Homology and homotopy for arbitrary categories
Suddhasattwa Das[1]
[1]Department of Mathematics and Statistics, Texas Tech University, Texas, USA

One of the prime motivation for topology was Homotopy theory, which captures the general idea of a continuous transformation between two entities, which may be spaces or maps. In later decades, an algebraic formulation of topology was discovered with the development of Homology theory. Some of the deepest results in topology are about the connections between Homotopy and Homology. These results are proved using intricate constructions. This paper re-proves these connections via an axiomatic approach that provides a common ground for homotopy and homology in arbitrary categories. One of the main contributions is a re-interpretation of convexity as an extrinsic rather than intrinsic property. All the axioms and results are applicable for the familiar notions of homology and homotopy for topological spaces. We also provide a recipe for making the axioms come to fruition in arbitrary categories.

§ INTRODUCTION

[scale=1.0, transform shape]
[style=rect2] (1) at (0, -0.2) Simplicial homology $\Homology : \sSet \to \AbelCat$  ;
[style=rect2] (2) at (, -0.2) Co-simplicial $\calC$ object $F:\Delta \to \calC$;
(3) at (2, -0.2) $\begin{tikzcd} F(0) \arrow[rr, bend left=30, "F d_{1,0}"] \arrow[rr, bend right=30, "F d_{1,1}"'] && F(1)   \end{tikzcd}$;
[style=rect5] (4) at (, 1.0) Nerve construction $\Nerve_F : \Delta \to \sSet$;
[style=rect6] (5) at (0, 2) Homology for category $\calC$ :  $\Homology_{F} : \calC \to \AbelCat$;
[style=rect5] (6) at (2, 1.0) Concept of homotopy in $\calC$;
[style=rect6] (7) at (, 2) Homotopy equivalence;
[style=rect2] (8) at (2, 2) Axioms <ref>, <ref>, <ref> and <ref>;
[style=rect7] (9) at (, 3) Homotopy invariance of homology;
[style=rect2] (10) at (2, 3) Axiom <ref>;
[-to] (1) to (5);
[-to] (4) to (5);
[-to] (2) to (4);
[-to] (2) to (3);
[-to] (3) to (6);
[-to] (6) to (7);
[-to] (8) to (7);
[-to] (7) to (9);
[-to] (5) to (9);
[-to] (10) to (9);

Outline of the theory. The paper presents a simple axiomatic approach to both homotopy and homology which guarantee that the homology preserves homotopy invariance. The chart above presents various assumptions and concepts, along with their logical dependence. The independent notions are in white boxes. This includes Simplicial homology, a means of converting the combinatorial structure of a simplicial set into an Abelian group. Homology is the confluence of simplicial homology and the nerve construction, as shown in detail in (<ref>). The nerve construction (<ref>) is an outcome of an arbitrary functor $F$ as in (<ref>). The basic concept of homotopy of the action of this functor on just the first two objects of the simplex category $\Delta$. This attains all the usual properties of homotopy equivalence under further axioms. See Figure <ref> for a detailed outline of this property. Also see Figure <ref> for a detailed outline of the derivation of homotopy invariance from more fundamental properties arising due to our axioms.

The two pillars of modern topology are Homotopy theory and Homology theory. The former studies the continuous deformation of maps and spaces, and the various equivalences they create. On the other hand, homology provides a means of embedding the arrangement of topological spaces and continuous maps into a system of Abelian groups and group homomorphisms between them. The axiomatic approaches for homotopy <cit.> and Homology <cit.> have mostly taken independent paths. The interconnections between these theories have been far and few <cit.>. Our goal is to establish these connections in an axiomatic manner, within the context of a general category.

Categories and category theory provide a panoramic views of each of the various branches of mathematics, such as Dynamical systems theory <cit.>, Measure theory <cit.>, Algorithms <cit.> or Game theory <cit.>. In fact, this discipline arose from a systematic study of topological objects and continuous map, and its connections with Algebra. The concepts of Homology and Homotopy were respectively established as functor from the category of topological spaces into the category of Abelian groups, and from the category of pointed topological spaces into the category of general Groups. The concept of a functor provides a concise description of a mapping that also preserves the numerous relations between two arrangements. Readers can find the basic definitions of category theory in standard sources such as <cit.>.

Simplex category The goal of the paper is to present how a simple functor

\begin{equation} \label{eqn:functorF}
F : \Delta \to \calC
\end{equation}

mapping the simplex category $\Delta$ into an arbitrary category $\calC$, leads to notions of homotopy and homology for the category $\calC$. See Figure <ref> for a concise outline of the paper. The simplex category $\Delta$ <cit.> has as objects the non-negative integers $\num_0 :- \braces{0, 1,2,\ldots }$. Each such integer $n$ is meant to represent the ordered sets $[n] := \{0, \ldots, n\}$. The morphisms are given by

\[ \Hom_{\Delta} (m;n) := \braces{ \mbox{ Order preserving maps } \phi:[m] \to [n] } . \]

The simplex category $\Delta$ is the most concise way of encoding combinatorial structures in other categories <cit.>. TThe functor $F$ thus creates an image of $\Delta$ within $F$. The objects $\SetDef{F(n)}{n\in\num_0}$ will be called cells. They are the basic building blocks of homology and homotopy.

Topological simplex One of the main motivating examples is the case when $\calC = \Topo$, the category of topological spaces and continuous maps. In that case $F$ is often taken to be standard topological simplex

\begin{equation} \label{eqn:def:StndrdTopoSmplx}
F_{stndrd, Topo} : \Delta \to \Topo, \quad F_{stndrd, Topo} (n) = \mbox{ convex span of } \braces{ e^{(n+1)}_0, \ldots, e^{(n+1)}_n } ,
\end{equation}

where $e^{(n+1)}_0, \ldots, e^{(n+1)}_n$ are $n+1$ independent eigenvectors in $\real^{(n+1)}$. The resulting homology is the singular homology for topological spaces. Note that the $n$-th cell of this functor is isomorphic to the $n$-dimensional closed disk. Our approach is not tied to to the specific example of (<ref>). We shall identify and isolate structural properties of the general functor (<ref>) and recreate the Homology and Homotopy.

Our first two axioms will be on the basic elements of $F$ :

The category $\calC$ has a terminal object $1_{\calC}$, and has finite products.

The functor $F$ maps $[0]$ into $1_\calC$ from Axiom <ref>.

Axiom <ref> simply states that the 0-th cell is the terminal object of $\calC$. Note that for the example of (<ref>), $1_{\calC}$ is the one-point topological space, which is also the $0$-th cell.

Generating morphisms In general, it is not easy to make an explicit construction of a functor of the form (<ref>), as the morphisms incoming at $n$ or outgoing at $n$ increase exponentially with $n$. The task becomes easier if one concentrates only on a special collection of morphisms known as face maps :

\begin{equation} \label{eqn:def:face}
0\leq i\leq n \;:\; \Shobuj{d_{n,i}} : [n-1] \to [n], \, \quad j \mapsto \begin{cases}
j & \mbox{ if } j<i	\\
j+1 & \mbox{ if } j\geq i
\end{cases}.
\end{equation}

and degeneracy maps :

\begin{equation} \label{eqn:def:degeneracy}
0\leq i\leq n+1 \;:\;  \Shobuj{s_{n,i}} : [n+1] \to [n], \quad j \mapsto \begin{cases}
j & \mbox{ if } j< i	\\
j-1 & \mbox{ if } j\geq i
\end{cases}.
\end{equation}

The face and degeneracy maps generate every morphism $\phi:[m]\to [n]$ in $\Delta$. The obey the following set of identities :

\begin{equation} \label{eqn:smplc_id}
\begin{split}
d_{n+1,j} d_{n,i} = d_{n+1,i} d_{n,j-1} & \mbox{ if } i<j \\
s_{n-1,j} d_{n,i} = d_{n-1,i} s_{n-2,j-1} & \mbox{ if } i<j \\
s_{n-1,j} d_{n,j} = 1 = s_{n-1,j} d_{n,j+1} \\
s_{n-1,j} d_{n,j} = d_{n-1,i-1} s_{n-2,j} & \mbox{ if } i>j+1 \\
s_{n-1,j} s_{n,i} = s_{n-1,i} s_{n,j+1} & \mbox{ if } i\leq j
\end{split}
\end{equation}

which are called the simplicial identities. Equations (<ref>), (<ref>) and (<ref>) provide a sufficient set of morphisms and composition relations that generate all the morphisms of $\Delta$. This means that given a collection of morphisms (<ref>) and (<ref>) which satisfy (<ref>), any morphism in $\Delta$ may be expressed as a composition of different $d_{n,i}$ and $s_{n',i'}$-s. Thus when trying to construct a functor $F$ as in (<ref>), one only needs to specify the objects $\SetDef{ F(n) }{ n\in \num_0 }$, the action of $F$ on the face and degeneracy morphisms, and ensure that $F$ preserves the simplicial identities of (<ref>).

