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Section complexes of simplicial height functions

A theory of sections of simplicial height functions is developed. At the core of this theory lies the section complex, which is assembled from higher section spaces. The latter encode flow lines along the height, as well as their homotopies, in a combinatorial way. The section complex has an associated spectral sequence, which computes the homology of the height functions domain. We extract Reeb complexes from the spectral sequence. These provide a first order approximation of how homology generators flow along height levels. Our theory in particular models topological section spaces of piecewise linear functions in a completely combinatorial way.


Introduction
It is a common theme in mathematics to study properties of a space X through the lens of a real-valued height function h : X → R. The best known example of this is probably Morse theory, where X is a smooth Riemannian manifold and one considers gradient flow lines of a Morse function h. There is also a discrete variant of Morse theory due to Forman [For98], in which X is a simplicial complex and h assigns a real value to every simplex of X . Given a height function h : X → R one can also investigate the space of all sections of h. A section is a local right inverse to the height function h. If for example X is a topological space and h : X → R a continuous function, then a section of h is a map σ : [a, b] → X such that h • σ = id. In [Try21] it is explained how these sections form the space of morphisms in a topological category-called the section category of h. It is then shown that under fairly mild regularity assumptions on h the classifying space of the section category is weakly equivalent to the base space X . As a consequence the spectral sequence associated to (the nerve of) the section category then computes the homology of X . A similar result for smooth Morse functions has been proven in unpublished work by Cohen, Jones and Segal [CJS92]. In their case, the associated spectral sequence reduces to the widely used Morse homology.
In this paper, we introduce a theory of sections for the case where X is a simplicial set and h : X → R a simplicial map taking values in a simplicial model of the real line given by the nerve of the poset category (R, ≤). The fibers of h assemble into a simplicial set ∏ a∈R h −1 a that we call the space of 0-sections. To capture how these fibers are connected across p levels, we introduce the p-sections. A 1-section from height a to height b is a 1-simplex in X starting in h −1 a and ending in h −1 b, a 2-section is a 2-simplex labeled by three height values and so on. There is a simplicial set (S h ) p , containing the psections as its vertices and their q-homotopies as q-simplices. We refer to it as the space of p-sections. It turns out that the section spaces are also simplicial in p. Thus, we may define a bisimplicial set S h , associated with the height function h, called the section complex of h. The section complex essentially splits up the homology information of X into two directions: the horizontal direction along h and the vertical one transversal to it. This is encapsulated in the following main result: Theorem 1.1. Let h : X → R be a height function. The diagonal of the bisimplicial set S h is homotopy equivalent to X : diag S h ≃ X .
The computational implications of this result come from the existence of a spectral sequence which computes the homology of the diagonal diag S h , from homological features of the section spaces (S h ) p (see e.g. [Seg68,GJ09]). We refer to the spectral sequence associated to S h as the section spectral sequence. From the first page of the section spectral sequence we extract for q = 0, 1, 2, . . . chain complexes G q by dividing out degeneracies. We refer to G q as the qth Reeb complex of h. One may think of it as an object that carries information about how generators of the q'th homology H q flow across height levels. Moreover, for a finite simplicial set X , the Reeb complexes can be computed in finite time.
Proposition 1.2. Let h : X → R be a height function and G q its qth Reeb complex. The associated section spectral sequence has H p G q appearing as the (p, q)th entry on the second page E 2 p,q ≃ H p G q and converges to the homology of X : In [VHT22] we define Reeb complexes G A q , that capture how generators of homology flow along sections of a continuous height function between heights A ⊂ R. Based on the theory in [Try21] we can often replace this object with a truncated Reeb complex T q that only incorporates sections between adjacent critical levels. The reason is that continuous sections (and their homotopies) always factorise into smaller sections between intermediate height levels. As a consequence the continuous section spectral sequence always converges on the second page. This is different from the simplicial setting of the paper at hand, where the presence of higher section spaces can lead to non-trivial higher differentials in the spectral sequence. We say that X is subdivided according to h if every 1-section traverses only successive height levels. In this case, the section spectral sequence collapses at the second page. We are then in a position to directly compare the topological and the simplicial Reeb complexes. Proposition 1.3. Let h : X → R be a simplicial height function, such that X is subdivided according to h. Then h associates to a piecewise linear function f : |X | → R and there is an induced isomorphism of chain complexes between G h and the truncated continuous Reeb complex T q .
This result shows in particular that simplicial section spaces can model the homology theory of topological sections of piecewise linear functions on CW-complexes.
