapped to generators in the associated monoid homomorphisms, completing the proof.
Remark. As is evident from the proof, the order of blow-ups of corners necessary to obtain an order structure consistent with (3.2) follows the rule of thumb that elements further away from the base point (X, Y ) ∈ M 0 (X) × M 0 (Y ) are to be blown up prior to those which are closer to it.
The boundary hypersurfaces of X ∼ ×Y are the lifts and/or front faces of the corners H ×G ⊂ X ×Y for (H, G) ∈ M 1,0 (X) × M 1,0 (Y ), and we denote these by In the next section we determine the structure of these boundary hypersurfaces.

Boundary hypersurfaces. We show below (and it is straightforward to see) that a face of the form
we record this below as Proposition 3.6. Far more important than this description, however, is the description of the product structure of L(H, G) obtained when H and G decompose into cartesian products, as is the case locally when the spaces have fibered corners, and we devote the remainder of the section to this latter situation. (3.10) In particular, if X and Y are both interior maximal or both interior minimal, these reduce to the ordered products X need to be considered since the others are either disjoint or induce a blow-up of a boundary hypersurface of H × G, having no effect.
First, the blow ups of Note that the blow ups of H × G ′ and H ′ × G for H ′ < H and G ′ < G induce blow ups of hypersurfaces in H × G and have no effect. Next comes the blow-up of (the lift of) H × G itself, introducing the product with an interval: Next come the blow-up of faces H × G ′ and H ′ × G for H < H ′ < X and G < G ′ < Y which induce the blow-ups M >,< 1 (H)×G×{0} and H ×M >,< 1 (G)×1, respectively. Next come the blow-ups of H ′ × G ′ for H < H ′ < X and G < G ′ < Y , which induce blow-ups of M >,< 1 (H) × M >,< 1 (G) × I, and finally come the blow-ups (in reverse order) of H ′ × G ′ for H ′ > X and G ′ > Y , which induce the blow-ups of is similar, proceding in the order indicated on the second line of (3.4). The case of (X, G) (or (Y, H)) is also similar, with the omission of the blow-up of X × G itself (since it is already codimension 1), as well as the omission of the blow-ups of the faces H × G ′ for H < X and G ′ > G or H > X and G ′ < G since these are incomparable as noted above.
We say that a boundary hypersurface H is product-type if it is a product, and hence an ordered product by Corollary 3.5, H = F × B = F min ∼ × B max of an interior minimal manifold F = F min and an interior maximal manifold B = B max . As noted, this holds locally when X is a manifold with fibered corners, where H is a fiber bundle H → B with fiber F (hence our choice of notation). Moreover this is always true locally in general; indeed, a manifold with ordered corners can always be decomposed locally as the product of an interior minimal and an interior maximal manifold. For a product-type hypersurface, the ordered corners structures on H, F and B are identified with the following subsets of M 1,0 (X): and we note that has the order structure while for H > X, M 1,0 (N + H) has the order structure into subsets of elements which are above or below X in the order with respect to the identifications (3.11).
Discarding the factor B in the first case or F in the second leads to the following 'relative cone' construction, which plays an important role below.
Definition 3.7. Let H = F × B be a product-type hypersurface of a manifold with ordered corners X. If H < X, the cone on F relative to X is the ordered corners manifold C X (F) := F × [0, ∞) with the order on M 1,0 (C X (F)) depicted by the following diagram: ) depicted by the following diagram: We call F × 0 or B × 0 the principal boundary hypersurface of the cone. If X is either maximal or minimal, then so is C X (B) or C X (F); in this case we use the alternate notation C max (F) or C min (B), respectively.
, is the boundary hypersurface of the product of relative cones associated to the principal hypersurfaces: In view of Proposition 3.6 these spaces have the form (1.4) and (1.5) in the case that X and Y are maximal or minimal, respectively.
Note that, in view of (3.12) and (3.14), or (3.13) and (3.15), the projection maps are ordered morphisms. In contrast, neither of the other factor projection maps The remedies for this are the following 'compressed projections', defined relative to the differentials, in the sense of §2.1, of a choice of boundary defining functions on X, lifted to N + H. These compressed projections play an important role at various points throughout the remainder of the paper. Lemma 3.9. Let H = F ×B be a product-type hypersurface of X, and fix a set of boundary defining Then the compressed projections We are now in a position to state the main result about product-type hypersurfaces, which identifies the product-type structure of their lifts in X ∼ × Y .
, has a product-type structure as follows.
with the projections coinciding with the restriction to L(H, with the projections coinciding with the restriction to L(H, . This map is clearly a diffeomorphism since H (resp. G) has diffeomorphic neighborhoods in X and N + H (resp. Y and N + G). Thus it suffices from now on to replace X and Y with N + H and N + G, respectively.
We consider in detail the case (H, G) > (X, Y ) as it has the most relevance in §4; the case that (H, G) < (X, Y ) is similar. From the universal property of the ordered product applied to the ordered morphisms (3.16) and (3.17) we obtain morphisms As morphisms these are simple and b-normal, and they are also easily seen to be b-surjective (for example using [16, Lemmas 2.5 and 2.7] applied to the b-fibrations , noting that all blow-ups of the domain are either lifts of blow-ups of the target or are transversal). In particular they are b-fibrations. We claim that the restrictions of these b-fibrations to the boundary hypersurface L(H, G) are in fact transverse surjective submersions of manifolds with corners, from which it follows that the product map is a surjective submersion of compact manifolds with corners of the same dimension. This latter map cannot be a nontrivial cover or else its descent to would be a nontrival cover as well; hence it must be a diffeomorphism. Moreover, since F H ∼ × F G is interior minimal and B H ∼ ⋆ X,Y B G is interior maximal, the target in (3.21) is an ordered product and the diffeomorphism is an isomorphism of ordered corners manifolds.
To show that (3.20) are surjective submersions we note first of all that they are simple b-fibrations (being restrictions to a boundary hypersurface of simple b-fibrations), and proceed to show that they have the property discussed in §2 that each boundary hypersurface of the target is mapped onto by a unique boundary hypersurface of the source. The boundary hypersurfaces of F H ∼ × F G are indexed by (H ′ , G ′ ) for H ′ ≥ H and G ′ ≥ G (except for (H, G) itself, which corresponds to the interior) and consist of the lifts of in the ordered product of cones, and each of these is mapped onto uniquely by the boundary hypersurface of L(H, G) given by its , and again each of these is mapped onto uniquely by the boundary hypersurface of L(H, G) given by its intersection with L(H ′ , G ′ ).
Third are those associated to (H, Y ) or (X, G), given by the intersection of B H Last are the boundary hypersurfaces associated to (H, Thus we have shown that (3.20) are surjective submersions and it remains to see that they are transverse (i.e., the kernels of their pointwise differentials intersect trivially). As this is a local property we can consider local coordinates, and we can essentially ignore any interior (meaning non-boundary defining) coordinates, since these are unaffected by boundary blow-ups and are carried along as products locally; morover transversality is clear for interior coordinates since they decompose into factors belonging to the interiors of the factors F • or C • (B • ), respectively. On the other hand, for a simple surjective submersion, the behavior of the boundary defining coordinates is entirely combinatorial: the kernel of the differential of L(H, G) → F H ∼ × F G is spanned by the coordinate vector fields of boundary defining coordinates for hypersurfaces mapping to the interior of the target and likewise for L(H, G) → B H ∼ ⋆ X,Y B G . As we have just shown above that these sets of boundary hypersurfaces are complementary, transversality follows, completing the proof.
The proof of the analogous product-type structure of L(H, G) for (H, G) < (X, Y ) is essentially the same but with orders reversed, so we omit the details.