The next axiom we need will be about the $1$-th cell :

There is an isomorphism $\text{swap} : F(1) \to F(1)$ such that the following commutation holds with boundary maps :

\begin{equation} \label{eqn:def:hmtpy:1}
\begin{tikzcd}
& & F(1) \arrow[d, "\text{swap}", bend left=49] \\
1_{\calC} \arrow[rr, "F\paran{ d_{1,1} }"'] \arrow[rru, "F\paran{ d_{1,0} }", bend left] & & F(1) \arrow[u, "\text{swap}^{-1}", bend left=49]
\end{tikzcd}
\end{equation}

We shall see later how the $0$-th cell $F(0)$ and $1$-th cell $F(1)$ alone create a notion of homotopy. Based on nomenclature used in the axiomatic theory of sets <cit.>, morphisms orginiating from the terminal element $F(0)$ are called elements of the target object. Thus according to (<ref>), the face maps $d_{1,0}$ and $d_{1,1}$ are two elements of $F(1)$. The 1-cell $F(1)$ acts as a bridge between these two distinguished elements. The terminal elements this play the role of entry and exit points. Axiom <ref> establishes a symmetry of these elements. Note that for the example of (<ref>), the morphism $\text{swap}$ is simply the map that reverses the orietnation of the unit interval, which is also the $1$-cell. Such a reversal also interchanges the endpoints, which are the images of the $0$-cell under the face maps.

To prove transitivity of homotopy, we shall need the following assumption

The diagram

\begin{equation} \label{eqn:dip3d}
\begin{tikzcd} [column sep = large]
F(1) & F(0) \arrow[l, "F\paran{d_{1,0}}"'] \arrow[r, "F\paran{d_{1,1}}"] & F(1)
\end{tikzcd}
\end{equation}

has a colimit, which is $F(1)$ itself.

Nerve construction The general functor $F$ induces a special functor called the nerve the category $\calC$ <cit.> as shown below :

\begin{equation} \label{eqn:def:Nerve}
\begin{split}
& \Nerve = \Nerve_F : \calC \to \sSet, \\
& \forall c\in ob(\calC) \;:\; \Nerve(c) := \Hom \paran{ F\cdot; c } : \Delta^{op} \to \SetCat .
\end{split}
\end{equation}

The functor $\Nerve_F$ assigns a simplicial set to each object of $\calC$. Thus it assigns a purely combinatorial identity to each object in $\calC$. This functor will be the key ingredient to creating a notion of Homology, as we present later in (<ref>). For the example of (<ref>), $\Nerve_F(X)$ is the collection of all possible mappings of the $n$-dimensional simplex into the topological object $X$. Thus $\Nerve_F(X)$ becomes a ledger for the topological “content" of $X$, and the entries of the ledger are all possible embeddings of $n$-dimensional disks.

In the example of (<ref>), it is easy to visualize the simplices being built inductively. $F(n+1)$ can be built by erecting a convex tent above $F(n)$. With this visual  interpretation, each simplex can be interpreted to be the base / bottom face of the next higher simplex. The $i$-th face of the $n$-simplex becomes the $i+1$-th face of the $n+1$-simplex. We make this nesting relation more precise for the general functor (<ref>).

Base maps Let $\tilde{\Delta}$ be the subcategory of $\Delta$ generated only by the face maps $\SetDef{ d_{n,i} }{ 0 \leq i \leq n }$. Let $\tilde{\Delta} \xrightarrow{i} \Delta$ denote the obvious inclusion of categories. The simplified simplex $\tilde{\Delta}$ has an in-built $\text{Shift}$ functor :

\begin{equation} \label{eqn:def:Shift}
\begin{tikzcd} \tilde{\Delta} \arrow[rr, "\text{Shift}"] && \tilde{\Delta}  \end{tikzcd}, \quad
\begin{tikzcd} \Holud{n} \arrow[d, "d_{n,i}"'] \\ \akashi{n+1} \end{tikzcd}
\begin{tikzcd} {} \arrow[r, mapsto] & {} \end{tikzcd}
\begin{tikzcd} \Holud{n+1} \arrow[d, "d_{n+1,i+1}"] \\ \akashi{n+2}	\end{tikzcd}
\end{equation}

A key observation is the existence of a natural transformation $\Base$ as shown below :

\begin{equation} \label{eqn:def:Base}
\begin{tikzcd}
& \Delta^{op} \arrow{rrr}{\Nerve_{F}(X)} &&& \SetCat \\
\tilde{\Delta}^{op} \arrow[dashed]{urrrr}[name=n1]{} \arrow[d, "\text{Shift}"'] \arrow[dashed]{drrrr}[name=n2]{} \arrow[d, "\text{Shift}"'] \arrow[ur, "\iota"] \\
\tilde{\Delta}^{op} \arrow[r, "\iota"] & \Delta^{op} \arrow{rrr}[swap]{\Nerve_{F}(X)} &&& \SetCat
\arrow[shorten <=1pt, shorten >=1pt, Rightarrow, to path={(n2) to[out=90,in=-90] node[xshift=20pt]{$\Base$} (n1)} ]{  }
\end{tikzcd} , \quad
\begin{tikzcd}
\Hom_{\calC} \paran{ F(n+1); X } \arrow{d}{ \Base_n } [swap]{ \circ F \paran{ d_{n+1,0} } } \\ \Hom_{\calC} \paran{ F(n); X }
\end{tikzcd} , \quad
\forall n\in \num_0 ,
\end{equation}

whose connecting morphisms are provided by right-composition with the zeroeth face maps. The natural transformation of (<ref>) always exsits uniquely, and will be verified in Section <ref>. We next formulate a purely categorical notion of convexity.

Convexity Any object $X$ will be called convex if it has the a natural transformation $\Cone$ which plays the role of a right inverse of $\Base$ :

\begin{equation} \label{eqn:def:ConeNat}
\begin{tikzcd}
&& & \Delta^{op} \arrow{rrr}{\Nerve_{F}(X)} &&& \SetCat \\
\Delta^{op} \arrow{d}[swap]{ \Nerve_{F}(X) } && \tilde{\Delta}^{op} \arrow[dashed, bend right=10]{dll}[name = n3]{} \arrow{ll}[swap]{\iota} \arrow[dashed]{urrrr}[name=n1]{} \arrow[d, "\text{Shift}"'] \arrow[dashed]{drrrr}[name=n2]{} \arrow[d, "\text{Shift}"'] \arrow[ur, "\iota"] \\
\SetCat && \tilde{\Delta}^{op} \arrow[r, "\iota"] & \Delta^{op} \arrow{rrr}[swap]{\Nerve_{F}(X)} &&& \SetCat
\arrow[shorten <=1pt, shorten >=1pt, Rightarrow, to path={(n2) to[out=90,in=-45] node[xshift=20pt]{$\Base$} (n1)} ]{  }
\arrow[shorten <=1pt, shorten >=1pt, Rightarrow, to path={(n3) to[out=-90,in=-90] node[yshift=-15pt]{$\Cone^X$} (n2)} ]{  }
\arrow[shorten <=1pt, shorten >=1pt, Rightarrow, to path={(n1) to[out=135,in=90] node[yshift=15pt]{$\Id$} (n3)} ]{  }
\end{tikzcd}
\end{equation}

The thick arrows in (<ref>) represent natural transformations. This arrangement is one of the major contributions of this paper. Convexity is usually interpreted as an intrinsic property of a space, and resulting form a closure property with respect to convex linear sums. Such a linear structure may not be available for a general object or category. Equation (<ref>) redefines convexity as a relational property. These relations concisely contained in the property of the transformation $\Cone$ being natural. For the topological example of (<ref>), $\Cone$ becomes the Cone-constructions, which are maps $\Cone^X_n : \Hom \paran{ F(n) ; X } \to \Hom \paran{ F(n+1) ; X }$. Recall that the standard topological $n$-simplex can be described by coordinates

\[ F_{stndrd, Topo} (n) := \SetDef{ \paran{ t_0, \ldots, t_n } }{ t_j \geq 0, \, \sum_{j=0}^{n} t_j = 1 } , \quad \forall n\in \num_0.\]