Because the topological Reeb complexes always compute the homology of the base space it also follows that in the case where X is subdivided according to h the Reeb complexes G q recover the homology of X directly. For field coefficients we then have: Outline. In Section 2 we start out by briefly reviewing some basic concepts from the theory of simplicial sets. This sets the stage to then define section spaces and describe how they assemble into the section complex. We proof Theorem 1.1 in Section 2.4. Subsequently Reeb complexes are introduced in Section 3. This is supplemented by a discussion of the more complicated section spectral sequence from which the Reeb complexes can be extracted. In section 3.2 we briefly provide some general background on spectral sequences of double complexes. Throughout section 3.1 numerous examples are computed to illustrate the theory. Finally Section 4 relates our simplicial theory back to the theory of topological sections in [Try21] by providing a proof of Proposition 1.3. The paper closes with some final remarks on the usefulness of simplicial sets in computational topology.
Computer code. A Python implementation for computing section complexes, as well as Reeb complexes, is found at https://github.com/paultrygs/Section-Complex/.
Notation. Categories of familiar objects are put inside parenthesis, e.g. (Simplicial Sets). The hom-set of maps X → Y is denoted Map(X ,Y ), while map(X ,Y ) refers to the simplicial set of maps. We will in general consider chain complexes over coefficients in some field k.

The section complex
We introduce the central object of our studies: the section complex S h associated to a simplicial height function h : X → R. We assemble it from the spaces of sections (S h ) p . As such it contains the topological information about sections between height levels of h but also the combinatorial information about how these fit together across different height levels. Theorem 1.1, that we proof at the end of this section, shows that we can combine these two pieces of information to recover the homology of X .
Let us start out with a quick review of some basics about simplicial sets. This is not intended to be very pedagogical but rather serves the purpose to quickly introduce some necessary notation. For a comprehensive introduction to the theory of simplicial sets see for example [GJ09] and for a more elementary treatment [Fri21].

Background on simplicial sets
A simplicial set X is a sequence X n of sets, ranging over n = 0, 1, 2, . . . , together with face maps d i : X n → X n−1 and degeneracy maps s j : X n → X n+1 satisfying certain relations [GJ09,p.4]. An element x in X n is interpreted as an n-simplex whose ith face is d i x, whereas s j x incorporates ways to consider x as an (n + 1)-simplex. In contrast to simplicial complexes, this for example implies that an (n + 1)-simplex y can have an (n − 1)-simplex x as its face; d i y = s j x. Moreover, two distinct n-simplices x and y can have equal faces d i x = d i y for all i. Equivalently, the data of a simplicial set X can be organized into a functor X : ∆ op → (Sets), where ∆ is the simplex category.
Example 2.1. We construct a circle from 0-simplices v 0 and v 1 and 1-simplices e 0 and e 1 , not counting degeneracies, by declaring d 0 e i = v 1 and d 1 e i = v 0 , for i = 0, 1. A sphere can be obtained from a single 0-simplex v and 2-simplex f . In this case, all faces of f must be equal to s 0 v, a degenerate 0-simplex. This means that the boundary of f is equal to the point v. A simplicial map f : X → Y is a series of maps f n : X n → Y n which commutes with face and degeneracy maps. Pictures as above are produced by labeling each n-simplex in X with the topological n-simplex ∆ n t and identifying appropriate simplices via the geometric realization |X | = ( ∏ n X n × ∆ n t )/ ∼. The quotient glues simplices along faces and collapses degenerate simplices. Note that the realization defines a functor | · | : (Simplicial Sets) → (topological spaces) Any small category C defines a simplicial set NC, called the nerve of C. The set of 0simplices, NC 0 , consists of objects in C, and NC n consists of tuples (m 1 , . . . , m n ), of composable morphisms within C. The ith face of (m 1 , . . . , m n ) is determined by composition (m 1 , . . . , m i+1 • m i , . . . , m n ) for i = 0, n, whereas d 0 and d n drops m 0 and m n , respectively. We depict a 2-simplex ( f , g): be the category generated by the directed graph 0 → 1 → · · · → n. Applying the nerve yields the standard simplicial n-simplex ∆ n = N[n]. It consists of a unique n-simplex coming from the tuple (0 → 1, 1 → 2, . . . , n− 1 → n) with n+ 1 distinct faces. We recover the topological n-simplex as |∆ n |.
There are simplicial inclusions δ i : ∆ n → ∆ n+1 which identify ∆ n with the ith face of ∆ n+1 . Observe that δ i (q) equals q if q < i and q + 1 otherwise. Conversely, there are simplicial collapses σ j : ∆ n+1 → ∆ n for which σ i (q) equals q if n ≤ i and q − 1 otherwise. Definition 2.3. Let (R, ≤) be the real line equipped with its usual ordering. We define the simplicial real line R = N(R, ≤).
An n-simplex in R is uniquely determined by a non-decreasing sequenceā = (a 0 , . . . , a n ) of real numbers.