3.3. Fiber products. We now show that the ordered corners category contains fiber products of appropriately transverse maps. In general, recall that interior b-maps f : X → Z and g : Y → Z are said to be b-transverse [22,18] provided that whenever f (x) = g(y) = z, Such maps are transverse in the ordinary sense over the interiors of X, Y , and Z, where the b-tangent bundles coincide with the ordinary tangent bundles, and here it is a standard result that is a smoothly embedded submanifold. On the other hand, the closure of this set in the cartesian product X × Y is typically singular, and even when it can be given a smooth manifold with corners structure, it is almost never a p-submanifold apart from extremely trivial cases. The main result below says that the closure of (3.23) in the ordered product X ∼ × Y is extremely well-behaved, and satisfies the universal property of the fiber product.
Remark. In [22] the notion of a 'binomial subvariety' is defined, generalizing sets like the closure of (3.23) in X × Y , and the authors determine a monoidal condition under which such a subvariety lifts to be smooth under a generalized blow-up of the ambient manifold, with applications to fiber products more specifically. In a related but slightly different direction, Joyce in [18] develops the differential topology of 'manifolds with generalized corners', an intrinsic structure that can be considered on a binomial subvariety without reference to the ambient manifold, and which always contains b-transverse fiber products; in particular the closure of (3.23) in X × Y can always be given a manifold with generalized corners structure when f and g are b-transverse. Finally, in [20] the theory of generalized blow-up is extended to manifolds with generalized corners, which could be used to characterize resolutions of the singular fiber product. While these theories apply to the present context, they involve quite a bit of machinery, so we opt for a self-contained treatment that can be understood independently. Proof. The short exact sequences b N x E → b T x X → b T x E for E ∈ M(X) and similar sequences for Y and Z fit into a commutative diagram Theorem 3.12. If f : X → Z and g : Y → Z be ordered morphisms which are b-transverse, then the lift of (3.23) to X ∼ × Y is a smooth interior p-submanifold Moreover, X ∼ × Z Y satisfies the universal property of the fiber product in the category of ordered corners: whenever there are morphisms W → X and W → Y commuting with the morphisms to Z, then there exists a unique morphism W → X Proof. To show that (3.24) is a p-submanifold, we work locally in coordinates (x, y) ∈ R k + × R l for X, (x ′ , y ′ ) ∈ R k ′ + × R l ′ for Y , and (x ′′ , y ′′ ) ∈ R k ′′ + × R l ′′ for Z, taken in normal form (2.6) for the b-normal maps f and g so that f (x, y) = x ν , a(x, y) = (x ′′ , y ′′ ) and By Lemma 3.11, the maps y → a(0, y) and y ′ → b(0, y ′ ) are transverse in the ordinary sense, from which it follows that d(a j − b j ), 1 ≤ j ≤ l ′′ , are independent over (x, x ′ ) = (0, 0) and hence in a suitably small neighborhood thereof. Thus y j = a j (x, y) − b j (x ′ , y ′ ), 1 ≤ j ≤ l ′′ , can be taken as the first l ′′ of l + l ′ local coordinates (x ′ , x ′′ , y) ∈ R k and it remains only to show that (x) ν i (x ′ ) µ i = 1 : 1 ≤ i ≤ k ′′ lifts to an interior p-submanifold. According to Proposition 2.4, this is the case provided that each monoid M ⊂ Z + h 1 , . . . , h k , g 1 , . . . , g k ′ , However, this subspace is spanned explicitly by all sums of the form h i + g j such that f ♯ (H i ) = g ♯ (G j ) ∈ M 1 (Z), and since M is generated by a totally ordered chain {h i + g j }, this intersection is precisely the face of M generated by those To see that it satisfies the universal property, suppose that W admits morphisms to X and Y forming a commutative diagram W Y X Z Then W factors uniquely through a morphism to X ∼ ×Y , with W • mapping into the lift of X • × Z • Y • . By continuity, the image of W in X ∼ × Y lies in the p-submanifold X ∼ × Z Y , giving the diagram (3.25).  In the other cases, we show that L(H, G) E := X ∼ × Z Y ∩ L(H, G) can be identified with the fiber product H ∼ × E G. If H = X, then E = Z by necessity, and since L(X, G) ∼ = X ∼ × G in this case, it is straightforward to see that the lift of Finally, if H, G, and E are all proper hypersurfaces, then our strategy is to show that L(H, G) E satisfies the universal property of the fiber product H ∼ × E G, namely that every commutative diagram formed by the outermost square of determines a unique morphism F → L(H, G) E filling in the remainder of the diagram. It suffices to replace X, Y , and Z by the normal models N + H, N + G and N + E, respectively. We proceed by first showing that the outer square of (3.28) can be 'thickened out' to a diagram of ordered corners morphisms of the form uniquely up to an automorphism of F × R + which restricts to the identity on F , whence the restriction of the map F × R + → N + H ∼ × N + E N + G to F × {0} provides the required unique morphism characterizing L(H, G) E as the fiber product H ∼ × E G. Indeed, with respect to trivializations of the normal models, df : N + H → N + E has the form Fixing boundary defining functions on F , it follows from commutativity of (3.28) that φ * σ = a H F ′ :F ′ →E ρ F ′ and ψ * σ ′ = a G F ′ :F ′ →E ρ F ′ for the same set {ρ F ′ : F ′ → E} of boundary defining functions on F , namely those associated to hypersurfaces mapped to the interior of E by ψ•g = φ•f in (3.28), but generally different positive functions a H , a G ∈ C ∞ (F ; (0, ∞)). But then (3.29) commutes if we define Moreover, these are the only possible extensions of φ and ψ up to an (0, ∞)-equivariant automorphism of F × R + of the form (q, t) → (q, b(q)t) for b ∈ C ∞ (F ; (0, ∞)), which in particular restricts to the identity on F × {0}, so the restriction F × {0} → L(H, G) E of the map F × R + → N + H ∼ × N + E N + G induced by any such choice is unique, giving the characterization of L(H, G) E as the fiber product H ∼ × E G. Note that the above strategy fails if E = Z, since in this csae the maps N + H ∼ = H × R + → Z and N + G ∼ = G × R + → Z are respectively independent of t ∈ R + , and while extensions φ : F × R + → N + H and ψ : G × R + → N + G may be defined, they are certainly not unique.
As an application of the previous result, note that if f : X → Y is any morphism, then the fiber product of f with the identity 1 : Y → Y (which b-transversal to every b-map) gives an embedding of the graph In particular this shows that every morphism can be factored as p-submanifold inclusion X ֒→ Another application, which will be important in the forthcoming work [24], is the following. Let ρ X = H∈M 1 (X) ρ H and ρ Y = G∈M 1 (Y ) ρ G be total boundary defining functions on interior minimal manifolds X and Y with ordered corners, respectively. These constitute ordered corners morphisms ρ X : X → [0, ∞) min and ρ Y : Y → [0, ∞) min , with target given the interior minimal ordered corners structure, and these are b-transverse in a sufficiently small neighborhood of the boundaries of X and Y where dρ X = 0 and dρ Y = 0.