Let $a_n$ be an arbitrary point in the $n$-th simplex. Using these coordinates, the action of the (topological) cone transformations may be described as

\begin{equation} \label{eqn:Cone_Topo}
\paran{ \Cone^X_n \sigma } \paran{ t_0, \ldots, t_n, t_{n+1} } :=
\begin{cases}
t_{n+1} a_n + \paran{1-t_{n+1}} \sigma \paran{ \frac{t_0}{1-t_{n+1}} , \ldots, \frac{t_n}{1-t_{n+1}} } & \mbox{ if } t_{n+1} < 1 \\
a_n & \mbox{ if } t_{n+1}  =1
\end{cases},
\quad \forall n\in \num_.
\end{equation}

In $\Topo$, the cells $F(n)$ are obviously convex. We formally state this as our final axiom :

All the cells $\SetDef{F(n)}{n\in\num_0}$ are convex.

This completes the formulation of our five basic axioms.

Main result The axioms we have stated are based the general functor (<ref>). Axiom <ref> is entirely about about the category $\calC$, while the next three axioms <ref>, <ref> and <ref> are entirely about the first two cells of the functor $F$. Axiom <ref> is the only assumption made on the functor $F$ as a whole. Our main result is :

Consider the functor $F$ as in (<ref>) mapping the simplex category $\Delta$ into an arbitrary category $\calC$.

* $F$ creates a homology functor for the category $\calC$, as defined in (<ref>).

* Under Axioms <ref>, <ref>, <ref> and <ref>, there is a notion of homotopy between morphisms of $\calC$, as defined in (<ref>), along with homotopy equivalence of objects.

* Now suppose that $F$ satisfies Axiom <ref>.
Then  homology is homotopy invariant.

Theorem <ref> is proved over the course of the next three sections.

Remark Theorem <ref> provides a separation of various concepts, as outlined outlined in Figure <ref>. The separate claims list the precise Axioms on which the notions of Homotopy and Homology depend. The construction of Homology requires no assumption other than the functor $F$ (<ref>). Homotopy requires Axioms <ref>, <ref>, <ref> and <ref>. Figure <ref> and Section <ref> further delineates the precise role played by each axiom in constructing the various components of homotopy. The fifth axiom <ref> of convexity is required to establish the inter-dependence of homology and homotopy. Also see Figure <ref> for a summary of this main result.

Remark There are constructs of homology that are purely combinatorial in nature, such as the homology of simplicial complexes. Those lie outside the scope of Theorem <ref>.

Remark The major innovation in Theorem <ref> is the Axiom <ref> on convexity. Convexity is usually interpreted via linear convex sums. Axiom <ref> extracts the structural essence of convex, using the language of natural transformations. See Theorem <ref> where convexity, a property borne by $\calC$, implies acyclicity (zero-homology), a property borne by the homology functor.

Remark While it is easy to conceive of functors (<ref>), it is a non-trivial task to satisfy its axioms. Axiom <ref> is the most challenging task to be verified for a general functor. Theorem <ref> in Section <ref> provides a recipe for constructing the functor $F$ in (<ref>) such that Axiom <ref> is satisfied.

Outline We first verify claim (i) of Theorem <ref> by reviewing how the nerve construction extends to a homology functor, in Section <ref>. Next, in Section <ref> we formulate a generalized notion of homotopy that originates from the functor $F$ in (<ref>), as claimed in Theorem <ref> (ii). There we also derive properties such as reflexivity, symmetry, transitivity and homotpy equivalence. Claim (iii) is proved in Section <ref>. There we identify several intermediary properties which guarantee the invariance of homology with homotopy. In Section <ref>	a recipe is provided by which a functor $F$ as in (<ref>) can be constructed, which satisfy all the axioms including Axiom <ref>.
The axiomatic approach that we present in this paper open several new directions of investigation. These are discussed briefly in Section <ref>. Finally, we provide some lemmas and proofs in Section <ref>.

§ HOMOLOGY

The goal of this section is mainly to understand the creation of the functor $\Homology_{\calG, F}$ shown in green below.

\begin{equation} \label{eqn:CatHmlgy}
\begin{tikzcd}[scale cd = 1.5]
\calC \arrow[dashed, Shobuj]{ddrr}[swap]{ \Homology_{\calG, F} } \arrow[Akashi]{rrrr}{ \Nerve_{F} } && && \sSet \arrow[dashed, Holud]{ddrr}[swap]{ \Chain_{\calG} } \arrow[dashed, Itranga]{ddll}[swap]{ \text{SimpHom}_{\calG} } \arrow{rr}{\otimes \calG} && \left[ \text{sGroup} \right] \arrow{dd}{ \text{alt} } \\
\\
&& \mathcal{A}^{\num} && && \ChainCmplx(\mathcal{A}) \arrow{llll}{ \Homology_{\calA} }
\end{tikzcd}
\end{equation}

All the dashed arrows in this diagram are defined via composition. Homology can thus be simply stated as the composition of two ingredients – the simplicial homology functor (red) with the nerve construction (blue). The category $\sSet$ is the category of simplicial sets. In general, given any category $\calX$, a simplicial $\calX$-object is a functor $x:\Delta^{op} \to \calX$. Note that simplicial $\calX$-objects are objects of the functor category $\Functor{\Delta^{op}}{\calX}$. See Table <ref> for various kinds of simplicial objects for various choices of $\calX$. $\sSet$ corresponds to the special case when $\calX=\SetCat$, the category of small sets. The first ingredient is the Nerve functor $\Nerve_F$ (blue) which we have defined previously in (<ref>). It encodes category $\calC$ into the category $\sSet$. Thereby every object in $\calC$ is assigned a combinatorial identity. The functor $\text{SimpHom}_{\calG}$ (red) converts this combinatorial information into algebraic form. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components. The construction of simplicial homology is completely independent of the category $\calC$ or the nerve construction. The details will be omitted and the readers are referred to classical sources such as <cit.>.

Examples of simplicial objects. Any functor $F:\Delta^{op} \to \calX$ is called a simplicial $\calX$-object. When the category $\calX$ are of some specific types, there are special labels attached to such objects.

$\calX$     Name
$\Topo$     Simplicial space
$\SetCat$     Simplicial set
$\GroupCat$     Simplicial group

This completes a very general discussion of homology, all stemming from the existence of the functor (<ref>). In summary, a homology functor embeds the category $\calC$ into the algebraic category of Abelian groups. Due to the simplistic nature of $\AbelCat$, it elucidates many properties within the category $\calC$. Suppose $Y$ is an acyclic object, and $X$ is a non-acyclic object. Then the arrangement on the left below is not possible :

\[\begin{tikzcd}
& Y \arrow[dr, "g"] \\
X \arrow[ur, "f"] \arrow[rr, dashed, "\Id_X"'] & & Z
\end{tikzcd} ,\;
\begin{tikzcd} {} \arrow[rr, mapsto] && {} \end{tikzcd} \;
\begin{tikzcd}
& \begin{array}{c} \Homology(Y) \\ =0 \end{array} \arrow[dr, "g"] \\
\Homology(X) \arrow[ur, "f"] \arrow[rr, dashed, "\Id"'] & & \Homology(X)
\end{tikzcd}\]

The diagram on the right is the image of the diagram on the left, under the homology functor. Because the intermediate Homology is zero, the composite group homomorphism map must also be zero. This violates the functorial property of Homology, which must map identity morphisms into identity morphisms. Thus acyclic objects cannot be retracts of non-acylic objects, a well known generalization of Brower's fixed point theorem <cit.>.

We next consider the notion of homotopy arising from this same functor.

