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Differential bundles and fibrations for tangent categories

Tangent categories are categories equipped with a tangent functor: an endofunctor with certain natural transformations which make it behave like the tangent bundle functor on the category of smooth manifolds. They provide an abstract setting for differential geometry by axiomatizing key aspects of the subject which allow the basic theory of these geometric settings to be captured. Importantly, they have models not only in classical differential geometry and its extensions, but also in algebraic geometry, combinatorics, computer science, and physics. This paper develops the theory of"differential bundles"for such categories, considers their relation to"differential objects", and develops the theory of fibrations of tangent categories. Differential bundles generalize the notion of smooth vector bundles in classical differential geometry. However, the definition departs from the standard one in several significant ways: in general, there is no scalar multiplication in the fibres of these bundles, and in general these bundles need not be locally trivial. To understand how these differential bundles relate to differential objects, which are the generalization of vector spaces in smooth manifolds, requires some careful handling of the behaviour of pullbacks with respect to the tangent functor. This is captured by"transverse"and"display"systems for tangent categories, which leads one into the fibrational theory of tangent categories. A key example of a tangent fibration is provided by the"display"differential bundles of a tangent category with a display system. Strikingly, in such examples the fibres are Cartesian differential categories demonstrating a -- not unexpected -- tight connection between the theory of these categories and that of tangent categories.


Introduction
Tangent categories provide an axiomatic setting for abstract differential geometry. They were first introduced in [27] as a category equipped with a "tangent functor" which associated to each object an Abelian group bundle with additional structure. It was shown in that paper that tangent categories encompass both standard differential geometry settings [10], algebraic geometry settings, and settings arising in synthetic differential geometry [15].
In [11], the present authors slightly generalized this notion so that the bundles were only assumed to be commutative monoids. This allowed for a key source of new examples of tangent categories arising from computer science and combinatorics. In computer science, the resource λ-calculus [4] and the differential λ-calculus [12] were developed in parallel. They were, eventually, unified [22] as being calculii with their semantics in Cartesian differential categories [6]. In combinatorics the differential of a combinatorial species [3] is also an important tool. This idea was developed more abstractly into a differential of polynomial functors which were then connected to the differential of datatypes (see [13], [1], and [2]). These provide examples of settings in which there is a notion of differentiation but in which negation has no natural meaning. An important aspect of this paper is to spell out in detail the connection between Cartesian differential categories, which provide a unifying framework for the settings above, and tangent categories. As we shall see, one aspect of this connection hinges on the notion of a differential bundle.
A fundamental structure in differential geometry is the (smooth) vector bundle. Vector bundles are important in differential geometry because they algebraically capture the notion of local coordinate systems for manifolds. They thus provide a way to describe additional structure on manifolds such as vector fields, symplectic forms, and differential forms. The tangent bundle of a manifold is, of course, itself a vector bundle, but one can also form the product (or the "Whitney sum") of two vector bundles and their tensor product. Significantly, one can pullback vector bundles along smooth maps allowing the transport of this local structure. One can also apply the tangent bundle functor to a vector bundle to produce another vector bundle.
In the abstract setting of a tangent category, it is not immediately obvious how to define vector bundles as, in particular, there is no assumption of any sort of "object of real numbers" from which one can define vector spaces with a scalar multiplication. However, quoting from the Wikipedia entry on vector bundles 1 , "smooth vector bundles have a very important property not shared by more general fibre bundles. Namely, the tangent space T v (E x ) at any v ∈ E x can be naturally identified with the fibre E x itself. This identification is obtained through the vertical lift...". It is this key structure which we use as the definition of the generalization of smooth vector bundle to arbitrary tangent categories. That is, rather than ask that each fibre of a map q : E G G M be a vector space (smoothly), we ask that q be an additive bundle with, in addition, a "lift" map which enjoys certain properties so that "the tangent space T v (E x ) at any v ∈ E x can be naturally identified with the fibre E x itself". Because at this level of generality there is no scalar multiplication -and in order to emphasize the connections we establish in this paper -we call such bundles differential bundles. Differential bundles, so defined, enjoy all of the key properties of ordinary smooth vector bundles. Namely that the tangent bundle is a differential bundle (example 2.4), applying the tangent bundle functor to a differential bundle produces another differential bundle (corollary 2.5), and the pullback of a differential bundle along any map is again a differential bundle (lemma 2.7). Moreover, we show that the obvious maps between such bundles, namely those that preserve the "lift" operation, suitably generalize ordinary linear maps between vector spaces. This observation thus gives an alternative perspective on the meaning of "linearity": linearity can be seen as the preservation of the lift map, rather than the preservation of an action 2 by real number objects.
With the basic definition of differential bundles and their properties, one can look at connections on such bundles, a topic which will be treated in a future paper by the authors.
However, there is more to say about these differential bundles as objects of interest in their own right. The authors' previous paper on tangent categories [11] defined the analog of vector spaces in a tangent category, calling these objects differential objects. Just as vector bundles are vector spaces in a slice category, it is then natural to ask whether differential bundles are differential objects in a slice tangent category.
Here, though, lies a difficulty which is at the heart of much of the rest of this paper. In [27] Rosický had proposed that the slice of a tangent category should again be a tangent category with respect to the "vertical" tangent bundle. Furthermore, he had suggested that, for this to be so, it sufficed that the construction of the vertical bundle (given by pulling back over the zero of the tangent bundle) needed to exist and be preserved by the tangent functor. Of course, to be a tangent category there are other pullbacks which need to be present. In particular, the pullback of the projection from the tangent bundle functor along itself -called here, and in the authors' previous paper, T 2 -needs to be present and preserved. While the condition Rosický had suggested was clearly necessary, it did not allow -as far as we could seea construction of this pullback. This led us to the view that the question of what pullbacks should exist (and be preserved), was more subtle than had been supposed and deserved careful treatment.
In a tangent category it is certainly not the case that arbitrary pullbacks exist let alone that the tangent functor preserves those pullbacks. Recall, for example, even in classical differential geometry the lack of pullbacks has given rise to a detailed theory -the theory of transverse maps and submersions -of when such pullbacks exist and are preserved. For tangent categories, therefore, it should be no surprise that a commensurate theory needs to be developed. Thus, in particular, for differential bundles to become differential objects in the slice, it should be expected that some -somewhat subtle -conditions on the pullbacks may arise.
To deal with these issues in tangent categories it is necessary to have explicit structural descriptions of which pullbacks must exist and be preserved. Toward this end we introduce two notions: transverse systems and display systems. A transverse system on a category specifies a class of pullbacks in the category subject to a small set of axioms. Every tangent category comes equipped with a minimal transverse system which comprises the pullbacks mandated by the tangent structure to exist and be preserved. However, it is quite possible that a given tangent category have a larger transverse system: in the category of smooth manifolds, there is a natural transverse system, first defined in [26], consisting of all pullbacks of a pair of maps which are transverse (in the sense of differential geometry) to each other. In addition to this, it is also often useful to specify a class of maps in the category along which all pullbacks exist and are in the transverse system. We call these display maps, as they are closely related to the display systems introduced by Paul Taylor [25] in the study of fibrations. A tangent category with a compatible display system is called a display tangent category.
With these ideas in hand, we can look at the subject of differential bundles and slice tangent categories from the prospective of fibrations. A tangent fibration is then a fibration of tangent categories satisfying certain additional axioms. A key result for these structures is that each fibre of a tangent fibration is again a tangent category (theorem 5.3). Moreover, if we have a display tangent category X, then the subcategory of the arrow category of X consisting of display maps is a tangent fibration. Thus, in a display tangent category, not only is the category of display maps over a fixed object a tangent category, but also the differential objects in this slice tangent category are exactly differential bundles whose projections are display maps (5.12). Another important example of a tangent fibration, which captures the idea of partial derivatives, is given in 5.6 (a).
Bringing the two main ideas of this paper together, for a display tangent category we can consider the fibration of differential bundles (with projec-tions display maps). Each fibre of this fibration has the very special property that every object is a differential object in a canonical way: this makes each fiber a Cartesian differential category. This then establishes the tight relationship between differential bundles in a tangent category and Cartesian differential categories.