Corollary 3.14. In the situation above, the fiber product

Fibered corners
We now turn to the consideration of manifolds with fibered corners. Definition 4.1 (c.f. [11]). A fibered corners structure (also known as a 'resolution structure' [5] or an 'iterated boundary fibration structure' [3]) on a manifold with corners X consists of the structure of a fiber bundle 7 of manifolds with corners φ H : H → B H for each (collective) boundary hypersurface H ∈ M 1 (X), with typical fiber denoted F H , satisfying the condition that whenever H ∩ H ′ = ∅ then dim(B H ) = dim(B H ′ ); moreover if dim(B H ) < dim(B H ′ ), say, then φ H ′ constitutes a coarser fibration on H ∩ H ′ compared to φ H in the following sense: This induces an ordering on M 1 (X) in the sense of Definition 3.1 (to be extended to M 1,0 (X) momentarily) defined by We require here in addition the structure of a fiber bundle φ X : X → B X on X itself, with fiber F X . Of course X ∩ H = H = ∅ for every H ∈ M 1 (X), and in order to extend the compatibility condition above, there are two possibilities. We require either: • H = X ∩ H is the restriction of φ X over a proper boundary hypersurface ∂ H B X of B X , which fibers over B H forming a commutative diagram in which case we set H < X, or else • H is trivially the restriction of φ H over the 'honorary boundary hypersurface' ∂ X B H = B H , which fibers over B X forming a commutative diagram in which case we set X < H. This extends the order on M 1 (X) to M 1,0 (X), and in particular every manifold with fibered corners is naturally a manifold with ordered corners in the sense of Definition 3.1. In the second case above we relax the strict dimension comparison on the base manifolds to require merely that dim(B X ) ≤ dim(B H ), to account for the possibility that φ XH = 1 has trivial fibers (which may occur). As a matter of notation, we denote the fiber of φ HH ′ :

Remark.
• For the definition we do not necessarily require that boundary hypersurfaces are connected; as in [5] we allow H to be a 'collective boundary hypersurface' i.e., a union of disjoint connected boundary hypersurfaces. One reason for this is that a fiber F H , a fibered corners manifold in its own right, may have distinct boundary hypersurfaces which are the restrictions to the fiber of a single connected boundary hypersurface of the total space, as in the example of a finite width Möbius strip fibering over a circle. Another reason comes from the connection to stratified spaces as discussed in §5.1, in which the B H and F H are closely related to the strata and associated links, respectively, of a stratified space. In the latter theory one often wants to consider stratifications in which the links, and hence the F H , may be disconnected; for example, a torus pinched along a non-bounding circle has a singular stratum whose link consists of two disjoint circles, and the associated manifold with fibered corners is a cylinder with both boundary components considered as a single boundary hypersurface fibering over a point. • In the existing literature it is typical to only consider fibrations only on the proper boundary hypersurfaces, and in the absence of an explicit fibration on X itself, there are two canonical choices: namely φ X can be either of the trivial fibrations: φ X = 1 : X → X, making X interior maximal, or φ X : X → * , making X interior minimal. We denote these by X max and X min , respectively. In Appendix A we characterize interior fibrations φ X : X → B X as certain surjective submersions in the category of fibered corners manifolds: see Definition A.3 and Corollary A.6.
As a locally trivial fiber bundle, the boundary hypersurfaces of H (given by H ∩ H ′ ) come in two types, either the preimage with respect to φ H of a boundary face ∂ H ′ B H of the base when H ′ < H, or a subbundle of φ H with fiber a (possibly disconnected) boundary face of ∂ H ′ F H when H ′ > H. From this the next result immediately follows. Prior to defining morphisms of manifolds with fibered corners, we first consider some natural conditions that may be imposed.
for some map f H : B H → B G , which, as a consequence of the fibered property of f for each H ′ < H, is itself fibered.
This condition is natural from the point of view of stratified spaces (see §5.1), since a fibered map f : X → Y descends to a well-defined map f : X → Y of the associated stratified spaces, obtained by collapsing the fibers of the respective boundary fibrations of X and Y . As it turns out, this condition alone is not sufficient to determine a category of fibered manifolds containing products; we need to impose some additional structure, in support of which we introduce the following definitions.
Equivalently, as a consequence of the normal form for b-normal maps discussed in §2, a map f is compatible if and only if it is rigid with respect to some representative boundary defining functions for the equivalence classes on X and Y .  It is a consequence of Proposition A.4 (see also [5,Prop. 3.7] or [11,Lem. 1.4]) that good boundary defining functions always exist. What is more, fixing an fc-equivalence class of good boundary defining functions, local coordinates on X can always be chosen of the form (y −n ′ , x −n ′ , y −n ′ +1 , x −n ′ +1 , . . . , y 0 , y 1 , x 1 , y 2 , x 2 , . . . , x n , y n+1 ) (4.4) where the x i are representative local boundary defining functions for the hypersurfaces H i forming a maximal chain H −n ′ < · · · < H −1 < X < H 1 < · · · < H n and y i are coordinates in R k i , such that the boundary fibration φ i = φ H i for −n ′ ≤ i ≤ n is given by the projection φ i : (y −n ′ , x −n ′ , . . . , x −1 , y 0 , y 1 , x 1 , y 2 , x 2 , . . . , x n , y n+1 ) x i =0 → (y −n ′ , x −n ′ , . . . , y i ) (4.5) including the case i = 0 which corresponds to φ X . This is proved as Corollary A.7 below. In particular (y −n ′ , x −n ′ , . . . , y i ) are coordinates for B H i and (y i+1 , x i+1 , . . . , y n+1 ) are coordinates for F H i . We say such coordinates are in standard form.
We are now in a position to define the category of fibered corners manifolds in which the ordered product will be shown to be the categorical product.
Definition 4.6. The category of manifolds with fibered corners is defined as follows.
• An object is a manifold with fibered corners X equipped with a fc-equivalence class of good boundary defining functions. The following result is a simple consequence of unwinding definitions. Remark. Note that the condition that the boundary defining functions on X are good is equivalent to the rigidity of each boundary fibration φ H : H → B H , H ∈ M 1 (X).

4.1.
Products and fiber products. We now proceed to show that the ordered product X ∼ × Y , equipped with a suitable set of boundary fibrations and a fc-equivalence class of boundary defining functions, is also a product in the category of manifolds with fibered corners. The Moreover, by the universal property of products in the ordered corners category, it follows that whenever Z is a manifold with fibered corners with morphisms f : Z → X and g : Z → Y thereby inducing an ordered morphism Z → X ∼ × Y , the restriction of this morphism to any E ∈ M 1,0 (Z) with E ≤ Z fits into a naturally commutative diagram In other words, the map Z → X ∼ × Y is canonically fibered over each E ≤ Z ∈ M 1,0 (Z). Note that none of the above makes any reference to an fc-equivalence class of boundary defining functions on X, Y , or Z; or to the compatibility of morphisms with respect to such classes.
In fact, it follows from this observation that, provided we restrict consideration to interior maximal manifolds with fibered corners, we can define a weaker category in which the objects are merely required to have fibered corners (without specifying equivalence classes of boundary defining functions) and morphisms are merely required to be ordered and fibered; what we have just shown is that X ∼ × Y is a product in this category: Theorem 4.8. The ordered product is a categorical product in the category whose objects are interior maximal manifolds with fibered corners and whose morphisms are ordered morphisms in the sense of    (3.12) and (3.13). An fc-equivalence class of good boundary defining functions on X induces an equivalence class of (0, ∞)-equivariant defining functions on N + H by taking their normal derivatives as in §2.1; for notational convenience we shall simply refer to these by the same notation as the boundary defining functions on X from which they are derived.