§ HOMOTOPY

[scale=1.0, transform shape]
[style=rect2] (1) at (0, 0) Functor $F$ from (<ref>);
[style=rect5] (2) at (0, -) Homotopy (<ref>) - terminal points and bridge;
[style=rect5] (3) at (, -) Reflexivity of homotopy;
[style=rect5] (4) at (2, -) Symmetry of homotopy;
[style=rect2] (5) at (, 0) Axioms <ref>, <ref> – initial element as the zero cell;
[style=rect2] (6) at (2, 0) Axiom <ref> – terminal swapping invertible endomorphism;
[style=rect5] (7) at (2, -2) Transitivity of homotopy;
[style=rect5] (8) at (0, -3) Preservation of homotopy under composition with morphisms homotopy;
[style=rect2] (9) at (2, -3) Axiom <ref> – colimit property;
[style=rect6] (10) at (, -2) Compositionality of homotopies;
[style=rect6] (11) at (, -3) Homotopy  equivalence;
[-to] (1) to (2);
[-to] (2) to (3);
[-to] (3) to (4);
[-to] (5) to (3);
[-to] (6) to (4);
[-to] (4) to (7);
[-to] (1) to (8);
[-to] (9) to (7);
[-to] (2) to (10);
[-to] (8) to (10);
[-to] (7) to (10);
[-to] (4) to (10);
[-to] (3) to (10);
[-to] (10) to (11);

Construction of homotopy.

The immediate consequence of Axioms <ref> and <ref> is the following diagram :

\[\begin{tikzcd}
F(0) \arrow[rr, "F\paran{ d_{1,0} }"] && F(1) && F(0) \arrow[ll, "F\paran{ d_{1,1} }"'] \\
1_{\calC} \arrow[urr, dashed] \arrow[u, "="] && && 1_{\calC} \arrow[u, "="'] \arrow[ull, dashed]
\end{tikzcd}\]

This diagram also serves as the basis for the notion of homotopy. Given two objects $X,Y\in ob(\calC)$, we say that two morphisms $f,g:X\to Y$ are homotopic if there is a morphism $H : X\times F(1) \to Y$ such that the following commutation holds :

\begin{equation} \label{eqn:def:hmtpy:2}
\begin{tikzcd}
X\times 1_{\calC} \arrow[Holud]{rrr}{ \Id_X \times F\paran{ d_{1,0} } } &&& X\times F(1) \arrow[Shobuj]{d}{H} &&& X\times 1_{\calC} \arrow[Akashi]{lll}[swap]{ \Id_X \times F\paran{ d_{1,1} } } \\
X \arrow[Holud]{rrr}[swap]{f} \arrow[Holud]{u}{\cong} &&& Y &&& X \arrow[Akashi]{u}[swap]{\cong} \arrow[Akashi]{lll}{g}
\end{tikzcd}
\end{equation}

The relation of being homotopic is reflexive. Consider the diagram on the left below, in which the commutation holds because of the terminal property of $1_{\calC}$ and Axiom <ref>. Taking a product of this diagram with $X$ gives the middle commuting diagram :

\[\begin{tikzcd} [scale cd = 0.7]
F(0) \arrow[dr, "\cong"', dashed] \arrow[r, "F\paran{d_{1,j}}"] & F(1) \arrow[d, "!"] \\
& F(0)
\end{tikzcd}
\; \begin{tikzcd} {} \arrow[mapsto, rr, "X\times"] && {} \end{tikzcd} \;
\begin{tikzcd} [scale cd = 0.7]
X\times F(0) \arrow[d, "\cong"'] \arrow[dr, "\cong"', dashed] \arrow[r, "F\paran{d_{1,j}}"] & X\times F(1) \arrow[d, "!"] \\
X & X\times F(0) \arrow[l, "\cong"]
\end{tikzcd}
\; \begin{tikzcd} {} \arrow[mapsto, rr, "f\circ"] && {} \end{tikzcd} \;
\begin{tikzcd} [scale cd = 0.7]
X\times F(0) \arrow[d, "\cong"'] \arrow[dr, "\cong"', dashed] \arrow[r, "F\paran{d_{1,j}}"] & X\times F(1) \arrow[d, "!"] \arrow[bend left=60, dd, dashed, "\bar{f}"] \\
X \arrow[dr, "f"'] & X\times F(0) \arrow[l, "\cong"] \\
& Y
\end{tikzcd}\]

In the rightmost diagram, we have appended the morphism $f:X\to Y$ to the middle diagram. Note that this commutation holds for $j=0,1$. This information can be redrawn as

\[\begin{tikzcd} [column sep = large]
X\times 1_{\calC} \arrow{r}{ \Id_X \times F\paran{ d_{1,0} } } & X\times F(1) \arrow{d}{\bar{f}} & X\times 1_{\calC} \arrow{l}[swap]{ \Id_X \times F\paran{ d_{1,1} } } \\
X \arrow{r}[swap]{f} \arrow{u}{\cong} & Y & X \arrow{u}[swap]{\cong} \arrow{l}{f}
\end{tikzcd}\]

This diagram is a special case of (<ref>) with $f=g$. This proves that $f=g$.

Symmetry At present, homotopy is just a relation within each Hom-set $\Hom(X;Y)$. By virtue of Axiom <ref>, the relation of homotopy turns out to be symmetric. This is because of the following commutation

\[\begin{tikzcd} [column sep = large]
& X\times I \arrow[d, "\Id_X \times \text{rev}" ] \arrow[Shobuj, bend right=30, dashed]{dd} \\
X\times \star \arrow{r}{ \Id_X \times F\paran{ d_{1,0} } } \arrow[bend left=20]{ur}{ \Id_X \times F\paran{ d_{1,1} } } & X\times I \arrow{d}{H} & X\times \star \arrow{l}[swap]{ \Id_X \times F\paran{ d_{1,1} } } \arrow[bend right=20]{ul}[swap]{ \Id_X \times F\paran{ d_{1,0} } } \\
X \arrow{r}[swap]{f} \arrow{u}{\cong} & Y & X \arrow{u}[swap]{\cong} \arrow{l}{g}
\end{tikzcd}\]

Thus $f$ being homotopic to $g$ is equivalent to saying that $g$  is homotopic to $f$. Homotopy is this a reflexive and symmetric property.

Transitivity The limiting co-cone assumed in Axiom <ref> is drawn below

\begin{equation} \label{eqn:A:3}
\begin{tikzcd}
&& F(1) \\
F(1) \arrow[urr, bend left=30, Shobuj, "\text{left}"] && F(0) \arrow[rr, "F\paran{d_{0,0}}"'] \arrow[ll, "F\paran{d_{0,1}}"] \arrow[u, "\text{center}", Shobuj] && F(1) \arrow[ull, bend right=30, Shobuj, "\text{right}"']
\end{tikzcd}
\end{equation}

The connecting morphisms of the co-cone has been shown in green, and assigned some distinguishing names. This diagram will be the key to proving transitivity. Let there be three morphisms $f, g, h :X\to Y$ such that $f$ is homotopic to $g$ via a homotopy $H$, and $g$ is homotopic to $h$ via a homotopy $H$. To establish transitivity, we first take a product of the diagram in (<ref>) with $X$ to get the diagram :

\[\begin{tikzcd} [column sep = large]
&& X\times F(1) \\
X\times F(1) \arrow[urr, bend left=30, "\Id_X\times \text{left}"] && X \arrow[rr, "\Id_X\times F\paran{d_{0,0}}"'] \arrow[ll, "\Id_X\times F\paran{d_{0,1}}"] \arrow[u, "\text{center}"] && X\times F(1) \arrow[ull, bend right=30, "X\times \text{right}"']
\end{tikzcd}\]

We now attend the morphisms and homotopies we assume in a second row below this diagram to get :

\[\begin{tikzcd} [column sep = large]
&& X\times F(1) \\
X\times F(1) \arrow[ddrr, "H"', Holud] \arrow[urr, bend left=30, "\Id_X\times \text{left}", Akashi] && X \arrow[dd, Holud, "g"] \arrow[rr, "\Id_X\times F\paran{d_{0,0}}"'] \arrow[ll, "\Id_X\times F\paran{d_{0,1}}"] \arrow[u, "\text{center}", Akashi] && X\times F(1) \arrow[ull, bend right=30, "X\times \text{right}"', Akashi] \arrow[ddll, "G", Holud] \\
\\
X \arrow[uu, "\Id_X\times F\paran{d_{0,0}}"] \arrow[rr, "f"'] && Y && X \arrow[uu, "\Id_X\times F\paran{d_{0,1}}"'] \arrow[ll, "h"]
\end{tikzcd}\]

Now observe that the morphisms colored in yellow are a co-cone above the diagram (<ref>). Thus this cone must factorize through the limiting cone (blue), via a morphism shown in green :

\[\begin{tikzcd} [column sep = large]
&& X\times F(1) \arrow[ddd, Shobuj, bend left=40, "G\circ H"] \\
X\times F(1) \arrow[ddrr, "H"', Holud] \arrow[urr, bend left=30, "\Id_X\times \text{left}", Akashi] && X \arrow[dd, Holud, "g"] \arrow[rr, "\Id_X\times F\paran{d_{0,0}}"] \arrow[ll, "\Id_X\times F\paran{d_{0,1}}"'] \arrow[u, "\text{center}", Akashi] && X\times F(1) \arrow[ull, bend right=30, "X\times \text{right}"', Akashi] \arrow[ddll, "G", Holud] \\
\\
X \arrow[uu, "\Id_X\times F\paran{d_{0,0}}"] \arrow[rr, "f"'] && Y && X \arrow[uu, "\Id_X\times F\paran{d_{0,1}}"'] \arrow[ll, "h"]
\end{tikzcd}\]

This green morphism is called the concatenation / vertical composition of the homotopies $H,G$. Note that the it serves as a homotopy from $f$ to $h$. Thus homotopy is a symmetric relation.