Organization
In section 2, we recall the definition of a tangent category, introduce differential bundles in tangent categories, and study their properties. In section 3 we begin the study of how differential bundles relate to differential objects and Cartesian differential categories. We recall that a Cartesian differential category is always a tangent category in which every object is a differential object. However, for a tangent category to be a Cartesian differential category not only must every object be a differential object but also there must be a global coherence between these structures. The key concept in this section is, thus, the notion of a tangent category with coherent differential structure. This idea becomes central when we later study the tangent category of differential bundles over a fixed base as these categories naturally have coherent differential structure. In section 4 we discuss the notions of transverse and display systems for arbitrary categories and for tangent categories. In this section we also revisit the definitions of morphisms of tangent categories and differential bundles when the tangent categories have transverse and display systems and we show how some earlier results can be obtained more conceptually from these notions. Finally, in section 5, we define and study fibrations between tangent categories, eventually concluding with the result that there is a tangent fibration of display differential bundles; in this fibration, every fibre is a tangent category with coherent differential structure and, thus, is a differential fibration.

Basic tangent categories and differential bundles
We shall begin with the definition of tangent category as a category with tangent structure consisting of a functor, some natural transformations, and certain pullbacks. The basic idea was introduced in [27]. In [11] those ideas were generalized slightly by allowing the tangent bundle to be a commutative monoid rather than an Abelian group bundle. This generalization was A category with tangent structure, (X, T), is a tangent category. A tangent category is said to be Cartesian if it has finite products which are preserved by T .
The requirement that each T n preserve the pullback expressing the universality of the vertical lift is, in fact, a consequence of the other requirements. This is because the canonical flip can be used to transform a universality diagram acted on by T n back into a "top-level" universality diagram by flipping the maps up to the top level (see [11] Lemma 2.15 for the preservation of the equalizer form of the condition). Here we wish to emphasize the fact that T n should preserve these pullbacks in order to hint at a more general pattern.
One can think of the vertical lift as a comultiplication for the tangent functor and the flip as a symmetry transformation. With this perspective ℓc = ℓ asserts that the comultiplication is commutative and ℓT (ℓ) = ℓℓ asserts that the comultiplication is coassociative. One might expect that as the vertical lift acts as a comultiplication that the tangent functor should be a comonad, however, significantly, this is not the case. It is the case, however, that (T, η.µ), where η m := 0 M : M G G T (M) and µ := p, T (p) + M : , is a monad (see [11], section 3.2). Tangent categories provide a strict generalization of traditional settings for differential geometry in a number of respects. An immediate and striking difference is that in tangent categories the tangent bundles are assumed only to be commutative monoids -that is they may lack negatives. Many of the results of classical differential geometry require that one has negatives. It should, therefore, be emphasized that, having negatives is a property (rather than structure) and all the basic results in this paper are, of course, true when one does have negation. Thus, importantly, this work strictly includes the settings of classical differential geometry. Another important difference (in contrast to settings like synthetic differential geometry) is that the definition assumes no "object of real numbers". (i) Finite dimensional smooth manifolds with the usual tangent bundle structure. For tangent vectors u and v at a point x, the vertical lift is given by (see [17], pg. 55). For smooth manifolds it is well known that the pullback of two maps f and g exists and is preserved by the tangent functor whenever the maps are transverse. This means that, at each point f (x) = g(y), the tangent space is generated by image of tangent spaces under T (f ) and T (g). A submersion is a map f which is surjective on the tangent spaces at each f (x): these are clearly transverse to every map. In particular, the projection p M : T (M) G G M is a submersion (eg., see [20, 7.1.e]) so that pullbacks along it exist and are preserved by the tangent functor.
It should be noted that many important examples of Cartesian differential categories (particularly those arising from Computer Science) do not have negatives.
For specific examples, see [5], section 2.5: by [6] section 3.2, the coKleisli categories of each of these ("monoidal") differential categories form Cartesian differential categories and hence also form tangent categories.
(iii) Any category is trivially a tangent category by setting the tangent functor to be the identity functor and p, 0, +, c, and ℓ all to be the identity natural transformation. The fact that every category is trivially a tangent category reminds one that tangent structure is certainly structure, rather than a property of the underlying category.
(iv) The infinitesimally linear objects in any model of synthetic differential geometry [15] gives an example of representable tangent structure (see [11], section 5.2 for a characterization of when the tangent functor T in a tangent category is representable). If D is the object of infinitesimals, then we take T M = M D . The vertical lift is then given by exponentiating the multiplication map D × D G G D. In models of SDG all pullbacks exist and, as ( ) D is a right adjoint, it preserves all pullbacks. Thus, in these settings every map has a pullback which is preserved by the tangent bundle functor and its powers.
(v) The opposite of the category of finitely presented commutative rings (or more generally rigs) is another standard example of a category with representable tangent structure: here D is the "rig of infinitesimals", (vi) A source of examples, from [27], uses the fact that if (X, T) is a tangent category then the functors from X to set which preserve both the wide pullbacks of T n (p), and the pullback from the universality of the lift with natural transformations as arrows forms a tangent category. The tangent functor T * is then given by T * (F ) := T F . In fact, this works for any category Y in place of set and functors X G G Y which preserve the required pullbacks. This source of examples includes, for example, C ∞ -rings (see [24] chapter 1) and more generally the product preserving functors from any Cartesian differential category. Viewing a Cartesian differential category as a (generalized) many-sorted theory of differentiation, it is pleasing to know that its category of algebras (in any category with sufficient limits) is necessarily a tangent category.
(vii) The category of functors, Cat(C, X), from any category to a tangent category inherits the tangent structure of X pointwise. Thus, for example the category of arrows in a tangent category, X 2 , is a tangent (viii) Convenient manifolds with the kinematic tangent bundle (see [18] section 28) form a tangent category, by combining the results of [7] and section 6 of [11].
As noted in some of the examples above, the fact that a given category can carry more than one tangent structure implies that being a tangent category is a structure on rather than a property of the category.