We also equip the relative cone C X (B H ) of Definition 3.7 with a fibered corners structure.  Though we will not use it explicitly below, this is an appropriate moment to record the fibered corners structure of C X (F H ) for H < X: is The map (4.6) is clearly simple, b-normal, and compatible with the fc-equivalence class of boundary defining functions on N + H and C X (B H ). To see that it is fibered, observe that the following diagrams commute: Theorem 4.13. If X and Y are manifolds with fibered corners, then X ∼ × Y has a canonical fibered corners structure with boundary fibrations of the form and fc-equivalence class of good boundary defining functions given locally by rational combinations of boundary defining functions from X and Y . The ordered product satisfies the universal property of a product in the category of manifolds with fibered corners, meaning that if f : Z → X and g : Z → Y are fibered corners morphisms, then the associated morphism Z → X restricting to G j uniquely so that it constitutes a local boundary defining function for L(H, G) on such a neighborhood in view of Corollary 2.3. Indeed, the coefficients {a i , b j } are precisely those for the basis of b N * F , F = i,j L(H i , G j ) which is dual to the basis {h i + g j } of b N F ; since the latter is unimodular, it follows that a i , b j ∈ {−1, 0, 1} (and in fact either all the a i are non-negative while the b j are non-positive, or vice versa). As fc-equivalence is a local property, the equivalence class of boundary defining functions on X ∼ × Y which are locally fc-equivalent to rational combinations as above is well-defined.
The fact that these rational boundary defining functions are good is equivalent to the rigidity of To see this, first note that N + H → C X (B H ) (resp. N + H → B H ) is rigid by definition and the fact that boundary defining functions on X are good, and hence N + H is rigid with respect to rational boundary defining functions on the domain. It then follows from the local coordinate structure of the unique lift N + H that this lift is rigid with respect to rational boundary defining functions on the domain and codomain, and restricting this to L(H, G) gives the desired result.
The projections X ∼ × Y → X and X ∼ × Y → Y are compatible with the fc-equivalence class on X ∼ × Y essentially by definition, and the fact that Z → X ∼ × Y is compatible follows from the general fact that the lift to a generalized blow-up as in Theorem 2.1.(b) of a locally rigid map is rigid with respect to rational local boundary defining functions.
We next address the fibered corners structure of the fiber products considered in §3.3.
• In all other cases, the boundary fibration has the form Moreover X ∼ × Z Y satisfies the universal property of the fiber product in the category of manifolds with fibered corners.
Proof. Suppose first that H < X and G < Y with f ♯ (H) = g ♯ (G) = Z ∈ M 0 (Z), and replace X and Y by N + H and N + G, respectively. From the commutative diagram and restricting over a point in the latter shows that the fiber may be identified with C X (F H ) In a similar manner, if H > X and G > Y are such that f ♯ (H) = g ♯ (G) = Z, then the commutative diagram , restricting over a point of which shows that the fiber may be identified with F H , giving the result in this case.
In all other cases, and consequently maps naturally to B H ∼ × B E B G , and restriction over a point in the latter shows that the fiber satisfies the universal property of F H ∼ × F E F G , proving the result in these cases. Note that in all cases, the boundary fibration just described coincides with the restriction to L(H, G) Z of the boundary fibration φ H,G on L(H, G). This is clear in all cases, except perhaps the last case when (H, G) > (X, Y ), but in that case the diagram of relevant maps , and arguing as in the proof of Theorem 3.13 shows that the relevant boundary face of C X (B H ) to which the restriction to L(H, G) Z maps. The fc-class of boundary defining functions of X ∼ × Z Y are inherited by restriction from the class on X ∼ × Y , and the fact that these are good with respect to the boundary fibrations described amounts to rigidity of all maps just described with respect to the fc-equivalence classes on N + H, N + G, and so on.
Returning to the application discussed in Corollary 3.14 in the fibered corners setting yeilds the following.
Corollary 4.15. If X and Y are manifolds with fibered corners with representative total boundary defining functions ρ X = H∈M 1 (X) ρ H : is a p-submanifold in a sufficiently small neighborhood of the boundary (where dρ X = 0 and dρ Y = 0), and has boundary fibrations of the form

Relation to other theories
In this section we discuss how products of fibered corners manifolds relate and/or generalize products in other categories.

Smoothly stratified spaces.
We briefly recall the definition of smoothly stratified spaces, referring to [1,30] for a more complete account. A stratified space is a topological space X decomposed as a disjoint union X = i s i of locally closed subspaces s i called strata, each of which is itself a manifold of some dimension, satisfying the frontier condition that s i ∩ s j = ∅ if and only if s i ⊂ s j , which defines a partial order s i < s j . In particular, there is a maximal principal stratum s 0 = X • which is dense, with the rest referred to as the singular strata. The closed strata s i are again stratified spaces with strata {s j : s j < s i }. A smoothly stratified space (aka 'Thom-Mather stratified space') is a stratified space X along with control data consisting of open neighborhoods T i ⊃ s i in X called tubes, equipped with retractions r i : T i → s i and distance functions ρ i : T i → R + such that s i = ρ −1 i (0) and satisfying the conditions that whenever s i < s j , As a consequence of Thom's isotopy lemma for smoothly stratified spaces (see [30, 3.9.2]), the T i are locally trivial bundles of cones over s i , the links of which are stratified spaces of lower depth. Associated to each interior maximal manifold with fibered corners X is a smoothly stratified space X obtained by iteratively collapsing the fibers over each boundary hypersurface in reverse order (starting with the largest base and ending with the smallest). The strata of X are the smooth manifolds s H = B • H , for H ∈ M 1,0 (X), with principal smooth stratum X • , and the control data are given by the passage to the quotient of tubular neighborhoods of the boundary hypersurfaces, their associated retractions, and the images of (good) boundary defining functions. The closure s H = B H is the stratified space obtained by fiber collapse of the fibered corners manifold B H itself, and the partial order s H < s H ′ on strata defined by s H ⊂ s H ′ coincides with the order H < H ′ induced by the fibered corners structure. The link of the cone bundle T H → s H is the stratified space F H obtained by collapsing the fibers of the boundary hypersurfaces of F H .
Conversely, as shown in [3,1] (a result originating in [32]; see also [30, 3.9.4]), every smoothly stratified space can be resolved to a manifold with fibered corners by iteratively replacing the cone bundle T H of each singular stratum with an associated cylinder bundle in order from smallest stratum to largest, with the tubes T H resolving to tubular neighborhoods of the boundary hypersurfaces, and distance functions resolving to boundary defining functions.
Remark. While this constitutes an equivalence of objects between smoothly stratified spaces on one hand, and (interior maximal) manifolds with fibered corners on the other hand, correspondingly little has been written about morphisms. In [3], Albin et. al. show that the resolution procedure above is functorial with respect to isomorphisms, meaning controlled isomorphisms of stratified spaces on one hand, and fibered diffeomorphisms of fibered corners manifolds on the other, but in general, pinning down a suitable class of morphisms on the stratified spaces side is difficult. In addition to the requisite condition that a map f : X → Y be stratified, meaning that it sends strata to strata and restricts to a smooth map of manifolds thereon (which corresponds to the fibered condition of Definition 4.3), the typical conditions discussed in the literature are inadequate for our purposes. Indeed, the weakest typical condition is that the map f is 'weakly controlled', meaning it maps tubes to tubes and commutes with the retractions. This is on the one hand too strong, since it amounts to extending the boundary fibrations to open neighborhoods of the boundary hypersurfaces and demanding an extension of the fiber bundle maps (4.3) to such neighborhoods, and it does not seem that every fibered corners morphism satisfies such a condition. On the other hand, it is also too weak, since it does not impose the kind of algebraic behavior with respect to boundary defining functions that a general b-map satisfies. From our point of view, the right class of morphisms are those maps of smoothly stratified spaces which are descended from fibered corners morphisms as defined here, but it seems difficult to characterize these directly using only the data of stratified spaces.
There is little discussion of products of stratified spaces in the literature, but the definition of a smoothly stratified structure on the cartesian product X × Y is obvious enough. It has strata s H × s G for (H, G) ∈ M 1,0 (X) × M 1,0 (Y ), with control data (T H × T G , r H × r G , ρ H + ρ G ) (it seems any homogeneous combination of ρ H and ρ G would also suffice, such as (ρ 2 H + ρ 2 G ) 1/2 ). This satisfies the universal property of the product for most reasonable notions of morphism of stratified spaces.