Two simplicial sets X and Y define a product X ×Y with n-simplices X n ×Y n , whose face and degeneracy maps are computed component-wise.
The simplicial mapping space map(X ,Y ) contain simplicial maps f : X → Y as its 0simplices. An n-simplex in map(X ,Y ) n is a simplicial map X × ∆ n → Y . Since ∆ 1 is a model of the interval, the 1-simplices are homotopies. Face and degeneracy maps, d i and s j , are obtained by pre-composing with component-wise maps id X × δ i and id X × σ j , respectively (Example 2.2). Applying d i to f : X × ∆ n → Y thus restricts ∆ n to its ith face, whereas s j adds appropriate identities.

Sections of height functions
We are now going to use the theory of simplicial sets from above to model the behaviour of sections of a height function. Before we do so lets quickly remind ourselves of the continuous theory of sections from [Try21].
Let f : T → R be a continuous function on a topological space. In [Try21], a section of f is defined as a map ρ : This makes it possible to define the section category of f , a category internal to topological spaces with Sect f as its space of morphisms. In [Try21] it is then shown that under fairly mild assumptions, the classifying space of this category is homotopy equivalent to T . These assumptions are for example met by piecewise linear functions. We will now describe how to obtain such piecewise linear functions from height functions on simplicial sets. To that end, recall the definition of the simplicial real line R (Definition 2.3).
Definition 2.6. Let X be a simplicial set. A height function h on X is defined as a simplicial map h : X → R.
We can equivalently characterize a height function, h, as a map between sets h : Conversely, assigning to every v in X 0 a height h(v) such that the orientation of the 1-simplices in X is respected, defines a unique height function h : X → R. We call the image h(X 0 ) the height levels of h.
We observe from the previous section that a point in the realization |R| is a class [ā,t] whereā = (a 0 , . . . , a n ) is a non-decreasing sequence of real numbers andt = (t 0 , . . . ,t n ) a point in the topological n-simplex. The dot productāt defines a continuous function c : |R| → R from the realization of R to the real line. Any height function h : X → R thus associates to a piecewise linear function f : |X | → R by composing |h| and c. It is shown in [Try21] that no homotopical information is lost if we only consider those sections that start and end at the height levels of h.
We now ask the following question: is it possible to construct a simplicial version of the section category directly from the simplicial height function h : X → R rather than from the associated piecewise linear function f ? This would render all the information involved combinatorial and in particular accessible to computational topology.
There is a natural choice for replacing the topological space of sections between two heights a 0 and a 1 with a simplicial set, S h [a 0 , a 1 ]. Namely, the subspace of the mapping space, map(∆ 1 , X ), carved out by the pullback: where we interpret the 1-simplex (a 0 , a 1 ) in R as the simplicial map ∆ 1 → R, with 0 → a 0 and 1 → a 1 . We may then define (S h ) 1 , the space of 1-sections of h as the disjoint union Example 2.7. We take as our simplicial set the standard 2-simplex ∆ 2 and define a height function by the labels of the figure: However, the following example illustrates why we cannot proceed as in the construction of the topological section category.
Example 2.8. We define a height function on the standard 2-simplex ∆ 2 as follows: There is a unique 1-simplex between each distinct pair of heights. This means that the 1section spaces are: Hence the space of 1-sections cannot be utilized to recover the 2-simplex s in ∆ 2 connecting e 0 , e 1 and e 2 . Contrast this with the corresponding topological situation. In that case, the sections |e 2 | and |e 0 | may be concatenated into |e 2 | * |e 0 |, a section from height 0 to height 2 running through 0 → 1 → 2. We would also find a continuous path from |e 2 | * |e 0 | to |e 1 | moving along the bottom of the triangle, given by continuous deformation through the realization of the 2-simplex. In particular, this process encodes the topological information provided by |s|.
To recover higher simplices across more than two height levels, we introduce higher sections.

Higher sections and the section complex
Definition 2.9. Given a height function h : X → R on a simplicial set X , we may construct for every p-simplexā : ∆ p → R, a simplicial set S h [ā] as the pullback Taking the disjoint union over all p-simplices in R, we obtain which we refer to as the space of p-sections of h.
By definition, the space of p-sections is a simplicial set. This is guaranteed by the fact that pullbacks and coproducts always exist in the category of simplicial sets. The 0simplices are the p-sections of h, i.e. the p-simplices in X spanning p height levels. The q-simplices are then the (p, q)-sections in (S h ) p corresponding to a simplicial map ρ : In Section 2.1, we explained how the face and degeneracy maps in map(∆ p , X ) are obtained by pre-composing with id × δ i and id × σ j : These diagrams characterize the face and degeneracy maps in the space of p-sections. As the superscript v indicates, we call these the vertical face and degeneracy maps.