Differential bundles
Vector bundles play an important role in differential geometry and this section defines the corresponding notion for tangent categories. We call this notion a differential bundle: it is an additive bundle with, in addition, a lift map satisfying properties similar to those of the vertical lift for the tangent bundle itself. The morphisms between these bundles will then just be commuting squares between the projections. However, we will identify an important subclass of morphisms, called the linear morphisms, which must in addition preserve the lift. Notably, neither ordinary morphisms nor linear morphisms require that the additive structure be preserved. However, we shall show that linear maps between differential bundles automatically preserve addition. (iii) A vector bundle, as defined in [10], consists of a smooth map between smooth manifolds q : E G G B which, for each b ∈ B, has a neighbourhood U b so that q| U b is smoothly equivalent to the trivial π 0 : U b × V G G U b for some fixed vector space V . Of course, this does not mean that the whole bundle is trivial (in the sense that E ≃ B ×V ): the locally trivial bundles of a vector bundle can glue together in a manner which incorporates twists which give important information about the properties of the space B.
Any vector bundle in smooth manifolds is a differential bundle: see [17], pg. 55 for the requisite structure. A differential bundle in smooth manifolds, however, is not necessarily a vector bundle for a rather simple reason: if B is not connected a differential bundle allows different components to have fibres which are vector spaces of different dimensions. In fact, a differential bundle in smooth manifolds is a vector bundle precisely when all of these fibres happen to have the same dimension: see corollary 31 of [23].
(iv) In synthetic differential geometry (SDG) the preferred notion of a "vector bundle" is a Euclidean module in X/M (for example, see [9], [16], and [19]). Theorem 3.9, below, will show that a Euclidean module in a model of SDG is precisely the same as a differential object. Proposition 3.4 will show that in a Cartesian tangent category, a differential object is the same as a differential bundle over the final object. Finally, in Section 5.2 we will show that, when the tangent functor preserves sufficient limits, differential bundles over a base M are the same as differential bundles over the final object in the slice category X/M. In models of SDG, the tangent functor preserves all limits as it is given by exponentiation. Thus, in a model of SDG, Euclidean modules in X/M are the same as differential bundles over M.

Two constructions of differential bundles
While the above examples show that differential bundles arise frequently in the standard examples, we will like to also show that differential bundles abound in an arbitrary tangent category. Towards this end we describe two fundamental ways of constructing new differential bundles from existing differential bundles. If q = (q, σ, ζ, λ) is a differential bundle, then define