Composition We next prove that the property of being homotopic is preserved udner pre or post composition with a fixed morphism. Throughout we assume the homotopy $H$ from (<ref>). Now suppose there is a morphism $h:Y\to Z$. Appending the morphism $h$ to (<ref>) gives :

\[\begin{tikzcd}
X\times 1_{\calC} \arrow{rrr}{ \Id_X \times F\paran{ d_{1,0} } } &&& X\times F(1) \arrow{d}{H} \arrow[dd, dashed, bend right=60] &&& X\times 1_{\calC} \arrow{lll}[swap]{ \Id_X \times F\paran{ d_{1,1} } } \\
X \arrow[drrr, dashed, "hf"'] \arrow{rrr}{f} \arrow{u}{\cong} &&& Y \arrow[d, "h"] &&& X \arrow{u}[swap]{\cong} \arrow{lll}[swap]{g} \arrow[dlll, dashed, "hg"] \\
&&& z
\end{tikzcd}\]

This creates a homotopy from $hf$ to $hg$. Next we assume a morphism $h:W\to X$. Then the diagram in (<ref>) can be expanded to :

\[\begin{tikzcd}
W\times 1_{\calC} \arrow[dr, dotted, "h\times 1_{\calC}"] \arrow[Holud]{rrrr}{ \Id_W \times F\paran{ d_{1,0} } } & &&& W\times F(1) \arrow[Holud]{d}{ h\times \Id_{F(1)} } &&& & W\times 1_{\calC} \arrow[Holud]{llll}[swap]{ \Id_W \times F\paran{ d_{1,1} } } \arrow[dl, dotted, "h\times 1_{\calC}"'] \\
& X\times 1_{\calC} \arrow{rrr}{ \Id_X \times F\paran{ d_{1,0} } } &&& X\times F(1) \arrow{d}{H} &&& X\times 1_{\calC} \arrow{lll}[swap]{ \Id_X \times F\paran{ d_{1,1} } } &  \\
W \arrow[uu, Holud, "="] \arrow[Holud, r, "h"'] & X \arrow{rrr}[swap]{f} \arrow{u}{\cong} &&& Y &&& X \arrow{u}[swap]{\cong} \arrow{lll}{g} & W \arrow[Holud, l, "h"] \arrow[uu, Holud, "="']
\end{tikzcd}\]

This creates a homotopy from $fh$ to $gh$. This proves that homotopy is preserved under pre and post composition. This bring us to the last and final property :

Composition of homotopies Suppose there are three objects $X,Y,Z$, and homotopic pairs of morphisms $f,g : X\to Y$ and $f',g':Y\to Z$. Let these homotopies be $H,G$ respectively. Then note that

\[\begin{split}
f'\circ f &\sim f'\circ g , \quad \mbox{ post-composition of } H \mbox{ with } f' ,\\
&\sim g'\circ g , \quad \mbox{ pre-composition of } G \mbox{ with } g .
\end{split}\]

The $\sim$ denotes a homotopy relation in the Hom-set $\Hom(X;Z)$, whose transitivity has already been established. Thus we can conclude that $f'\circ f$ is homotopic to $g'\circ g$. This relation can be summarized by saying that homotopies can be composed.

Homotopy equivalence A morphism in $\Hom(X;X)$ is said to be null-homotopic if it is homotopic to $\Id_X$. Two objects $A,B$ will be called homotopy equivalent <cit.> if there are morphisms $f:A\to B$ and $g:B\to A$ such that $f\circ g \sim \Id_B$ and $g\circ f \sim \Id_A$. In other words, both $f\circ g$ and $g\circ f$ are null-homotopic.

Under Axioms <ref>, <ref>, <ref> and <ref> the relation of being homotopic is symmetric, reflexive and transitive. Thus this relation can be extended to an equivalence relation on $\Hom_{\calC}(X;Y)$. One of the most important consequences of homotopy is that one can derive a new category $\tilde{\calC}$ from $\calC$, which has the same object-sets, but in which the hom set $\Hom_{\tilde\calC}(X;Y)$ are the homotopy equivalence classes of $\Hom_{\calC}(X;Y)$. This is a well defined operation which retains the compositionality of $\calC$. Note that homotopy equivalence in $\calC$ is the same as isomorphism in $\tilde\calC$.

This completes the construction of the notion of homotopy. See Figure <ref> for an overview of this process. We have now seen the constructions of homotopy and homology along two different routes. We next see how they are inter-linked, due to ground assumptions we make on $F(0)$ and $F(1)$.

§ HOMOTOPY INVARIANCE OF HOMOLOGY

[scale=1.0, transform shape]
[style=rect2] (ES) at (0, 0) Eilenberg Steenrod axioms;
[style=rect6] (P1) at (, 0) Dimension axiom <ref>;
[style=rect5] (A1toP1) at (2, 0) Lemma <ref>;
[style=rect2] (A1) at (2, ) Axiom <ref>;
[style=rect2] (AC) at (, ) Axiom <ref>;
[style=rect6] (P3) at (0, -) Homotopy axiom <ref>;
[style=rect5] (P4toP3) at (, -) Lemma <ref>;
[style=rect6] (P4) at (2, -) Face-maps homotopy property <ref>;
[style=rect5] (P5toP4) at (2, -2) Lemma <ref> ;
[style=rect6] (P5) at (, -2) Cross morphisms property <ref>;
[style=rect5] (P2toP5) at (0, -2) Lemma <ref>;
[style=rect6] (P2) at (0, -3) Zero homology of cells <ref>;
[style=rect5] (A4toP2) at (, -3) Theorem <ref> ;
[style=rect2] (A4) at (2, -3) Axiom <ref>;
[-to] (P1) to (ES);
[-to] (P3) to (ES);
[-to] (A1toP1) to (P1);
[-to] (A1) to (A1toP1);
[-to] (AC) to (A1toP1);
[-to] (P4toP3) to (P3);
[-to] (P4) to (P4toP3);
[-to] (P5toP4) to (P4);
[-to] (P5) to (P5toP4);
[-to] (P2toP5) to (P5);
[-to] (P2) to (P2toP5);
[-to] (A4toP2) to (P2);
[-to] (A4) to (A4toP2);

Axiomatic basis of the homotopy invariance of homology.

We now examine the axioms and mechanisms which guarantee that homotopic maps induce the same homomorphisms between homology groups. We shall follow the outline presented in Figure <ref>.

[Dimension axiom]
Homology of $F(0)$ is zero.

If the functor from (<ref>) satisfies Axiom <ref>, then the homology it induces satisfies the dimension axiom <ref>.

The lemma follows from the fact that $F(0)$ is the terminal object of $\calC$. Thus for each $n\in \num$ and $c\in \calC$, $\Nerve_F(c)(n) = \Hom_{\calC} \paran{ F(n); F(1)}$ is a singleton set. The proof is algebraic and can be found in <cit.>.

[Homotopy axiom]
The notion of homology is homotopy invariant.