The above use of 'vertical' hints to the fact that there is a second, horizontal, simplicial structure. Indeed, we can pre-compose a (p, q)-section with a simplicial inclusion or collapse applied to the first component in ∆ p × ∆ q to obtain commutative diagrams These characterize maps of sets which we refer to as the horizontal face and degeneracy maps. Alternatively, the horizontal face maps can be induced from the universal property of the pullback via: and similarly for horizontal degeneracy maps. This shows that the horizontal face maps are in fact simplicial maps going from p-sections to (p − 1) and (p + 1)-sections, respectively. The intuition is that d h i restricts a p-section in S h [a 0 , . . . , a p ] to a (p − 1)-section in S h [a 0 , . . . ,â i , . . . , a p ], whereas s h j adds a degenerate label S h [a 0 , . . . , a j , a j , . . . , a p ]. The set (S h ) p,q is therefore simplicial in both p and q, defining a bisimplicial set.
Definition 2.10. The section complex of a height function h : X → R is the bisimplicial set S h with (p, q)-simplices given by (S h ) p,q , i.e. the (p, q)-sections ρ : ∆ p × ∆ q → X . It has horizontal and vertical face and degeneracy maps as defined above.
Remark 2.11. We didn't explain what a bisimplicial set in general is. Similar to a simplicial set it is given by a sequence X p,q of sets that now ranges over pairs of natural numbers. The first component of these indices is sometimes called the horizontal direction and the second one is called the vertical direction. Correspondingly there are also two types of face-and degeneracy-maps for bisimplicial sets. The horizontal ones: d h i : X p,q → X p−1,q and s h j : X p,q → X p+1,q and the vertical ones: These are both required to satisfy the simplicial relations. We can also define a bisimplicial set as a functor where ∆ denotes the simplex category. The reader familiar with adjunctions will agree that this information is then equivalently presented as a functor Postcomposing with the realization functor exhibits the intimate relationship of bisimplicial sets and simplicial spaces. In particular we may turn the section complex S h from Definition ?? into a simplicial space TS h : Example 2.12. Consider once more the standard 2-simplex with height function like in . All these components consist of a single point determined by the 0-simplices at the corresponding heights. If we don't count degeneracies, the simplicial set . Again, all of the components are singletons corresponding to the 1-simplices e 2 , e 0 and e 1 , respectively. Lastly, (S h ) 2 = S h [0, 1, 2], containing the 2-section corresponding to s. In this example, the horizontal face maps of s corresponds to the ordinary face maps of the standard 2-simplex; d h 0 s = e 0 , d h 1 s = e 1 and d h 2 s = e 2 . Notice how the higher section space (S h ) 2 makes it possible to recover the topology of the 2-simplex.
Example 2.13. Consider the product of two standard 1-simplices as in Example 2.4.
We obtain a height function by projecting the labels of the vertices to their first component h : , with two components corresponding to the two 1-simplices e 0 and e 1 . The space of 1simplices is (S h ) 1 = S h [0, 1]. Consider the (1, 1)-section defined in terms of the identity id ∆ 1 ×∆ 1 : ∆ 1 × ∆ 1 → ∆ 1 × ∆ 1 . It has two horizontal faces e 0 and e 1 and two vertical faces given by the two 1-sections ρ 0 and ρ 1 . We interpret this as id ∆ 1 ×∆ 1 being a homotopy from ρ 0 to ρ 1 .

Proof of Theorem 1.1
We will now prove that the diagonal of the section complex S h , associated to a height function h : X → R, is homotopy equivalent to X (Theorem 1.1). This justifies to use the spectral sequence of S h for extracting homological features of X , which will thoroughly discussed in Section 3.
The diagonal of S h , (diag S h ) n , is defined to have (n, n)-sections ρ : ∆ n × ∆ n → X as nsimplices. Since horizontal and vertical face maps are independent, we can safely de- Let us understand how to relate diag S h and X : the n-simplices in diag S h define a subset of map(∆ n , X ) n = Map(∆ n × ∆ n , X ), while the n-simplices of X are given by the set Map(∆ n , X ). There are maps (id ∆ n , id ∆ n ) : ∆ n → ∆ n × ∆ n , i → (i, i) and, conversely, the projection onto the "section component" is defined: pr 0 : ∆ n × ∆ n → ∆ n , (i, j) → i.