Lemma 2.5
If q is a differential bundle then so is T (q).
PROOF: While it is immediate that this is an additive bundle and that pullbacks of T (q) along itself exist and are preserved by T n , it is not immediate that T (λ)c acts as a lift. We start by checking (T (λ)c, 0 T ) : are additive bundle morphisms. For the first, as (λ, 0) : as an additive bundle morphism. For the second, similarly we apply T to obtain the additive bundle morphism as an additive bundle morphism.
For the coherence of T (λ)c, we need to check the following commutes: First note that the last coherence diagram for tangent structure may be re-expressed as using the fact that c 2 = 1. This gives For universality of lift we need to show that is a pullback. We may re-express this square as the outer square of Since T preserves the universality of q, the outer square is a pullback which is preserved by T n , as required. ✷ Observe also that there are some obvious linear morphisms of bundles associated with this construction: Another important way of constructing differential bundles is by pulling back. The pullbacks involved, however, not only have to exist but they also must be preserved by T n , neither of which is guaranteed in an arbitrary tangent category. The question of which pullbacks exist and are preserved by T n is a topic to which we will return when we introduce the notion of transverse and display systems. In the meantime we shall deal with these matters in a rather ad hoc fashion in order to get this basic construction of differential bundles on the table early.
If q = (q : E G G M, ζ, σ, λ) is a differential bundle, and f : X G G M is a map then to say the pullback of the bundle along f exists and is preserved by T n means that the pullback along f of all the wide pullbacks of q along itself must also exist and be preserved by T n . It is clear that pulling back q along f under these assumptions will provide an additive bundle. However, as the following cube (whose front and back faces are pullbacks) shows this additive bundle also has the data for being a differential bundle as there is a candidate for the lift, f * (λ), given by We shall suggestively call this structure f * (q). Notice that the cube also immediately shows two further useful facts. First, as the right face is an additive bundle morphism, the left face must also be an additive bundle morphism.
Secondly, if f * (q) is indeed a differential bundle, then there is a linear mor- This will clearly be a Cartesian map sitting above f for the functor P : DBun(X) G G X.
We now confirm very concretely that f * (q) is indeed a differential bundle. Later, when we consider "display" tangent categories for which q is a "display differential bundle", we shall see a more conceptual proof of this result (remark 5.13).

Lemma 2.7
In any tangent category, when the pullback of q along f exists and is preserved by T n (in the sense described above), f * (q) is a differential bundle and the linear morphism f * q is a Cartesian morphism (in the fibrational sense) for the functor P : DBun(X) G G X sitting above f .
G G X exists by assumption and, furthermore, pullback powers of this map exist and are preserved by T n by assumption. The pullback of an additive bundle is always an additive bundle, thus, the only issue is the behaviour of the lift f * (λ). Above we have already seen that (f * (λ), 0) is an additive bundle morphism. We must also show that (f * (λ), f * (ζ)) is an additive bundle morphism. The map f * (ζ) is the unique map to the pullback in the lower back square of the following diagram. Just to obtain the basic data of a bundle morphism, we must start by showing that the left square face commutes.
However this square is a map to the apex of a pullback. Thus, the square Here are the calculations: is an additive bundle morphism. However, the codomain of this morphism is the apex of a pullback of additive bundle morphisms: are additive bundle morphisms. The first expression equals (f * (q), 1 X )(0, 1 X ) -the final bundle morphism followed by the zero bundle morphism -which is certainly additive. The second expression equals (f * E , f )(λ, ζ) which is also certainly an additive bundle morphism.
Next we must show that f * (λ)ℓ = f * (λ)T (f * (λ)). As these maps have codomain T 2 (f * (E)), which is the apex of a pullback, they are equal if postcomposing with T 2 (f * E ) and T 2 (f * (q)) makes them equal. Here are the calculations: Finally we must prove the universality of the lift: this is given by the following in which the front, back, and left faces are pullbacks (all preserved by T n ) so that there is a unique f * (µ) completing the cube making all the vertical squares pullbacks (preserved by T n ). It is easily checked that f * (µ) is the µ map for f * (q). ✷ An example of an application of this lemma is in forming the "Whitney sum" of two differential bundles. Given two differential bundles q, q ′ over the same base M in a Cartesian tangent category, the Whitney sum may be formed by taking the pullback of q × q ′ along the diagonal: It is easy to see that in a Cartesian tangent category q × q ′ is a differential bundle. However, we must know that pullbacks of differential bundles along the diagonal map exist and are preserved by T n in order to form the pullback bundle. In general this is not guaranteed. There are however more general situations in which the Whitney sum exists and we shall return to this later when we discuss display systems (see section 4.2).

Properties of differential bundles
Many of the basic properties of the tangent bundle in a tangent category (which were described in [11], section 2.5) can be generalized to differential bundles. This section collects these generalizations, as they are very useful when concretely working with differential bundles.
To begin, note that for a differential bundle q on E, T (E) has two addition operations: T (σ) and +. Our first observation concerns the conditions under which one can interchange these operations: (i) Whenever both sides are defined, we can interchange the additions T (σ) and +: share a common zero) and when both sides are defined, then: (i) The requirement that both sides of this equation be defined amounts to requiring that the following equations hold: We have: For the last step, using the naturality of +, + T 2 T (π 0 ) = T 2 (π 0 )+ T and (ii) The requirement that both sides of this equation be defined amounts to We then use an Eckmann-Hilton argument: Next, we develop some useful identities concerning the map µ: Lemma 2.9 If q is a differential bundle then µp = π 1 and λ = 1, qζ µ.
PROOF: For the first claim: For the second claim: The following result generalizes Lemma 2.12 in [11]: In the presence of the other axioms for a differential bundle q, the universality of the lift may be equivalently expressed by demanding either of the following: is an equalizer; (ii) for any map f : X PROOF: (i) Assuming the pullback, first note that the µT (q) = π 1 q0 = µpq0 so µ equalizes the two maps. Now given g with gT (q) = gpq0 then this gives a unique map g| µ which mediates the pullback.
Conversely, assuming the equalizer then µT (q) = µpq0 = π 1 q0 and so the square commutes. If gT (q) = g ′ 0 then gT (q) = g ′ 0 = g ′ 0p0 = gT (q)p0 = gpq0 so there is a unique map to the equalizer which makes the square a pullback.
The uniqueness of {f } follows from the uniqueness of f | µ as ✷ Note that for a trivial differential bundle M 1 G G M and a map f : Because T (and all powers of T ) preserves the pullback expressing the universality of lift, it follows that: Corollary 2.11 For a differential bundle q in a tangent category, T and all powers of T preserve the equalizer We shall have occasion to use the bracketing operation introduced in Lemma 2.10 and so it is useful to establish some of its key properties: and pq0 and have f pq = gpq: PROOF: equalizes T (q ′ ) and pq ′ 0: To show that {x}h = {xT (h)}, we need to show that {x}h satisfies the same universal property as {xT (h)}: (g is a linear bundle morphism and naturality of p0) (iv) We need to show qζ has the same universal property as {0}: qζλ, 0p0 T (σ) = q0T (ζ), 0 T (σ) (additivity of λ) = 0T (q)T (ζ), 0 T (σ) (naturality of 0) = 0 T (q)T (ζ), 1 T (σ) = 0 (addition of a zero term) so {f } and similarly {g} is defined. Finally, by (ii) Thus one side is defined if and only if the other side is.