If we knew that the projections X × Y → X and X × Y → Y lifted to fibered corners morphisms upon resolution, as well as the map into X × Y induced by a good map Z → X and Z → Y, then an easy application of the universal property of the product in both categories would give a quick proof of the following result. However, since this is unavailable at present, we settle for a coordinate based proof.
Theorem 5.1. Let X and Y be interior maximal manifolds with fibered corners, with associated smoothly stratified spaces X and Y. Then the ordered product X ∼ × Y is equivalent to the resolution of the product X × Y of stratified spaces.
The resolution procedure consists of first replacing r 1 and r ′ 1 by r 1 + r ′ 1 and s 1 = r 1 /(r 1 + r ′ 1 ) (or s ′ 1 = 1 − s 1 = r ′ 1 /(r 1 + r ′ 1 )), dividing all coordinates to the right by r 1 + r ′ 1 . Then, near the lift of (say) s ′ 1 = 0 and r 2 = 0, the next step is to replace s ′ 1 and s 2 = r 2 /(r 1 + r ′ 1 ) by s ′ 1 + s 2 and s 2 /(s ′ 1 + s 2 ) = r 2 /(r ′ 1 + r 2 ), divide all coordinates to the right by s ′ 1 + s 2 , and so on. However, as is frequently the case with blow-up computations, it is more convenient here to use equivalent rational coordinates, in which (r 1 , r ′ 1 ) is replaced instead by (r 1 , r ′ 1 /r 1 ) and all coordinates to the right are divided by r 1 ; then (r ′ 1 /r 1 , r 2 /r 1 ) is replaced by (r ′ 1 /r 1 , r 2 /r ′ 1 ) and all coordinates to the right are divided by r ′ 1 /r 1 , and so on. The end of this process results in standard form coordinates y 1 , y ′ 1 , x 1 , y 2 , for the resolution of X × Y which are identical to the rational coordinates near L(H 1 , G 1 ) ∩ L(H 1 , G 2 ) · · · ∩ L(Y, H n ) for the ordered product X ∼ × Y based on (5.1) and (5.2). Other coordinate charts based on other totally ordered chains of strata involve similar computations.
If X is a manifold with fibered corners, then the natural fibered corners structure on the cone C max (X) according to Definition 4.11 consists of boundary fibrations It is clear that the stratified space associated to C max (X) is the cone with link X . From the well-known fact that the product of cones is a cone over the join of the links, we obtain the following.
Corollary 5.2. For fibered corners manifolds X and Y , the stratified space associated to C max (X) y, 0) for all x, x ′ ∈ X and y ∈ Y and (x, y, 1) ∼ (x, y ′ , 1) for all x ∈ X , y, y ′ ∈ Y.

5.2.
Many body spaces. Many body compactifications of vector spaces go back at least to [31], and have been discussed more recently in [21, 6]. Using notation from [21], recall that given a finite dimensional vector space V and It is clear that the stratified space associated to M (V ) by collapsing the fibers of all boundary fibrations except the interior is the radial compactification V , with principal stratum V , and (closed) singular strata consisting of the boundaries ∂S of the subspaces in S V , the links of which are the normal quotients V /S with strata ∂S ′ /S for S ′ > S. In this category, a natural alternative to the usual product of V and W as stratified spaces (which does not resolve to a many body compactification of a vector space) is the space V × W with closed singular strata ∂(S × T ) for (S, T ) ∈ S V × S W , the resolution of which is the many body compactification M (V × W ).
Note that, as spheres, the closed singular strata ∂(S × T ) are homeomorphic to the topological joins ∂S ⋆ ∂T of the singular strata of the factors, a property which generalizes to the interior minimal fibered corners setting since B H ∼ ⋆ min B G has associated stratified space the join B H ⋆ B G of the stratified spaces associated to B H and B G as a consequence of the results of §7. The restriction to a hypersurface h : which has full rank down to {x = 0} for r ∈ (0, 1).

Geometric structures
In this section we consider certain classes of metrics on interior minimal and maximal manifolds with fibered corners. A convenient and equivalent way to discuss these classes of metrics is in terms of their associated geometric structures, as encoded by associated rescaled tangent bundles, on which a Riemannian metric of the given type on the interior extends to a uniformly bounded inner product up to all boundary faces. Both structures to be considered begin with the following. Definition 6.1. The algebra of edge vector fields is defined as H (C ∞ (B H ))) = 0} or those vector fields whose restriction to every H ∈ M 1 (X) is tangent to the fiber of φ H . This definition has appeared in various forms in [3,4,2] and others sometimes under the name 'iterated edge' or 'ie', and extends the definition due to [25] in the depth 1 case. This is a Lie subalgebra of V b (X) and forms a locally free sheaf of constant rank, defining the edge tangent bundle by e T X → X, where V e (X) = C ∞ (X; e T X). (6.1) In local standard form coordinates, a frame for e T X is furnished by the vector fields Since these vector fields will be of interest in both the interior maximal and minimal cases, and since the interior fibration won't play a role, we enumerate boundary hypersurfaces exclusively with positive integers in all cases throughout this section.
The vector fields of particular interest in the setting of interior maximal manifolds are the following. Definition 6.2. A wedge vector field (called an 'iterated incomplete edge' vector field in [3,4], with the name 'wedge' coming from [14] in the depth 1 case) is one of the form 1 ρ X V , where V ∈ V e (X) and ρ X = H∈M 1 (X) ρ H is a total boundary defining function. Such a vector field is singular at the boundary faces of X, and the set V w (X) of wedge vector fields does not form a Lie algebra. Instead the wedge tangent bundle may be defined as a rescaling (see [29] for a general discussion of this procedure) of e T X by the requirement ρ X C ∞ (X; w T X) = C ∞ (X; e T X).
(See [2] for an alternate definition via the cotangent space w T * X.) Note that V w (X) and w T X are independent of the choice of ρ X . A local frame for w T X is given in standard form coordinates by ∂ ∂y 1 , As for b T X, both e T X and w T X bundles are canonically isomorphic to the usual tangent bundle over the interior of X, and a wedge metric on X is then by definition the Riemannian metric on X • induced by an inner product on w T X. Wedge metrics are incomplete on X (hence the term 'incomplete iterated edge metric' of [3,4]), and contain the iterated conic metrics studied by Cheeger in [9] as special cases. While a general wedge metric is defined by an arbitrary inner product on w T X, a model wedge metric in standard form coordinates which shows off the iterated conic aspect of the geometry (and an example of what is called a 'rigid wedge metric' in [3]) is one of the form where dy 2 i is shorthand for an inner product in the variables y i ∈ R n i , g B 1 denotes a Riemannian metric on B 1 and κ F 1 denotes a family of wedge metrics on F 1 .
In contrast, the following structure is of interest particularly in the setting of interior minimal manifolds. Definition 6.3. The algebra of quasi fibered boundary (QFB), or simply Φ vector fields is the set , consisting of those edge vector fields which annhilate a total boundary defining function ρ X to second order [10]. While V e (X) and V w (X) only depend on the fibered corners structure, V Φ (X) also depends on the choice of total boundary defining function ρ X up to equivalence, where two total boundary defining functions ρ X and ρ ′ X are equivalent if their ratio is basic [10]. In particular, V Φ (X) is well-defined by a fc-equivalence class of boundary defining functions on X.
As with edge vector fields above, V Φ (X) is a Lie subalgebra of V b (X) and forms a locally free sheaf of constant rank, defining the Φ tangent bundle (which is again canonically isomorphic to T X in the interior) by In local standard form coordinates a local frame for Φ T X is given by v k = x k · · · x n . (6.5) Note that the basis element v 1 x 1 ∂ ∂x 1 can be replaced by v 1 x k ∂ ∂x k for any k by taking appropriate linear combinations.