Pre-composition defines maps
(id, id) * : diag S h → X and pr * 0 : X → diag S h which will be proven mutual homotopy inverses. The latter map is well-defined. Indeed, if τ : ∆ n → X is any n-simplex in X , then the composition , whereā is defined by the n-simplex h•τ. Furthermore, these maps are clearly simplicial and the composition (id, id) * • pr * 0 is the identity. The proof of the theorem is thus reduced to finding a simplicial homotopy To do so we first introduce for every n ≥ 0, two families of simplicial maps and Pulling these maps back along sections in (diag S h ) n will then provide us with the components of our homotopy. Note that the parameter s will be necessary to make these components fit together into a simplicial map.
We specify how the maps (1) and (2) act on 0-simplices: Note that the so defined assignments preserve the preorder on 0-simplices in ∆ n × ∆ n . Because ∆ n × ∆ n is the nerve of the category [n] × [n], and the nerve functor preserves products, (3) and (4) uniquely determine the families (1) and (2), respectively.
The following figure depicts the maps φ 1,s and ψ 1,s in terms of their image.
(0, 0) In general, the map φ n,0 is always the identity and ψ n,n = ψ n,n+1 = (id, id) • pr 0 . These maps will respectively correspond to the start and end of our final homotopy.
We proceed by looking at how the maps φ 2,s act on the three diagonal faces of ∆ 2 × ∆ 2 .
Lemma 2.14. The family of simplicial maps {φ } n,s satisfies for every n ≥ 1 and likewise for the family {ψ} n,s .
Proof. The proof is straightforward: We compute the images of 0-simplices along both sides of the asserted diagrams. As previously mentioned, this suffices because morphisms between products of standard simplices are uniquely determined by the image of 0-simplices. We first consider the left-hand diagram in (5). It should commute whenever l ≤ n − s. An arbitrary 0-simplex (i, j) is mapped to The commutativity of all the other diagrams is shown in the same manner. No complications arise in the corresponding computations and we will thus not spell them out here.
With this recursive description of the families {φ n,s } and {ψ n,s } we can now give the homotopy which finishes the proof.
Using the right-hand diagrams in (5) and (6) we can show compatibility with the degeneracy maps in the same way. This concludes the construction of the homotopy and thus the proof.

The section spectral sequence
We apply homology to the section spaces (S h ) p and assemble the resulting vector spaces into chain complexes that we term with the name Reeb complexes. Due to Corollary 4.4 of Section 4, these Reeb complexes serve as a completely combinatorical model for the Reeb complexes associated to continuous height functions, as defined in [VHT22]. As a bisimplicial set, the section complex has an associated spectral sequence. We name it section spectral sequence. Theorem 1.1 then implies that the section spectral sequence computes the homology of the height functions base space in general. The Reeb complexes can be extracted from the first page of the section spectral sequence (Proposition 3.8) and thus provide a first order approximation of the homology of X in terms of the homology of section spaces.

Reeb Complexes
Recall that the section complex, S h , consists of all section spaces (S h ) p . We can thus apply any homology functor H q to (S h ) p and induce H q d h i : H q (S h ) p → H q (S h ) p−1 . This defines a simplicial vector space H q S h , because every set of p-simplices, H q (S h ) p , is a vector space. Furthermore, for any simplicial vector space V , there is a complex CV , called the Moore Complex. Its pth entry CV p is equal to the vector space V p , and its differential is induced by the alternating sum of face maps, ∂ = ∑(−1) i d i . Denote by DV the subcomplex of CV whose pth entry only consists of the degenerate p-simplices in V p . The differential induces a well-defined differential CV p /DV p → CV p−1 /DV p−1 from which we define the non-degenerate complex CV /DV . Definition 3.1. For a height function h : X → R and integer q ≥ 1, we define the qth Reeb complex G q as the chain complex C(H q S h )/D(H q S h ).
Example 3.2. The qth Reeb complex associated to a height function h : X → R has as its pth entry, ranging over all increasing sequencesā = (a 0 , . . . , a p ) in R p .
Reeb complexes provide an approximate tool to better understand homological features of the underlying space X . This is achieved by understanding how homology generators flow between height levels and along sections.
Example 3.3. Recall the standard 2-simplex ∆ 2 , with heights as indicated by the labels: We glue two copies of ∆ 2 together along their boundary ∂ ∆ 2 to obtain ∆ 2 ∏ ∂ ∆ 2 ∆ 2 , a simplicial model for the 2-sphere. Label 0 and 1-simplices by their integers, e.g. 01 is the 1-simplex from 0 to 1. The two 2-simplices sharing a common boundary, are denoted a and b. To determine a basis for H q S h , we identify the homotopy types of all section spaces, indexed by increasing sequences: Hence, all Reeb complexes with q ≥ 1 are trivial. For q = 0, however, we determine the boundary maps Picking the evident bases from the above Elementary linear algebra gives H 0 G 0 = k and H 2 G 0 = k, whereas other homology groups are trivial. In this particular example, the zeroth Reeb complex carries the homology of the underlying space ∆ 2 ∏ ∂ ∆ 2 ∆ 2 .