✷
The above generalizes most of lemma 2.14 of [11] to differential bundles, with in particular (vii) of lemma 2.14 generalizing to (ii) in the above. The only result not generalized is 2.14(vi). This is generalized by the following result, which relates the bracketing operation for the differential bundle T (q) (see Lemma 2.5) to the bracketing operation in the differential bundle q: Lemma 2.13 If q is a differential bundle and f : Indeed, We now check T ({f }) has the same universal property as {T (f )c}: The next result generalizes lemma 2.13 of [11].
Lemma 2.14 For a differential bundle, q, the following is a joint-equalizer diagram: and thus, in particular, λ is monic.
PROOF: ζ is a section so monic, thus it suffices to show that if x0 = xλ that x factors through ζ. To this end we have:

Properties of linear bundle morphisms
Morphisms of differential bundles are not required to preserve the additive structure; they simply commute with the projections of the bundles. Bundle morphisms which preserve the lift, that is, the linear bundle morphisms, are of fundamental importance. They generalize the linear maps of Cartesian differential categories (we shall see this later in proposition 3.7), and so also generalize linear maps in the ordinary sense.
In this subsection we prove some basic properties of these linear bundle morphisms.
Proposition 2.16 Linear morphisms of differential bundles are additive.
PROOF: If (f, g) : (q, σ, ζ, λ) G G (q ′ , σ ′ , ζ ′ , λ ′ ) is a linear morphism of bundles (so that λT (f ) = f λ ′ ) then to say it is additive is to require that σf = π 0 f, π 1 f σ ′ and ζf = gζ ′ . To show the first equality, we postcompose by λ ′ and use the fact that λ ′ is monic (see Lemma 2.14): For the preservation of zero, we use the fact that q is epic: ✷ This generalizes the fact that linear maps in a Cartesian differential category are always additive (lemma 2.2.2(i) in [6]).
The following shows that how linearity of a bundle morphism is related to preservation of µ: is linear, then the identity µT (f ) = π 0 f, π 1 f µ ′ holds. Conversely, if this identity holds and the bundle morphism preserves zero, that is ζf = gζ ′ , then the bundle morphism is linear.
PROOF: If (f, g) is a linear bundle morphism it is additive so that: Conversely, if (f, g) satisfies the above identity and preserves zero, then precomposing with 1, qζ gives so that (f, g) is a linear bundle morphism.

✷
Another useful result is that the inverse of a linear bundle morphism is automatically linear: a retract and (h, k) :

PROOF:
(i) Let (f ′ , g ′ ) be a section of (f, g). Then since (f, g) and (f h, gk) are linear we have (ii) This follows immediately from (i) since the identity bundle morphism is linear.

Differential structure
In a Cartesian tangent category one can consider differential bundles over the final object. As we shall see, these special differential bundles have an alternate description as a differential object, first defined in [11]. When the tangent structure behaves well with respect to slicing, as is the case in synthetic differential geometry (see Example 2.4 (4)), it is possible to use the structure of a differential object in the slice X/M as the definition of a differential bundle over M. However, as discussed in the introduction, securing the necessary behaviour with respect to slicing requires some delicacy, and will be returned to in later sections of this paper. Differential objects were developed in order to relate tangent categories to Cartesian differential categories. In [11] it was shown that the differential objects of any Cartesian tangent category always form a Cartesian differential category. In section 5.2, we will need to know when a Cartesian tangent category is a Cartesian differential category. Sorting this out is the main and rather technical objective of this section. The problem is this: clearly a Cartesian tangent category which is a Cartesian differential category must have associated to every object a "canonical" differential structure. However, objects may possess multiple differential structures and an arbitrary choice of differential structure for each object will not necessarily be compatible with extant tangent structure. There must, therefore, be a manner of choosing from the possible differential structures which "fits" with the extant tangent structure. We call such a choice a coherent choice of differential structure. Furthermore, we describe precise requirements for a choice to be coherent. We then show that a Cartesian tangent category with such a coherent choice of differential structure is a Cartesian differential category. This rather technical result underpins the transition from tangent categories to Cartesian differential categories which is needed to establish one of the main results of this paper, Theorem 5.14, which shows that there is a tangent fibration of differential bundles, and in this fibration, every fibre is a Cartesian differential category.

Differential objects
In [11], an important structure an object of a tangent category can possess, called differential structure, was described. Objects with differential structure are called differential objects and they play an analogous role to vector spaces in the category of smooth manifolds. Of course, since a general tangent category has no analogue of the real numbers there is no notion of "vector space" as such. Instead differential structure is defined based on another special property enjoyed by vector spaces: namely, that of having a trivial tangent bundle in the sense that T (V ) ∼ = V × V . Since the tangent bundle already has one projection map, p : T (A) G G A, in order to define a differential object, it suffices to demand the existence of a further projection p : T (A) G G A satisfying certain properties.

Definition 3.1 For an object
A in a Cartesian tangent category, differential structure on A consists of a commutative monoid structure σ : is an additive bundle morphism, that is the following diagrams commute: is an additive bundle morphism, that is the following diagrams commute: •p is coherent with respect to the vertical lift in the sense that the following commutes: An object together with a specified differential structure is called a differential object.
The requirement that the map T (ζ) is preserved is actually implied by the fact that the map 0 A is preserved: PROOF: Consider the calculation: Thus, the form of this definition has some redundancy, which was exploited in the original definition given in [11]. Here we have chosen a more natural presentation which displays the additive bundle morphisms involved. It should also be noted that the definition in [11] omitted the important axiom ℓT (p)p =p.