The homotopy axiom can be derived from a simpler property involving the boundary maps :

[Face-maps homotopy property]
An object $X\in \calC$ has this property if the following composite morphisms in $\calC$

\[\begin{tikzcd}
&& X \arrow[d, "\cong"] \arrow[dashed, drr, "\lambda^{X}_0"] \arrow[dll, dashed, "\lambda^{X}_1"'] \\
X\times F(1) && X\times \star \arrow[rr, "\Id_x \times F\paran{d_{1,0}}"'] \arrow[ll, "\Id_x \times F\paran{d_{1,1}}"] && X\times F(1)
\end{tikzcd}\]

induce the same morphisms under Homology.

<cit.> Suppose each object $X$ has Property <ref>. Then the Homotopy axiom <ref> is satisfied. In other words,  if $f,g : X\to Y$  are homotopic, then their induced homologies are the same.

A homotopy between morphisms $f,g:X\to Y$ is a morphism $H:X\times F(1) \to Y$ such that $f = H\circ \lambda^{X}_0$ and $g = H\circ \lambda^{X}_1$. Then note that

\[\begin{split}
\Chain\paran{ f }_{\#} &= \Chain\paran{H\circ \lambda^{X}_0}_{\#} = \Chain\paran{ H }_{\#} \circ \Chain\paran{ \lambda^{X}_0 }_{\#} \\
& = \Chain\paran{ H }_{\#} \circ \Chain\paran{ \lambda^{X}_1 }_{\#} , \quad \mbox{by Property \ref{P:4}} , \\
&= \Chain\paran{H\circ \lambda^{X}_1}_{\#} = \Chain\paran{g}_{\#} .
\end{split}\]

In the equalities above, we have repeatedly used the composition preserving property of the functor $\Chain$.

Thus to prove Homotopy property <ref>, we are presented with the task of proving Property <ref>. This shall follow from :

[Cross morphisms property]
An object $X\in \calC$ has this property if there are homomorphisms

\[ P^{X}_{n} : \Chain(X)_n \to \Chain\paran{ X\times F(1) }_{n+1} \]

such that

\begin{equation} \label{eqn:P:5a}
\Chain\paran{ \lambda^{X}_1 }_n - \Chain\paran{ \lambda^{X}_0 }_n = \partial_{n+1} P^{X}_{n} + P^{X}_{n-1} \partial_{n} .
\end{equation}

<cit.> For any object $X\in \calC$, Property <ref> implies Property <ref>.

Let $\Boundary(X)_n$ denote the subgroup of $\Homology(X)_n$, created by boundaries. Let $z$ be a cycle in $\Homology(X)_n$, i.e., $\partial_n z = 0$. By the definition of the homology maps

\[ \Homology(f)_n : z \oplus \Boundary(X)_n \mapsto \Chain(f)_n (z) + \Boundary(Y)_n \]

But by Property <ref>, we have

\[ \paran{ \Chain(f)_n - \Chain(g)_n } (z) = \partial^Y_{} P z + P \partial z  = \partial^Y_{} P z . \]

Since the RHS is a boundary element, so is the LHS. As a result, we can write

\[ \Chain(f)_n(z) \oplus \Boundary(Y)_n = \Chain(g)_n(z) \oplus \Boundary(Y)_n . \]

This means that $\Homology(f)_n$ must be equal to $\Homology(g)_n$.

Thus to prove Homotopy property <ref>, we are presented with the task of proving Property <ref>. This shall follow from :

[Zero homology of cells]
The cells $\SetDef{F(n)}{n\in\num_0}$ have zero homology.

An object $X\in \calC$ will be called acyclic if its homology group is zero. Thus Property <ref> says that the cells $F(n)$ are all acyclic.

Suppose Axioms <ref> and <ref> hold. Then Property <ref> implies Property <ref> for every object $X\in \calC$.

Lemma <ref> is an outcome of the theory of acyclic models <cit.>. We provide the proof in Section <ref>, for the sake of completeness. The zero homology property <ref> is a consequence of the following important result

Let Axioms <ref>, <ref>, <ref>, <ref> and <ref> hold, and recall the definition of convexity. Then the collection of convex objects of $\calC$

* are closed under finite products;

* are acyclic.

Theorem <ref> is proved in Section <ref>. Also see Figure <ref> for an outline of the chain of reasoning. By Axiom <ref>, the cells $\SetDef{ F(n) }{ n\in\num_0 }$ are convex. Thus by Theorem <ref> (ii), they are acyclic too, as claimed in Property <ref>. This completes the verification of the Homotopy axiom.

Remark Similar to classic Homotopy theory, an object $X$ of $\calC$ will be called contractible if its identity morphism $\Id_X$ is homotopic to some constant endomorphism. A constant endomorphism is a composite of morphisms of the form $X \xrightarrow{!} 1_{\calC} \xrightarrow{x} X$. Here $x$ is an element of $X$ and the composite morphism can be interpreted to be constant of value $x$. Theorems <ref> (iii) and <ref> (ii) together imply that contractible objects are acyclic.

§ CONVEXITY

[scale=0.7, transform shape]
[style=rect6] (P2) at (0, -3) Zero homology of cells <ref>;
[style=rect5] (P7toP2) at (, -3) Lemma <ref>;
[style=rect6] (P7) at (2, -3) Equation (<ref>);
[style=rect5] (P6toP7) at (2, -4) Lemma <ref>;
[style=rect6] (P6) at (, -4) Equation (<ref>);
[style=rect5] (A4toP6) at (0, -4) Equations (<ref>) and (<ref>);
[style=rect2] (A4) at (-1, -4) Axiom <ref>;
[-to] (P7toP2) to (P2);
[-to] (P7) to (P7toP2);
[-to] (P6toP7) to (P7);
[-to] (P6) to (P6toP7);
[-to] (A4toP6) to (P6);
[-to] (A4) to (A4toP6);

Convexity and acyclicity. The flowchart is an outline of the proof of Theorem <ref> (ii).

The main purpose of this section is to examine the consequences of convexity by verifying the claims of Theorem <ref>. Amjor part of the discussion that follows is outlined in Figure <ref>. Convexity is borne by the natural transformations in (<ref>) and . The first one indicates the following commutations :

\begin{equation} \label{eqn:Base:1}
\begin{tikzcd}
\Hom_{ \calC } \paran{ F(n+1); X  } \arrow{d}[swap]{ \circ F \paran{ d_{n+1,0} } } \arrow{rrr}{ \circ F \paran{ d_{n+1,i+1} } } &&& \Hom_{ \calC } \paran{ F(n); X  } \arrow{d}{ \circ F \paran{ d_{n,0} } } \\
\Hom_{ \calC } \paran{ F(n); X  } \arrow{rrr}{ \circ F \paran{ d_{n,i} } } &&& \Hom_{ \calC } \paran{ F(n-1); X  }
\end{tikzcd}
\end{equation}

The commutation in (<ref>) follows from (<ref>) (i). Thus the $\Base$ transformation exists and is independent of $X$. We next examine the consequences of (<ref>). It implies the existence of a special collection of maps

\begin{equation} \label{eqn:cone:a}
\Cone^X_n : \Hom_{\calC} \paran{ F(n); X } \to \Hom_{\calC} \paran{ F(n+1); X } , \quad n\in \num_0 .
\end{equation}

which satisfy

\begin{equation} \label{eqn:ConeNat:3}
\begin{tikzcd}
\Hom_{ \calC } \paran{ F(n); X  } \arrow[d, "\Cone^{X}_n"'] \arrow{rrr}{ \circ F \paran{ d_{n,i} } } &&& \Hom_{ \calC } \paran{ F(n-1); X  } \arrow[d, "\Cone^{X}_{n-1}"] \\
\Hom_{ \calC } \paran{ F(n+1); X  } \arrow{rrr}{ \circ F \paran{ d_{n+1,i+1} } } &&& \Hom_{ \calC } \paran{ F(n); X  }
\end{tikzcd} , \quad
0\leq i\leq n
\end{equation}

\begin{equation} \label{eqn:Base:2}
\begin{tikzcd}
\Hom_{\calC} \paran{ F(n); X } \arrow{d}[swap]{ \Cone^X(n) } \arrow[r, "="] & \Hom_{\calC} \paran{ F(n); X } \\
\Hom_{\calC} \paran{ F(n+1); X } \arrow[ur, "\circ F\paran{ d_{n+1,0} }"']
\end{tikzcd},
\quad \forall n\in\num_0 .
\end{equation}