Definition 3.4. We say that a simplicial set X is subdivided according to a height function h : X → R if all section spaces S h [a, b] are empty whenever there is an intermediate Whenever X is subdivided with respect to a height function h : X → R, the Reeb complexes only have two non-zero entries (G q ) p . Indeed, the pth entry, ⊕S h [a 0 , . . . , a p ], of the formula in Example 3.2, is zero for p ≥ 2. Thus, choosing coordinates in this case reduces the information contained in G q to a single matrix. Interpreting this matrix as an incidence matrix provides a graph which gives insight as to how homology generators flow across height levels. This is illustrated with an example.
Example 3.5. We subdivide ∆ 2 according to the heights given in Example 3.3: The subdivided 2-simplex still maps to R by also sending 1 ′ → 1 in R 0 . We construct a space as in the previous example, by gluing two copies of the subdivided 2-simplex together along the boundary defined by the cycle 0 → 1 → 2 ← 1 ′ ← 0. It is not difficult to determine the homotopy types of the section spaces: For instance, the homotopy type of S h [0, 1] is given as follows. There are two 0-sections represented by the edges 01 and 01 ′ . Each copy of the subdivided 2-simplex provides a (1, 1)-section, i.e. a homotopy 0 1 1 ′ between the two sections 01 and 01 ′ , but no higher simplices connect them. Thus, S h [0, 1] is isomorphic to two 1-simplices glued tail to tail and head to head. The horizontal face maps used to calculate Reeb complexes can be depicted: To draw the associated graphs, we think of the basis elements in (G 0 ) 0 as vertices whereas the basis in (G 0 ) 1 defines edges. Then the first column in ∂ tells us that the edge given by the first basis vector in k 2 connects the first and second vertices (basis elements) in k 3 .
For G 1 we have to take a bit care as we have two edges and one vertex. One edge starts in the vertex, the other ends in it. This is due to face maps being sent to zero maps in H 1 , a phenomenon that does not occur for H 0 . We can assemble this information in a barcode-like diagram: In the previous example, the reader familiar with Reeb graphs may have observed that the graph determined by introducing coordinates to G 0 is the Reeb graph of the given height function. This observation is true in general, if X is subdivided according to h : X → R.
Proposition 3.6. If X is subdivided according to a height function h : X → R. Then the simplicial set π 0 S h is the Reeb graph of h. In particular, the zeroth Reeb complex G 0 computes the homology of the associated Reeb graph.
Proof. The result is an immediate consequence of Proposition 4.2, to be proved in Section 4, and Theorem 1.2 in [Try21].

Background on spectral sequences
In the present Section 3.2 and the following Section 3.3 we make the simplifying assumption of taking homology with field coefficients. Everything works as well for arbitrary generalised homology theories and our only aim is a slight simplification of the presentation.
A double chain complex C is a collection C p,q of vector spaces together with horizontal and vertical boundary maps ∂ h : C p,q → C p−1,q and ∂ v : C p,q → C p,q−1 . The maps are further required to satisfy ∂ 2 We always assume a double chain complex to be contained within the first quadrant, so that all entries with p or q negative are zero. To a double complex C, we can functorially associate a chain complex TotC, the total complex of C with TotC n = ⊕ p+q=n C p,q .
There is a functor F : (Bisimplicial Sets) → (Double Complexes). It sends a bisimplicial set X to the double complex FX with (FX ) p,q = FX p,q , the free vector space on X p,q . The horizontal and vertical boundary maps are induced by the horizontal and vertical face maps: Total complexes thus define a functor (Bisimplicial Sets) → (Chain Complexes), by mapping a bisimplicial set X to the total complex TotFX . A theorem of Dold and Puppe [DP61,GJ09] tells us that TotFX is naturally homology equivalent to diag X , the diagonal on X : Therefore in order to understand the homology of diag X , one may rather consider the homology TotFX . One advantage of the total complex, is that it comes with a spectral sequence for computing its homology. The following is a brief recap of how this computational tool works. We refer to [McC01] for a more in-depth introduction.
Given a double complex C, we define the zeroth page of the spectral sequence E 0 p,q = C p,q and remember only the vertical boundary maps ∂ v = ∂ 0 : Applying homology produces the first page E 1 p,q = H q C p,q with induced differentials ∂ 1 from the horizontal differentials of C. Thus, if C = FX for some bisimplicial set X , then we have described a procedure to compute H * diag X from TotFX .

The section spectral sequence
The previous Section implies the existence of a spectral sequence associated to S h which calculates the homology of diag S h .