Example 3.3
(i) In the category of finite-dimensional smooth manifolds, the differential objects are vector spaces, R n , as their tangent bundle is T (R n ) ≃ R n × R n .
(ii) Similarly, in the category of convenient manifolds, the convenient vector spaces are differential objects.
(iii) In the category of affine schemes, that is (cRing op ), polynomial rings Z[x 1 , x 2 . . . x n ] are differential objects.
(iv) In a model of SDG the ring of line type is always a differential object, as the requirement that R D ≃ R × R is part of its definition. More generally, in models of SDG differential objects are exactly Euclidean R-modules (see section 3.3).
(v) In a Cartesian differential category (see section 3.4) every object is canonically a differential object.
An alternative viewpoint to take on the definition of differential objects is as follows. Since A is a commutative monoid and T preserves products, T (A) is also a commutative monoid. The first two requirements of a differential object then ask that T (A) be a product in the category of commutative monoids in X, cMon(X). However, cMon(X) is an additive category and so in fact the first two requirements are equivalent to asking that T (A) be a biproduct in cMon(X). Thus, by general results about biproducts in an additive category (see, for example, [21], VIII.2), this part of the definition could equivalently be given by asking that T (A) be a coproduct in cMon(X) (involving, in particular, a map λ from A to T (A)), or by asking that there be ap and a λ satisfying certain equations. These observations will be useful in the proof of: In a Cartesian tangent category, to give a differential object is precisely to give a differential bundle over the final object.
PROOF: Given a differential object (A,p, σ, ζ), we use the universal property of the product T (A) to set λ = 1, !ζ . (Note that the other injection, !ζ, 1 , is 0 : A G G T (A) since 0p =!ζ.) As mentioned above, by general results about biproducts in an additive category, the universality of λ and one of the additivity requirements for λ follows automatically.
The final coherence, λℓ A = λT (λ), asks for the equality of two maps into T 2 (A). Since T (A) is a product and T preserves products, T 2 (A) is also a product with projections Thus, it suffices to check the two sides of the equality are equal when postcomposed by each of these four maps. For the first, by the last coherence for p, For the second, Checking for post-composing with the other two maps is straightforward. Conversely, if we have a differential bundle over 1 (A, λ, σ, ζ), we set p := {1}. The universality ofp and the first additive requirement follows from general biproduct results.

Corollary 3.5 In any Cartesian tangent category, if q : E
G G M is a differential bundle and a : 1 G G M is any point such that the pullback E a of a along q itself is preserved by T n , then E a is a differential object.
PROOF: By 2.7, E a is a differential bundle over 1, and so by the previous result is a differential object. ✷ Note that this observation allows an alternate characterization of differential objects and it subsumes a key result of [11] (theorem 4.15).

Additional properties of differential objects
One may wonder whether any coherence with the canonical flip c should be required of differential objects (or differential bundles over 1). In fact, some basic coherence is automatic: Proposition 3.6 For any differential bundle over 1, (A, σ, ζ, λ), cT (p)p = T (p)p.
PROOF: The proof is by calculation. It repeatedly uses the identity on T (A) that we established in proposition 3.4: where m T × = T (π 0 , T (π 1 ) −1 . We start by proving an auxiliary fact: in which it is useful to observe the following commutation: Here is the calculation: where λ 2 and c 2 are the maps defined by: A useful observation for the next calculation is that the following diagram commutes: It then follows that T 2 (σ)T (p)p = T 2 (π 0 )T (p)p, T 2 (π 1 )T (p)p σ.
Now we use all the above in: Proposition 3.7 Linear morphisms between differential bundles over 1 are the same as maps which are differentially linear.
PROOF: If f : A G G A ′ is linear bundle morphism andp = {1} is the first projection for the differential bundle, then p, p is the inverse of µ; asking that f be a linear bundle morphism implies by lemma 2.17 that:

Post-composing byp then gives
Conversely, suppose we have a differentially linear map; that is, T (f )p = pf . Then the above proves that on the first component, and on the second component, asking T (f )p = p, pf µp = p, pf π 1 = pf is just naturallity of p. However, we also have that f preserves the zeroes of the bundles, since so since ! is epic, ζ ′ = ζf . Thus by lemma 2.17, f is a linear differential bundle morphism. ✷

Differential objects in representable tangent categories
One aspect of differential objects that was not investigated in [11] was what form they take when the tangent category is representable; that is, when there is an object D for which T M = M D . Section 5 of [11] discusses what conditions on an object D are required so that defining T M = M D gives a tangent category. In particular, D has: • maps ⊙ : D × D G G D and ℘ : 1 G G D ("comultiplication" and "zero"); • the pushout of 1 ℘ G G D along itself exists; this pushout will be denoted by D ⋆ D, with injections ı 0 : D G G D ⋆ D, ı 1 : D G G D ⋆ D; • a map δ : D G G D ⋆ D.
satisfying various axioms (see definition 5.6 of [11]). Such a D is referred to as an "infinitesimal" object. Now, suppose that we have a differential object A in a representable tangent category with corresponding infinitesimal object D. By the previous section, A is a differential bundle over 1 and hence has a lift map λ : A G G T A. But since T A = A D , this is equivalent to giving an "action" The following theorem characterizes the differential object axioms in terms of this action, written in the term logic used in section 5 of [11].
Theorem 3.8 Suppose that (X, T) is a representable tangent category with infinitesimal object D. Then to give a differential object is equivalent to giving an "infinitesimal module": a commutative monoid (A, σ, ζ) with a map ⊲ : D × A G G A so that: PROOF: The result is a straightforward application of uncurrying the differential bundle axioms. ✷ If the representable tangent category is in fact a model of SDG, then more can be said. Recall that in SDG one has a "line object R", and it is often useful to consider "Euclidean R-modules": R-modules A with the property that for any f : D G G A there exists unique b ∈ A such that f (b) = db + a (for example, see [19], pg. 5).