A Φ metric on X is by definition a (complete) Riemannian metric on X • induced by a uniformly bounded inner product on Φ T X. A particularly nice model Φ metric in local standard form coordinates is one of the form where dy 2 i is shorthand for an inner product in the variables y i ∈ R n i . Note that such a metric can support additional terms of the form , and in particular restricts to a Φ metric on any interior fiber of a boundary hypersurface.
6.1. Products of Φ structures. We first consider the product of Φ structures on interior minimal manifolds X and Y with fibered corners, proving an isomorphism between Φ T X ⊕ Φ T Y and Φ T (X ∼ × Y ). In fact we give two proofs, one based on direct analysis of Φ-tangent bundles using Proposition 6.5 below, which decomposes Φ T X with respect to the bundle structure of the compressed projection φ H : N + H → C X (B H ) of Lemma 4.12. The second proof is based on an analysis of Φ metrics taken to be in a convenient local form.
The Φ structure on X ∼ ×Y is well-defined by the fc-equivalence class of boundary defining functions given by local rational combinations of representative boundary defining functions on X and Y , as discussed in §4.1. However, the following characterization of total boundary defining functions on X ∼ × Y is conceptually satisfying. Proposition 6.4. Let ρ X and ρ Y be representative total boundary defining functions on X and Y . Then the reciprocal p-sum Proof. Consider a neighborhood in X ∼ × Y of maximal depth, meeting a maximal totally ordered chain of hypersurfaces, which without loss of generality (exchanging the role of X and Y if necessary) contains hypersurfaces of the form L(H, Y ) but none of the form L(X, G). Denote by ρ H and ρ ′ G representative boundary defining functions for H ∈ M 1 (X) and G ∈ M 1 (Y ), respectively, and denote by h = ρ H ∂ ρ H and g = ρ ′ G ∂ ρ ′ G the generators of the associated monoids. We claim that in such a neighborhood, ρ X = H∈M 1 (X) ρ H is a representative total boundary G is basic. Indeed, both are rational combinations, and b dρ X pairs to 1 with every generator h + g or h + 0 of the associated monoid characterizing this neighborhood of X ∼ × Y , so it is locally a total boundary defining function. Likewise, b d(ρ X /ρ Y ) pairs to 1 with every generator of the form h+ 0 and to 0 with every generator of the form h + g, so it is locally the product of boundary defining functions for faces of the form L(H, Y ). On any such face ρ X /ρ Y vanishes identically, while at any face of the form L(H ′ , G), the boundary defining functions for L(H, Y ) are all basic since L(H, Y ) < L(H ′ , G j ), thus ρ X /ρ Y is basic.
The result now follows from the observation that is a smooth function of ρ X /ρ Y , hence basic, so ρ X and ρ X,Y,p are locally equivalent for any p.
Proposition 6.5. Let H ∈ M 1 (X) where X is an interior minimal manifold with fibered corners, with normal model N + H. Then the differential of the compressed projection φ H : N + H → C X (B H ) of Lemma 4.12 extends by continuity from the interior to a short exact sequence of vector bundles.
By direct computation, the basis (6.5) transforms as follows: The first four lines of the above constitute a basis for Φ T C min (B k ), with coordinates (y 1 , x 1 , . . . , x k−1 , y k , t) while the rest, spanning the kernel, constitute a basis for Φ T F k , with coordinates (y k+1 , . . . , y n , x n , z).
Theorem 6.6. If X and Y are interior minimal fibered corners, then the lifted projection maps In particular, if g X and g Y are Φ-metrics on X and Y , then π * Proof. The isomorphism is unique if it exists, since it extends the canonical isomorphism T (X • × Y • ) ∼ = π * X T X • ⊕ π * Y T Y • over the interior. To show existence, we proceed by induction on depth of X ∼ × Y , starting with the isomorphism T (X × Y ) ∼ = T X ⊕ T Y in depth 0. In general, a point of maximal depth in X ∼ × Y lies in an intersection of boundary hypersurfaces which must include a minimal hypersurface, without loss of generality of the form L(H, Y ), where H is a minimal boundary hypersurface of X; in particular B H has depth 0. The projection X ∼ ×Y → X induces via Lemma 4.12 a diagram which we can arrange to be the identity along the bottom row since we may take π * X ρ ≥H = ρ ≥L(H,Y ) , and the restriction to the fibers of which is the lifted projection × Y has strictly lower depth), and along with a splitting Φ T (N + L(H, Y )) ∼ = Φ T (F H ∼ × Y ) ⊕ Φ T C(B H ) of (6.6) this gives an isomorphism leading to the desired isomorphism upon identifying the normal models N + L(H, Y ) and N + H with neighborhoods in X ∼ × Y and X, respectively.
Alternate proof of Theorem 6.6. For the second proof, we use the characterization of Φ T X as the unique vector bundle (up to isomorphism) which extends T X • and to which any Φ metric g X on X • extends to be nondegenerate and nonsingular. We take advantage of the freedom to choose g X to be an a particularly convenient form.
When X or Y is of depth 0, the result is trivial, and proceeding by induction on the depth of X ∼ × Y , we may assume that X and Y are of positive depth. Letting H and G be minimal boundary hypersurfaces of X and Y , it suffices by induction to prove the result near the lift of H × G. Working locally, we may assume φ H and φ G are trivial, and choose the defining Φ metrics g X and g Y to have the simple form where g B H and g B G are smooth metrics on B H and B G , while κ H and κ G are Φ metrics associated to F H and F G with total boundary defining functions ρ >H = H ′ >H ρ H ′ and ρ >G . Since ρ H = ρ X ρ >H is a boundary defining function for H, the condition ρ X < ρ >H is equivalent to ρ H < 1. With u X = ρ −1 X and u Y = ρ −1 Y , this suggests using the polar coordinates r = u 2 with respect to which the cartesian product g of g X and g Y takes the form By induction, the cartesian product κ H + κ G is a Φ metric associated to the manifold with fibered corners F H ∼ × F G with total boundary defining function (ρ −2 >H + ρ −2 >G ) −1/2 , while dθ 2 + sin 2 θg B H + cos 2 θg B G is manifestly a wedge metric on B H ∼ ⋆ B G = B H × B G × [0, π/2] θ . It follows that (6.8) is locally a Φ metric on X ∼ × Y near L(H, G) with respect to the total boundary defining function ρ, and hence by induction, is a Φ metric globally on X ∼ × Y .
6.2. Products of wedge structures. We now turn to the product statement for wedge structures on interior maximal manifolds, in the form of Theorem 6.8 below. An initial attempt along the lines of the previous section would begin by observing that the map φ H : which while true, turns out not to be immediately useful since it is not obvious that the lifted projections off of X ∼ × Y implement the correct rescalings necessary to identify the rescaled bundles ρ −1 . Instead, we implement a dual approach involving a somewhat unnatural map from N + H to C(F H ), assuming H = F H × B H is a product. Since H is a product locally, and since we need not be concerned with uniqueness-the desired isomorphism w T (X ∼ × Y ) ∼ = w T X ⊕ w T Y being the unique extension of T (X • ×Y • ) ∼ = π * X T X • × π * Y T Y • as long as it exists-this unnaturality is tolerable. Proposition 6.7. Let H = F H ×B H be a product-type hypersurface of an interior maximal manifold X with fibered corners. Then the compressed projection We compute in standard form local coordinates supposing H = H k , in which the map is given by (y 1 , x 1 , . . . , y k , x k , y k+1 , . . . , y n , x n , z) → (t, y k+1 , x k+1 , . . . , y n , x n , z) t = x 1 · · · x k .