Definition 3.7. The section spectral sequence of a height function h : X → R is the spectral sequence naturally associated to S h .
Entries on the zeroth page are determined by the free double complex FS h . Explicitly, E 0 p,q = (FS h ) p,q is the free vector space on ∏ ¯a∈R p S h [ā] q , ranging over all nondecreasing real-valued sequencesā = (a 0 , . . . , a p ). Differentials ∂ 0 p,q : E 0 p,q → E 0 p,q−1 are induced from the alternating sum of vertical face maps ∑ i (−1) i d v i in the spatial qdirection. Computing homology vertically (in the q-direction) thus produces the entries of the first page E 1 . Differentials on the first page are then induced in homology from the alternating sum of the horizontal face maps in the section p- Proceeding as in Section 3.2, the section spectral sequence tells us how to recover the homology of diag S h , and thus of X .
Proposition 3.8 (Proposition 1.2). Let h : X → R be a height function. The associated section spectral sequence satisfies E 2 p,q ≃ H p G q and converges to the homology of X : Proof. Theorem 1.1 tells us that diag S h is homotopy equivalent, hence homology equivalent, to X . So it only remains to verify that E 2 p,q ≃ H p G q . The pth entry of the qth Reeb complex G q is ranging over all increasing sequences in R p , whereas E 1 p,q ranges over all non-decreasing sequences. Hence we observe that G q is the non-degenerate complex E 1 −,q /DE 1 −,q . It is well-known that E 1 −,q is chain homotopic to E 1 −,q /DE 1 −,q , see e.g. [GJ09,p.150]. In particular, E 2 To summarize Proposition 3.8: It does not matter if we exchange the qth row E 1 −,q , including all non-decreasing sequences, with the Reeb complex G q (including only increasing sequences). Note the importance of this fact for making computations with section complexes feasible within finite time. This is perhaps best illustrated through some simple examples: Example 3.9. In Example 3.3 we identified the first page of the section spectral sequence of X = ∆ 2 ∏ ∂ ∆ 2 ∆ 2 with a single row -the Reeb complex G 0 . Hence, the differentials on the second page must be zero and we conclude that the homology of X and G 0 coincide. In Example 3.5, where the 2-simplices were subdivided prior to gluing, we are left with two non-zero rows on the first page: G 0 and G 1 . Moreover, the only non-trivial entries of both G 0 and G 1 are in p = 0, 1, implying that the differentials on the second page must be equal to zero. We calculate H p G q and thus the second page in coordinates: Since the sequence has converged, we extract H 0 X = k, H 2 X = k and H n X = 0 otherwise.
It was not a coincidence that the section spectral sequence from Example 3.5 converged on the second page, as we shall make precise.
Definition 3.10. For a height function h : X → R, we introduce the subdivision number as the biggest n for which there is an increasing sequenceā = (a 0 , . . . , a n ) such that the section space S h [ā] is non-empty.
Proposition 3.11. Let h : X → R be a height function with subdivision number s. The section spectral sequence collapses at the (s + 1)st page: E n+s p,q ≃ E s p,q for all n ≥ 0.
Proof. From the assumption, it follows that every Reeb complex G q (Example 3.2) has trivial entries above s. Hence, the first page consists of zeros for p ≥ s + 1 and the differentials on the (s + 1)st page must all terminate outside of the first quadrant; be equal to zero.
Thus, the number of pages we need to compute is bounded by how subdivided X is relative to h : X → R.
Example 3.12. Construct a simplicial cylinder X by gluing together the leftmost and rightmost vertical 1-simplices in:  The sequence must collapse on the next page, and, as the homology of X is not calculated yet, the differential cannot be zero. The representative in E 2 2,0 = k is given by the difference of the two gray 2-simplices. The alternating sum of the surrounding 1-simplices is in the image of ∂ v . Geometrically, this happens by applying ∂ v to the sum of all 2simplices (i.e. 1-simplices in the section spaces S h [a 0 , a 1 ]) not colored gray. Applying ∂ h to this sum gives the difference of the generators in E 2 0,1 ≃ k 2 . As an example, the top generator is obtained from the sum of the target of the purple and orange 2-simplices. We can thus conclude that ∂ 2 2,0 is the transpose of 1 −1 . The third page only has two non-zero entries: E 3 0,0 ≃ k and E 3 0,1 ≃ k. In particular, we calculate H 0 X = H 1 X = k and H n X = 0 otherwise.