Theorem 3.9
Suppose that X is a model of SDG with line object R. Then to give a differential object in the corresponding tangent category (X, T) is precisely to give a Euclidean R-module.
PROOF: Let D be the corresponding infinitesimal object of R. By the theorem above, we need to show how to translate between an infinitesimal module and an R-module. Recall that in SDG the object R is the pullback and there is an inclusion j of D into R (which is given by the unique map D G G R given by currying the multiplication ⊙ : D × D G G D and using the above pullback description of R). Now, given an infinitesimal module A we can define an action of R by the composite Conversely, given a Euclidean R-module A with action m : R × A G G A, we can define an action of D on R by It is now straightforward to check that the axioms for an infinitesimal module translate directly to the axioms for a Euclidean R-module and vice versa, with one exception: axiom (iv) for an infinitesimal module. This axiom is equivalent to the differential bundle axiom This does not come directly from any axiom for a Euclidean R-module, but in fact is automatic given the other axioms. Indeed, while in an arbitrary tangent category T 2 (A) is given by A D⋆D , in a model of SDG, T 2 (A) is also given by A D (2) , where D(2) = {(d 1 , d 2 ) ∈ D × D : d 1 d 2 = 0}. In this case, the map π 0 λ, π 1 λ is given by the map since post-composing this map by the projections π 0 , π 1 : T 2 A G G T A gives π 0 λ and π 1 λ. Then since + is given by δ : which then by (iii) equals d. (d ⊲ σ(a 1 , a 2 )), as required. ✷

Cartesian differential categories
Recall that a Cartesian differential category [6] is a left-additive category 4 with a combinator (called "differentiation") satisfying certain axioms. Here, we give an alternative form of the axioms as explained in proposition 4.2 of [11]. This form of the axioms allows one to align tangent structure more conveniently with differential structure. As noted earlier, a Cartesian differential category is always a tangent category with T (A) := A × A and T (f ) = D(f ), π 1 f . Furthermore, it is straightforward to check that every object in a Cartesian differential category is, with respect to the structure A := (! A , 0, π 0 + π 1 , 1, 0 ), canonically a differential bundle over the final object. For this, in particular, one must check the universal property of the lift, that the diagrams (1),(2),(3) and (6) in the proof of Proposition 3.4 commute, and the coherence with the lift is satisfied.
The results of the previous section imply that when the lift is preserved by f : A G G B then f should be linear in the sense that D[f ] = π 0 f . For a 4 These are examples of so called skew enriched categories following [28]. Concretely, this means each homset is a commutative monoid and with respect to composition satisfies the left distribution law -f (h + k) = f h + f k and f 0 = 0 -but not necessarily the right distribution law.
Cartesian differential category, we may verify this with a direct proof. The preservation of the lift tells us implying that 0f = 0 and 1, 0 D[f ] = f . We want to prove that D[f ] = π 0 f . As we see below, it actually suffices to know that 1, 0 D[f ] = f . Here is the calculation:

Coherent differential structure
To determine, conversely, whether a Cartesian tangent category is, in fact, a Cartesian differential category one must clearly be able to assign to each object a "canonical" differential bundle structure. There is no reason, however, why an arbitrary assignment of such structures will allow one to reconstruct a Cartesian differential structure for the category which is compatible with the existing tangent structure. Clearly some additional compatibility is required to ensure the choice of bundle structure at each object is compatible with the Cartesian tangent structure. Definition 3.10 A Cartesian tangent category has coherent differential structure if every object A has an associated structure as a differential bundle over 1, A = (! A , ζ A , σ A , λ A ), which we refer to as its canonical differential structure, such that: [CDS.1] the canonical differential structure on A × B is the product of the canonical structures of its components: 2] the canonical differential structure on T (A) is given by: Our first objective is to translate these conditions into equivalent requirements for differential (object) structure. Preliminary to this we prove two useful equations which we shall use repeatedly in the proofs below: Lemma 3.11 In a Cartesian tangent category with a coherent differential structure µ A×B T (π 0 ) = (π 0 × π 0 )µ A and µ A×B T (π 1 ) = (π 1 × π 1 )µ B .
PROOF: We shall prove the first: the other identity is similar. ✷ Proposition 3.12 A Cartesian tangent category has coherent differential structure if and only if each object has an associated differential structure such that [CDS.1] ′ A × B has coherent differential structure satisfying the following equations: has coherent structure satisfying the following equations: PROOF: We shall check only the second condition, leaving the first to the reader.
[CDS.2] ′ We need to show that for a differential objectp T (A) = cT (p A ) if and only if λ T (A) = T (λ A )c. Starting with a differential object we define λ T (A) as We must check that T (λ A )c works: Conversely for a differential bundle over 1 we must checkp . Using 4.23 we have: ✷ It is straightforward now to observe: Lemma 3.13 Cartesian differential categories always have a coherent differential structure given by A = (! A , +, 0, 1, 0 ).

The main theorem of the section is:
Theorem 3.14 Every Cartesian tangent category with a coherent differential structure is a left additive category and has an associated differential which makes it a Cartesian differential category.