The basis (6.3) is transformed as follows: The right hand sides of the first two lines, constituting the kernel of the map over H k = {x k = 0}, form a basis for w T B k while the right hand side of the remaining lines form a basis for w T C(F H ).
Theorem 6.8. For X and Y interior maximal manifolds with fibered corners, the lifted projections induce an isomorphism

9)
In particular, if g X and g Y are wedge metrics on X and Y , respectively, then π * X g X ⊕ π * Y g Y is a wedge metric on X ∼ × Y .
Proof. Such an isomorphism is unique if it exists, being the continuation of the standard isomorphism T (X • × Y • ) ∼ = T X • ⊕ T Y • over the interior, which also constitutes the result when X and Y both have depth 0. Proceeding then by induction on the depth of X ∼ × Y , it suffices to consider a point in X ∼ × Y of maximal depth, lying in a maximal boundary hypersurface, which without loss of generality we may assume has the form L(H, Y ) where H ∈ M 1 (X) is a maximal boundary hypersurface of X. Restricting attention to a sufficiently small neighborhood, we may assume H = F H × B H and L(H, Y ) = (F H ) × (B H ∼ × Y ) are products. Then since we may take ρ ≤L(H,Y ) = π * X ρ ≤H by maximality, and since B H ∼ ×Y has lower depth, Proposition 6.7 and induction give a diagram of short exact sequences, in which the left and right hand vertical maps are isomorphisms, assembling to the desired isomorphism w T (X ∼ × Y ) ∼ = w T X ⊕ w T Y in the middle.
Alternate proof of Theorem 6.8. For the second proof, we use the characterization of w T X as the unique vector bundle (up to isomorphism) which extends T X • and to which any Φ metric g X on X • extends to be nondegenerate and nonsingular. The result is trivial when X or Y is of depth 0, and we then proceed by induction on the depth of X ∼ × Y . Let H and G be minimal hypersurfaces of X and Y respectively; by induction it suffices to prove the result near the lift of H × G. In particular, B H and B G are closed manifolds of depth 0, and, by localizing as necessary, we may assume that φ H and φ G are trivial, so In order to prove the isomorphism (6.9) we may freely choose the defining wedge metrics g X and g Y . Hence on neighborhoods B H × F H × [0, 1) and B G × F G × [0, 1) we may assume that g X and g Y have the form where κ H and κ G are respective wedge metrics on F H and F G , and where g B H and g B G are respective smooth metrics on B H and B G . Using the polar coordinates r = ρ 2 H + ρ 2 G , θ = arctan ρ H ρ G , the cartesian product g = π * X g X + π * Y g Y takes the form g = dr 2 + g B H + g B G + r 2 dθ 2 + sin 2 θκ H + cos 2 θκ G .
Clearly, this is a wedge metric on the neighborhood provided we know that the geometric join dθ 2 + sin 2 θκ H + cos 2 θκ G lifts to a wedge metric on F H ∼ ⋆ F G . However, by our inductive assumption on the depth, g is known to be a wedge metric away from r = 0, which in particular forces this geometric join to be a wedge metric on F H ∼ ⋆F G .

Equivalence of minimal and maximal joins
As noted previously, the ordered products X max ∼ × Y max and X min ∼ × Y min are generally not diffeomorphic, nor is either typically diffeomorphic to X ∼ × Y for any other non-trivial interior fibrations φ X and φ Y . By contrast, in this section we prove a rather remarkable diffeomorphism between the minimal and maximal joins X ∼ ⋆ max Y and X ∼ ⋆ min Y . Note that since X ∼ ⋆ max Y is interior minimal while X ∼ ⋆ min Y is interior maximal, we don't expect (nor do we obtain) a natural fibered corners isomorphism from one to the other; however it happens that the diffeomorphism does interwtine all the boundary fibrations except for the interior; in particular it follows that these two extremal versions of the join product have the same underlying stratified space, namely the topological join of the two associated stratified spaces.
Theorem 7.1. Let X and Y be manifolds with fibered corners with total boundary defining functions ρ X = H∈M 1 (X) ρ H and ρ Y = G∈M 1 (Y ) ρ G . Then the b-map lifts to a b-map (not a fibered corners morphism) the restriction of which to the principal boundary hypersurface is a diffeomorphism In addition, (7.2) is fibered over boundary hypersurface, and constitutes a fibered corners isomorphism X ∼ ⋆ max Y ∼ = (X ∼ ⋆ min Y ) min . In particular, it follows from Corollary 5.2 that the stratified space associated to X ∼ ⋆ min Y is also the topological join X ⋆ Y of the stratified spaces associated to X and Y .
Remark. Note that the fibered corners structure on C min (X) or C max (X) doesn't refer to the interior fibration on X itself. Viewing the joins as blow-ups of X ×Y ×[0, 1], the isomorphism (7.2) identifies boundary hypersuraces in a manner which is somewhat surprising, namely: • Proof. Enumerate the hypersurfaces of X and Y by H 1 , . . . , H m and G 1 , . . . , G n , respectively, and suppose that they are ordered in the sense that H i < H j implies i < j and similarly for G i . On C max/min (X) (and its product with C max/min (Y )) denote by h i the monoid generator associated to the boundary hypersurface H i ×R + and by ξ the generator associated to the boundary hypersurface X × 0. Likewise, on C max/min (Y ) (and its product with C max/min (X)) denote by g i the monoid generator associated to the boundary hypersurface G i × R + and by η the generator associated to the boundary hypersurface Y × 0. On the unresolved product C max (X) × C max (Y ), the map (7.1) is associated to monoid homomorphisms generated by ξ → η, η → ξ, h i → h i + η, g j → ξ + g j .

(7.3)
We proceed to show that the map lifts to the ordered products. The ordered product of the domain, C max (X) ∼ ×C max (Y ) is characterized as a blow-up via Theorem 2.1 by monoids generated by ordered chains of sums a + b where a ∈ {ξ < h 1 < · · · < h m < 0} , b ∈ {η < g 1 < · · · < g n < 0} . Such a chain must begin with ξ + η, and may have terms of the form h i + η or ξ + g j but not both, and also of the form h i + 0 or 0 + g j but not both. Though it is not really a generator, it is useful to imagine 0 + 0 at the end of the chain. On the other hand, the ordered product of the range, C min (X) ∼ × C min (Y ) is characterized by monoids generated by ordered chains of sums a + b where a ∈ {0 < h 1 < · · · < h m < ξ} b ∈ {0 < g 1 < · · · < g n < η} .
Such an ordered chain must end with ξ + η, and may have terms of the form h i + η or ξ + g j but not both, and also of the form h i + 0 or 0 + g j but not both. Again, though it is not a generator, it is useful to imagine 0 + 0 at the beginning of the chain.
To each maximal ordered chain of the source C max (X) ∼ ×C max (Y ), we associate a maximal ordered chain of the target C min (X) ∼ × C min (Y ) by exchanging ξ with 0 and η with 0 in the list of generators (in particular ξ + η is exchanged with 0 + 0 at the start or end), and we claim that the monoid homomorphism (7.3) induces a homomorphism from the monoid generated by this source chain into (but not onto) the monoid generated by the target chain. Indeed, the generators map as ξ + η → ξ + η h i + g j → h i + g j ξ + 0 → 0 + η 0 + η → ξ + 0 h i + 0 → h i + η 0 + g j → ξ + g j h i + η → (h i + 0) + (ξ + η) ξ + g i → (0 + g i ) + (ξ + η) It follows from Theorem 2.1 that (7.1) lifts to a b-map C max (X) ∼ × C max (Y ) → C min (X) ∼ × C min (Y ). Note that the monoid homomorphism is not an isomorphism (it is injective but not surjective) owing to the last line; in particular the lifted map is not b-normal.