Comparison to the continuous case
In Section 2.2, we saw that a height function h : X → R always associates to a piecewise linear function f : |X | → R. Example 2.8 illustrated that, in general, the topological space of sections Sect f from [Try21] and the simplicial space of 1-sections (S h ) 1 are significantly different. A topological section of the form [a, b] → T factorizes into smaller sections defined on [a, c] and [c, b] for any real number a ≤ c ≤ b. Conversely, two sections ρ : [a, c] → T and τ : [c, b] → T compose to a section on [a, b] via a canonical concatenation. This means that in contrast to the simplicial sections in (S h ) 1 , the topological sections are automatically subdivided. In particular, the spectral sequence obtained from Sect f terminates on the second page, reflecting the fact that all information about the homology of T is contained in Sect f . This is not true for (S h ) 1 in general, which led us to introduce higher sections. Consider now the case where the simplicial set X is subdivided according to h (Definition 3.4). Then we observed, in Section 3.3, that the spectral sequence associated to S h terminates on the second page as well. Whenever X is subdivided according to h, we can thus expect the space of 1-sections (S h ) 1 to contain the same homological information as the topological section space Sect f . The rest of this section is about making this observation into a formal statement which finally leads to a proof of Proposition 1.3.
Generally, for fixed real values a ≤ b we can define a map from the realization |S h [a, b]| to the space Sect f [a, b] as follows. A point in |S h [a, b]| is a class [ρ,t] with ρ : ∆ 1 × ∆ n → X an n-simplex in S h [a, b] andt a point in the standard topological n-simplex |∆ n |. If we realize ρ, then we obtain a continuous function |ρ| : |∆ 1 | × |∆ n | → |X | which hinges upon the existence of a homeomorphism |∆ 1 × ∆ n | ≃ |∆ 1 | × |∆ n |. For a fixedt, the restriction of |ρ| to |∆ 1 | ×t is a section of f up to the linear orientation-preserving homeomorphism L a,b : [a, b] → |∆ 1 |. Indeed, the composition h • ρ maps the unique nondegenerate 1-simplex in ∆ 1 to a ≤ b in R regardless of its second component. See Definition 2.9. It follows that |h| • |ρ|| |∆ 1 |×t identifies |∆ 1 | with the 1-cell labeled by a ≤ b in |X |. Whence we define a continuous function Φ h : |S h [a, b]| → Sect f [a, b] from the formula Φ h [ρ,t] = |ρ| • (L a,b ,t).
Example 4.1. Consider the height function h : ∆ 2 ∏ ∂ ∆ 2 ∆ 2 → R from Example 3.3. The section space S h [0, 1] only consists of a single point represented by the 1-simplex 0 → 1, whereas the topological version Sect f [0, 1] is a circle. Hence, the map Φ h cannot be a weak equivalence. While, if we subdivide ∆ 2 ∏ ∂ ∆ 2 ∆ 2 as in Example 3.5, then S h [0, 1] is a circle and Φ h is a weak equivalence.   Proof. Postcomposing a section ρ ∈ S h [0, 1] with the inclusion ∆ 1 × ∆ nX X ρ ρ yields a sectionρ ∈ Sh. Moreover, starting with anyρ ∈ Sh it always factorizes like that. Indeed, if we assume that for a given section ρ ∈ Sh[0, 1] such a factorization does not exist. Then the image of this ρ contains a simplex that is not in X . This simplex must then lie either in f −1 (0) or in f −1 (1) and thus be a horizontal face of ρ. Furthermore it contain one of the two vertices inX which are not in X . But as this vertex is clearly no horizontal face of any section we get a contradiction. Thus every section ρ ∈ Sh[0, 1] factors through X giving us the desired isomorphism.
We can now proof Proposition 4.2 by reducing to the case of contractible fibers.
a theory of sections for simplicial sets. While this theory is certainly inspired by the continuous version, it can also stand on its own feet. This is possible because simplicial sets provide a model for spaces that is equal in power to the continuous one. There is a rich simplicial homotopy theory and often a clean categorical treatment in terms of universal constructions is made possible by the fact that simplicial sets are presheaves. If need be, one can translate to topological spaces in terms of the Quillen equivalence given by the realization functor. Furthermore, we note that the theory of simplicial sets contains that of simplicial complexes as a special case. All this makes us believe that it can be beneficial to formulate theories of relevance to computational topology directly in simplicial sets. While still modelling the homotopy type of all spaces, such a theory is inherently combinatorial. This can make the subsequent development of implementable algorithms straightforward and streamlined with the theory. Furthermore, many of the intended applications may be of a discrete nature anyways. Using simplicial sets as the preferred modelling language might reveal important phenomena intrinsic to these discrete systems. An example that came up in the present paper are the higher section spaces. These were important through the presence of higher differentials in the general section spectral sequence. Contrast this with the continuous theory, where the spectral sequence converges on the second page. It is conceivable that higher section spaces prove to be more then a technicality and provide a valuable tool in the modelling and analysis of discrete height functions.