The set maps $\Cone^X_n$ from (<ref>) extend uniquely to homomorphisms on the freely generated Abelian groups

\[ \Cone^X_n : \Chain \paran{ X}_n \to \Chain \paran{ X}_{n+1} ; \quad \forall n\in\num_0 . \]

Now observe that (<ref>) and (<ref>) together imply

\begin{equation} \label{eqn:P:6}
\forall \begin{tikzcd} F(n) \arrow[ "\sigma"', d ] \\ X	\end{tikzcd} , \quad
\Cone^{X}_n \paran{ \sigma } \circ F\paran{ d_{n+1,i} } \,=\,
\begin{cases}
\Cone^{X}_{n-1} \paran{ \sigma \circ F \paran{ d_{n,i-1} } } & \mbox{ if } 1\leq i \leq n+1 \\
\sigma & \mbox{ if } j=0
\end{cases} ; \quad \forall n\in \num_0 .
\end{equation}

Suppose that an object $X$ has a sequence of maps $\Cone_n$ as in (<ref>). Suppose that the maps satisfy (<ref>). Then

\begin{equation} \label{eqn:P:7}
\partial_{n+1} \Cone_n + \Cone_{n-1} \partial_{n} = \Id_n; \quad \forall n\in \num_0 .
\end{equation}

The lemma follows from the following simple algebraic calculation :

\[\begin{split}
\partial_{n+1} \paran{ \Cone_n \sigma } &= \sum_{i=0}^{n+1} (-1)^i \paran{ \Cone_n \sigma } \circ F \paran{ d_{n+1,i} } = \sigma + \sum_{i=1}^{n+1} (-1)^i \paran{ \Cone_n \sigma } \circ F \paran{ d_{n+1,i} } \\
&= \sigma + \sum_{i=1}^{n+1} (-1)^i \Cone_{n-1} \paran{ \sigma \circ F \paran{ d_{n+1,i-1} } } = \sigma + \Cone_{n-1} \paran{ \sum_{i=1}^{n+1} (-1)^i \sigma \circ F \paran{ d_{n+1,i-1} } } \\
&= \sigma + \Cone_{n-1} \paran{ \partial_{n} \sigma } . \qed
\end{split}\]

Equation (<ref>) distills the essential identity from the natural transformations $\Base$ and $\Cone$ needed to guarantee acyclicity :

Suppose that an object $X$ has a sequence of maps $\Cone_n$ as in (<ref>), such that their extensions satisfy (<ref>). Then Property <ref> is satisfied.

It has to be shown that any cycle in $\Chain(X)_n$ is also a boundary. Take any $\sigma\in \Chain(X)_n$. Suppose $\sigma$ is a cycle, i.e., $\partial_{n} \sigma = 0$. Then according to (<ref>), $\gamma = \partial_{n+1} \Cone_n \sigma$, i.e., $\gamma$ is a boundary, as required for the completion of the proof.

Thus we have established Theorem <ref> (ii) We next focus on Claim  (i). For that purpose, suppose that $X$ and $Y$ are two convex objects, with cones $\Cone^X$ and $\Cone^Y$ respectively. Note that any morphism in the Hom-set $\Hom_{\calC} \paran{ F(n) ; X\times Y }$ is a unique categorical product of morphisms in $\Hom_{\calC} \paran{ F(n) ; Y }$ and $\Hom_{\calC} \paran{ F(n) ; X }$ respectively. Thus it is enough to prescribe $\Cone_n^{X\times Y}$ on product morphisms only. Consider the following commuting diagram :

\[\begin{tikzcd}
&& && X\\
&& F(n+1) \arrow{rru}{ \Cone^X_n (\alpha) } \arrow{rrd}[swap]{ \Cone^Y_n (\beta) } \arrow[Shobuj]{rrrr}{ \left\langle \Cone^X_n (\alpha), \Cone^Y_n (\beta) \right\rangle } && && X\times Y \arrow[ull, "\proj_1"'] \arrow[dll, "\proj_2"] \\
&& && Y
\end{tikzcd}\]

The green arrow  is the product of two cones, and the commutations represent its relationship with its two coordinate projections. We are going to set

\begin{equation} \label{eqn:ConeProd}
\forall \alpha \in \Hom_{ \calC } \paran{ F(n); X }, \; \forall \beta \in \Hom_{ \calC } \paran{ F(n); Y }, \quad \Cone_n^{X\times Y} \paran{ \alpha \times \beta } = \left\langle \Cone_n^X(\alpha) , \Cone_n^Y (\beta) \right\rangle , \quad \forall n\in \num_0 .
\end{equation}

To check that this is indeed a cone construction for $X\times Y$, (<ref>) has to be verified. For that purpose, we append a face map to the left of the commutation diagram to get :

\[\begin{tikzcd}
&& && X\\
F(n) \arrow[rr, "F(d_{n+1,i})", Shobuj] \arrow[Holud, dashed, bend left=20, urrrr] \arrow[Holud, dashed, bend right=20, drrrr] && F(n+1) \arrow{rru}{ \Cone^X_n (\alpha) } \arrow{rrd}[swap]{ \Cone^Y_n (\beta) } \arrow[Shobuj]{rrrr}{ \left\langle \Cone^X_n (\alpha), \Cone^Y_n (\beta) \right\rangle } && && X\times Y \arrow[ull, "\proj_1"'] \arrow[dll, "\proj_2"] \\
&& && Y
\end{tikzcd}\]

The dashed yellow arrows represent compositions. Thus the composite of the green arrows is the product of the yellow arrows. Now if $i=0$ then by (<ref>) the yellow arrows are $\alpha$ and $\beta$ respectively. This means that

\[ \left\langle \Cone_n^X(\alpha) , \Cone_n^Y (\beta) \right\rangle \circ F \paran{ d_{n+1,0} } = \left\langle \alpha , \beta \right\rangle .\]

If $1\leq i \leq n+1$, then again by (<ref>) the yellow arrows are $\Cone^{X}_{n-1} \paran{ \alpha \circ F \paran{ d_{n,i-1} } }$ and $\Cone^{Y}_{n-1} \paran{ \beta \circ F \paran{ d_{n,i-1} } }$ respectively. This means that

\[\begin{split}
& \left\langle \Cone_n^X(\alpha) , \Cone_n^Y (\beta) \right\rangle \circ F \paran{ d_{n+1,i} } = \left\langle \Cone^{X}_{n-1} \paran{ \alpha \circ F \paran{ d_{n,i-1} } } , \Cone^{Y}_{n-1} \paran{ \beta \circ F \paran{ d_{n,i-1} } } \right\rangle \\
& \quad \quad = \Cone^{X}_{n-1} \paran{ \left\langle \alpha \circ F \paran{ d_{n,i-1} } , \beta \circ F \paran{ d_{n,i-1} } \right\rangle }, \quad \mbox{by definition}, \\
& \quad \quad =  \Cone^{X}_{n-1} \paran{ \left\langle \alpha , \beta \right\rangle } \circ F \paran{ d_{n,i-1} } .
\end{split}\]

Thus (<ref>) indeed provides a cone construction for $X\times Y$. So $X\times Y$ is convex, and Claim (ii) is true. This completes the proof of Theorem <ref>. This completes the discussion on how convexity implies acyclicity. Thus all the ingredients of Theorem <ref> are in place. While it is easy to conceive of functors (<ref>), it is a non-trivial task to satisfy its axioms. Axiom <ref> is the most challenging task to be verified for a general functor. We next consider a particular construction of $F$ that fulfills the requirement that cells be convex.

§ CREATING CONVEX CELLS

[scale=0.7, transform shape]
[style=rect2] (N1) at (0, 0) Axioms <ref>, <ref> and <ref>;
[style=rect2] (N2) at (, 0) Inductive procedure (<ref>), (<ref>), and (<ref>);
[style=rect5] (N3) at (2, 0) Theorem <ref>;
[style=rect2] (N4) at (3, 0) Functor $F:\Delta \to \calC$;
[-to] (N1) to (N2);
[-to] (N2) to (N3);
[-to] (N3) to (N4);

Convexity and acyclicity. The flowchart is an outline of the proof of Theorem <ref> (ii).

We introduce two new axioms on the category $\calC$. The first is about pushouts :

The category $\calC$ has pushouts a.