Transverse and display systems
The theory of transverse maps in classical differential geometry -as embodied by smooth manifolds -indicates that the issue of which pullbacks exist and are preserved by the tangent functor can be quite delicate. In the general setting of tangent categories the question of which pullbacks exist is, as a consequence, also important. In particular, considering Cartesian differential categories as a source of examples of Cartesian tangent categories, we cannot require many pullbacks -outside those implied by having products -to exist. Yet, the presence of pullbacks is clearly important in the development of the fibrational properties of tangent categories. Thus, the purpose of this section is formalize these issues in the context of arbitrary tangent categories.
Definition 4.10 A display tangent category is a transverse tangent category equipped with a display system on that transverse system such that T preserves display maps.
It is important to note that we do not necessarily require that the projection maps p M : T M G G M be a display map in a display tangent category. This is important not only for some examples we would like to consider, but also for the development of fibrations of tangent categories (see the note after 5.3).
Nonetheless, when the projection maps are in the transverse system, then the transverse system generated by the display maps alone will already include all the transverses required of a tangent category. Thus, in this case, requiring preservation of the display maps will imply all the preservation requirements of the transverse system. (ii) Any Cartesian tangent category has a display system generated by projections from products and the maps to the final object.
Explicitly, we say that a map f : X G G A is a projection if there exists a map f ′ : X G G A ′ such that the pair (f, f ′ ) makes X the product of A and A ′ . In a Cartesian tangent category, we can form a display system in which the display maps are the projections in this sense and the transverses required to make these maps quarrable are all the multiple pullbacks in which all but one leg are projections. The only axiom that takes a bit of work to check is the naturality axiom. Suppose we have a natural transformation α : P G G Q between such transverses: so that each α 01 , α 10 , α 11 are projections, as well as the maps P 01 G G P 11 and Q 01 G G Q 11 . Then if we have P 01 = Q 01 ×P ′ 01 and P 10 = Q 10 ×P ′ 01 then one can easily check that P 00 is the product of Q 00 , P ′ 01 , and P ′ 01 , with α 00 one of the projections.
Note that in this display system the projections from the tangent bundle need not be display maps.
(iii) Every Cartesian differential category is an example of a Cartesian tangent category, as was discussed in Section 3, thus the projections form a natural display system. In this case the projections from the tangent bundle are display maps.
(iv) In a model of SDG, all pullbacks exist and are preserved by the tangent functor, so the standard display system on a model of SDG consists of all maps in the category.
Definition 4.12 A display functor between categories with a display system: is a transverse functor that sends elements of D to elements of D ′ .
In particular, this means explicitly that if d ∈ D then F (d) ∈ D ′ and, furthermore, whenever is a pullback then is a pullback.
Example 4.13 As noted above, well-adapted models of SDG consist of, in particular, a transverse functor from smooth models to a model of SDG; such a functor is in addition always a display functor relative to the natural display structure on the model of SDG (which consists of all maps in the category).

Morphisms of tangent categories revisited
In this section we revisit morphisms of tangent categories which, in addition, may have a transverse and/or display system associated to them. We also consider comorphisms of tangent categories, which were not considered in [11]. consists of a transverse functor F : X G G X ′ and a natural transformation α : T F G G F T ′ such that the following diagrams commute: If the categories also have display systems, a morphism of display tangent categories simply asks in addition that F be a display functor.
The other possibility is to consider comorphisms between tangent categories: these consist of a functor F together with a transformation β : F T G G T ′ F satisfying the same requirements as above but with the functor order reversed; if the tangent categories have display systems the functor is again required to preserve these. It is worth noticing that to make sense of comorphisms, the assumption that F is transverse is crucial. Consider the coherence diagram for the addition: To obtain a map β 2 : T ′ 2 (F (M)) G G F (T 2 (M)) one needs the codomain to be a limit.
Definition 4. 15 We say a morphism of tangent categories is strong if α is an isomorphism and strict if α is the identity.
Note that a strong morphism is also a comorphism.

Definition 4.16 A morphism of transverse tangent categories is
Cartesian if for each f , the naturality square is a pullback which is transverse in the codomain category X ′ .
Clearly strong (and strict) morphisms are always cartesian, and the identity functor is a strict morphism of tangent categories. An important example of a strong morphism is T itself: Lemma 4.17 If X is a (display) tangent category, then the pair (T, c) : X G G X is a strong morphism of tangent categories.
PROOF: The equational axioms for F (q) to be a differential bundle and for (F f, F g) to be a linear bundle morphism follow in a straightforward fashion from the equations for (F, α) being a morphism of tangent categories and naturality of α. The only slight difficulty is the universal of the lift for F (q); this asks that be a pullback; however, we may re-express this diagram as the outer square of the composite the leftmost square has isomorphisms as the top arrows, the middle square is a pullback since F preserves the pullback diagram for T (q), and the rightmost square is a pullback (and transverse) as the morphism is cartesian. Thus the entire square is a pullback, as required. ✷ The following result relates the bracketing operation of a differential bundle q to the bracketing operation of the differential bundle F (q) (for F a Cartesian morphism).

Lemma 4.23
Suppose that q is a differential bundle, f : X G G T E equalizes T (q) and pq0 and (F, α) : X G G Y is a Cartesian morphism of tangent categories. Then F ({f }) = {F (f )α}.
PROOF: We first need to show {F (f )α} is well-defined; that is, we need to show that F (f )αT (F (q)) = F (f )αT (F (q))p0.

Tangent fibrations
This paper's main contribution so far is the abstract formulation of differential bundles for tangent categories. This section describes some basic aspects of the general theory of tangent fibrations in order to place this development in a wider context. It will be assumed for this section that the reader is familiar with the general theory of fibrations. For the reader less familiar with these notions we recommend [14] or [8].
Before we start it is necessary to make some remarks about the closure of transverse systems in fibrations. Suppose we have a fibration ∂ : (X, Q) G G (B, Q ′ ) between categories with transverse systems such that ∂ is a transverse functor. Suppose also that q : 2 n G G X is transverse in X, q ′ : 2 n G G B is transverse in B, and α : q ′ ⇒ q∂ is a natural transformation of transverses. From this data we can define a functor α * (q) : 2 n G G X with α * (q)∂ = q ′ : each vertex v ∈ 2 n determines the vertex α * (q)(v) by setting it equal to α * v (q(v)), and the maps between the vertexes are then determined by requiring that they sit above the corresponding map of q ′ .
By construction α * (q) is a functor and preserves all the limits that q, q∂, and q ′ preserve. (For transverses it suffices to show that pullbacks will be preserved by condition (d) of the definition of a transverse system.)