To understand what the map looks like when restricted to X ∼ ⋆ max Y ⊂ C max (X) ∼ ×C max (Y ), recall that the b-normal monoids of a boundary hypersurface are the quotient of those from the ambient space by the generator of the said hypersurface, which is this case is ξ + η. Thus the monoids of X ∼ ⋆ max Y are generated by ordered chains as described above with the omission of ξ + η at the start, while the monoids of X ∼ ⋆ min Y are generated by ordered chains as for C min (X) ∼ × C min (Y ) with the omission of ξ + η at the end. Moreover, the induced map on the quotient monoids is generated as follows: (7.4) Thus each monoid of X ∼ ⋆ max Y is mapped isomorphically to a related monoid of X ∼ ⋆ min Y obtained by exchanging ξ with 0 and η with 0. In particular it follows that X ∼ ⋆ max Y → X ∼ ⋆ min Y is simple, b-normal and ordered if the domain is made interior minimal (or the target maximal).
To see that it is also fibered (and to check that the diffeomorphism on the interior extends up to the boundary faces, which does not follow automatically from the above), we resort to using coordinates. Before recording the general result, it is illustrative to consider some explicit examples setting n = m = 2 for simplicity. Begining with standard form coordinates (t, y 1 , x 1 , y 2 , x 2 , y 3 ) on C max (X) and (s, y ′ 1 , x ′ 1 , y ′ 2 , x ′ 2 , y ′ 3 ) on C max (Y ) taken so that the x i and x ′ i agree with the boundary defining functions used to construct the map, standard form coordinates on C max (X) interior coordinates (y j or y ′ j for some j) for the base of the boundary fibration on {σ i = 0} which belong to the fiber of the boundary fibration on {σ i−1 = 0}. Likewise, standard form coordinates on C min (X) ∼ × C min (Y ) associated to the ordered chain given by exchanging ξ ↔ 0 and η ↔ 0 in the source chain have a similar form (ζ 1 , σ ′ 1 , ζ 2 , σ ′ 2 , . . . , ζ m+n , σ ′ m+n , ζ m+n+1 , σ ′ m+n+1 ) (7.10) where (σ ′ 1 , . . . , σ ′ m+n+1 ) are rational boundary defining coordinates made up from (x 1 , . . . , x m , σ) and (x ′ 1 , . . . , x ′ n , τ ), with σ ′ m+n+1 = 0 defining the principal hypersurface X ∼ ⋆ min Y , and with the same ordered sequence of interior coordinates ζ j . The form of the boundary defining coordinates is entirely algebraic, and since (7.4) is an isomorphism, it follows that τ = sx 1 · · · x m and σ = tx ′ 1 · · · x ′ n lifts to an identification σ i = σ ′ i for 1 ≤ i ≤ m + n. The interior coordinates ζ j are identical between the source and target, and it follows that the induced map from {σ 0 = 0} in (7.9) to σ ′ m+n+1 = 0 in (7.10) is a diffeomorphism which intertwines the boundary fibrations. fibered corners Y = Y max which is ordered and fibered, and which has the property that whenever f ♯ (H) = G ∈ M 1 (Y ), the bottom row of the commuative diagram Proposition A.4. Let X be a manifold with fibered corners with representative boundary defining functions {ρ H } for the given fc-equivalence class and f : X → Y an interior fibration. Then there exists a tube system on X with the following properties: Remark. Observe that X → * and 1 : X → X are always interior fibrations according to Definition A.3 and neither imposes any additional conditions on a tube system. Thus the result with either of these interior fibrations furnishes the existence of tube systems in general.
Proof. The proof is by induction on the depth of X; if X has depth 0 then the statement is vacuuous. Thus assume the result holds for all spaces of depth strictly less than that of X; in particular each B H is equipped with a tube system satisfying properties Either case is consistent with the condition that ξ H (ρ H ) = 1 over H ′ since ρ H is transverse to φ H ′ for H ′ < H as well as to f as noted above. Note that the flow of ξ H just constructed preserves the distribution ker df by integrability of the latter. If on the other hand f ♯ (H) = G ∈ M 1 (Y ), then we may simply take ξ H to be transversal to H, tangent to the fibers of φ H ′ for each H ′ < H, as well as f -related to some vector normal field to G; this again ensures that the flow of ξ H preserves ker f by the identity f * ([ξ H , η]) = [f * ξ H , f * η] = 0 for η tangent to the fibers of f . Again the condition that ξ H (ρ H ) = 1 at every H ′ ∈ M 1 (X) can be imposed since ρ H is transverse to φ H ′ for H ′ < H.
Proceeding now by induction, suppose that ξ H ′ has been chosen for all H ′ > H to satisfy the conditions of Definition A.2 as well as conditions (a) and (b) above. We first define ξ H | H ′ for each H ′ > H to be a lift by φ H ′ of the normal vector field ξ H,H ′ for ∂ H B H ′ ⊂ B H ′ ; such a lift is automatically tangent to the fibers of f in case f ♯ (H) = Y since f : H ′ → Y factors through φ H ′ in this case and ξ H,H ′ is already tangent to the fibers of B H ′ → Y . Likewise, since ρ H | H ′ is the pull-back of a boundary defining function ρ H,H ′ for ∂ H B H ′ ⊂ B H ′ and we may assume that ξ H,H ′ (ρ H,H ′ ) = 1, the condition that ξ H (ρ H ) = 1 is satisfied here. Note that such lifts may be chosen consistently over H ′ ∩ H ′′ by Definition A.2.(c) applied to ξ H,H ′ and ξ H,H ′′ . We then extend ξ H over the tubular neighborhoods U H ′ of each H ′ > H by flowing out along ξ H ′ ; by Definition A.2.(a) these flow-outs are consistent on U H ′ ∩ U H ′′ . Since the ξ H ′ preserve ker f , the extensions of ξ H remain tangent to the fibers of f if applicable. Finally, away from the neighborhoods U H ′ for H ′ > H we proceed as in the base case, taking ξ H to be normal to H, tangent to the fibers of f if f ♯ (H) = Y (and f -related to a normal vector field for f ♯ (H) ∈ M 1 (Y ) otherwise), tangent to the fibers of φ H ′ for H ′ < H, and such that ξ H (ρ H ) = 1 over H ′ ≤ H; then we use a partition of unity to assemble ξ H globally to satisfy Definition A.2.(a)-(c) and with respect to the vector fields already constructed, as well as conditions (a) and (b) above. This completes the inner induction on H ∈ M 1 (X), at the end of which X is equipped with a tube system having the required properties, completing the outer induction.
Proposition A.5. Let f : X → Y be an interior fibration of manifolds with fibered corners. Then there exists a connection on f : X → Y with respect to which the lift η of any vector field η ∈ V(Y ) satisfies the following properties with respect to each H ∈ M 1 (X) such that f ♯ (H) = Y : Having defined the connection on a neighborhood of H, we now consider the double, X H , of X across H; this consists of two copies of X glued along H with opposite orientations, and f : X → Y extends to a smooth map f : X H → Y (provided the gluing is performed with respect to a choice of normal direction along ker df | H ) which is again an interior fibration according to Definition A.3. The complement of U H ⊂ X has a neighborhood which may be identified with an open set in X H , and since f : X H → Y has strictly lower depth fibers, the connection on U H may be combined with a suitable connection on this neighborhood by the inductive assumption.
Corollary A.6. An interior fibration in the sense of Definition A.3 is a locally trivial fiber bundle of manifolds with fibered corners. More precisely, for every p ∈ Y there exists a neighborhood O ∋ p