Suppose Axioms <ref> and <ref> hold. Then one can create pushouts as follows :

\begin{equation} \label{eqn:def:Wedge}
\begin{tikzcd} [scale cd = 0.8]
& & 1_{\calC} \\
X \arrow[r, "\cong"] & X \times 1_{\calC} \arrow[ur, "!"] \arrow[dr, " X \times \tilde{d}_{1,1} "'] \\
& & X \times F_1
\end{tikzcd} \;
\begin{tikzcd} {} \arrow[rr, mapsto, "\text{pushout}"] && {} \end{tikzcd} \;
\begin{tikzcd} [scale cd = 0.8]
& & 1_{\calC} \arrow[dr, Itranga] \\
X \arrow[r, "\cong"] & X \times 1_{\calC} \arrow[ur, "!"] \arrow[dr, " X \times \tilde{d}_{1,1} "'] && \Shobuj{ \Wedge(X) } \\
& & X \times F_1 \arrow[ur, Itranga]
\end{tikzcd}
\end{equation}

This operation is functorial, as we argue below :

If Axioms <ref>, <ref> and <ref> hold, then the pushout construction in (<ref>) creates a functor $\Wedge : \calC \to \calC$.

When constructing a functor $F$ as in (<ref>) we shall take the zeroth cell to be $1_{\calC}$. The 1-cells exist by assumption

There is an object $F_1$ in $\calC$ along with morphisms $\tilde{d}_{1,0}, \tilde{d}_{1,j} : 1_{\calC} \to F_1$ .

We thus have analogs of $0$ and $1$-th cells, along with boundary maps. The successive cells are created iteratively via the wedge construction :

\begin{equation} \label{eqn:Wedge_cells:1}
\forall n\in\num  \,: \quad F_{n+1} := \Wedge \paran{ F_n } .
\end{equation}

We next construct the face maps between these cells. For that purpose we redraw (<ref>) but for the cells :

\begin{equation} \label{eqn:Wedge_cells}
\begin{tikzcd} [scale cd = 0.8]
& & F_0 \\
F_n \arrow[r, "\cong"] & F_n \times F_0 \arrow[ur, "!"] \arrow[dr, " F_n \times \tilde{d}_{1,1} "'] \\
& & F_n \times F_1
\end{tikzcd} \;
\begin{tikzcd} {} \arrow[rr, mapsto, "\text{pushout}"] && {} \end{tikzcd} \;
\begin{tikzcd} [scale cd = 0.8]
& & F_0 \arrow[dr, Itranga] \\
F_n \arrow[dd, Shobuj] \arrow[r, "\cong"] & F_n \times F_0 \arrow[ur, "!"] \arrow[dr, " F_n \times \tilde{d}_{1,1} "'] && F_{n+1} \\
& & F_n \times F_1 \arrow[ur, Itranga, "\pi_n"'] \\
{} \arrow[rrr, Shobuj, " \tilde{d}_{n+1,n+1} "{pos=0.2}] & & & {} \arrow[uu, Shobuj]
\end{tikzcd}
\end{equation}

The newly defined face map $\tilde{d}_{n+1,n+1}$ (green) will be one of the face maps, as shown below :

\begin{equation} \label{eqn:Wedge_cells:2}
\forall n\in\num  \,:
\begin{tikzcd} F_{n} \arrow[d, " \tilde{d}_{n+1,j} "', Shobuj] \\ F_{n+1} \end{tikzcd} :=
\begin{cases}
\Wedge \paran{ \tilde{d}_{n,j} } & \mbox{ if } 0\leq j< n+1 \\
\mbox{Equation \eqref{eqn:Wedge_cells}} & \mbox{ if } j=n+1
\end{cases}
\end{equation}

Next, we construct the degeneracy maps. We use the construction shown below :

\begin{equation} \label{eqn:wik0m3}
\forall \begin{tikzcd} 1_{\calC} \arrow[d, "x"'] \\ X \end{tikzcd} , \quad
\begin{tikzcd} [scale cd = 0.8]
& 1_{\calC} \arrow[dddr, Holud] \arrow[rrr, "="] &&& 1_{\calC} \arrow[dddr, Holud] \\
X \times 1_{\calC} \arrow[ur, "!", Akashi] \arrow[dr, " X \times \tilde{d}_{1,1} "', Akashi] &&& 1_{\calC} \arrow[dr, "x"', Akashi] \arrow[ur, "=", Akashi] \\
& X \times F_1 \arrow[dr, Holud] \arrow[rrr, "X \times ! "'{pos=0.8} ] &&& X \arrow[dr, Holud] \\
& & \Wedge(X) \arrow[Shobuj]{rrr}{\exists ! \beta(X,x)} &&& X
\end{tikzcd}
\end{equation}

The starting point of the construction if  is an element $x$ of an object $X$. The diagram shows two vee-shaped diagrams in blue, and the completion of their pushout squares in yellow. The pushout diagram on the right is a cone below the vee diagram on the left. Thus by the universality of pushouts, there pushout on the right factors through the pushout on the left. This is made possible by the morphism $\beta(X,x)$ in green, connecting the pushouts. This morphism $\beta(X,x)$ will be used to construct some of the degeneracy maps. For that purpose we need elements in each cell. This will be guaranteed by Axiom <ref>. Consider the diagram on the left :

\begin{equation} \label{eqn:def:a1}
\begin{tikzcd}
& F_1 \arrow[dr, Itranga] \\
1_{\calC} \arrow[rr, Shobuj, "a_1", dashed] \arrow[ur, "\tilde{d}_{1,0}"] \arrow[dr, "\tilde{d}_{1,1}"'] && F_1 \\
& F_1 \arrow[ur, Itranga] \\
\end{tikzcd} , \quad
\begin{tikzcd}
& F_1 \\
1_{\calC} \arrow[dashed, Shobuj, dd] \arrow[dr, "a_n"'] \arrow[ur, "a_1"] \arrow{rr}{ \left\langle a_1, a_n \right\rangle } && F_{n} \times F_{1} \arrow[r, "\beta_n"] \arrow[dl, "\proj_1"] \arrow[ul, "\proj_2"'] & F_{n+1} \\
& F_n \\
{} \arrow[dashed, Shobuj, "a_{n+1}", rrr] &&& {} \arrow[dashed, Shobuj, uu]
\end{tikzcd}
\end{equation}

The pushout diagram is a colimit by Axiom <ref>. This creates a unique element marked $a_1$ of $F_1$, and labeled as $\text{center}$ in the statement of Axiom <ref>. The diagram on the right shows how the elements $a_n$ of cells $F_n$ are created inductively. We can now define the degeneracy maps as

\begin{equation} \label{eqn:Wedge_cells:3}
\forall n\in\num  \,:
\begin{tikzcd} F_{n+1} \arrow[d, " \tilde{s}_{n,j} "', Shobuj] \\ F_{n} \end{tikzcd} :=
\begin{cases}
\Wedge \paran{ \tilde{s}_{n-1,j} } & \mbox{ if } 0\leq j< n+1 \\
\beta \paran{ F_n, a_{n} } \mbox{ Equation \eqref{eqn:wik0m3}} & \mbox{ if } j=n+1
\end{cases}
\end{equation}

So far we have created a sequence $\braces{F_N}_{n\in\num_0}$ of objects on $\calC$ called cells, and morphisms $\tilde{d}_{n,j} : F_{n-1} \to F_n$ and $\tilde{s}_{n,j} : F_n \to F_{n-1}$. By virtue of their construction, they turn out to be the images of a functor from $\Delta$ :

Suppose Axioms <ref>,  <ref>, <ref> and <ref> hold.

* $F_n$ is isomorphic to $F_n \times F_1$.

* The objects $F_n$ defined inductively in (<ref>), morphisms $\tilde{d}_{n+1,j}$ defined in (<ref>), and morphisms $\tilde{s}_{n+1,j}$ defined in (<ref>) assemble to create a functor $F$ as in (<ref>).

* If $F_1$ is convex, then so is $F_n$ for every $n$.

Claim  (i) follows from the following elementary result :

For every $X\in \calC$, $\Wedge(X) \cong X \times F_1$.

Proving Claim (ii) involves verifying that the constructions in (<ref>), (<ref>) and (<ref>) the simplicial identities <ref>. This is a routine task and will be omitted.

§ CONCLUSIONS

We have thus seen that in any category $\calC$, a concept of a 1-cell $F_1$ and boundary  maps from the terminal object into the 1-cell creates a notion of homotopy. Section <ref> shows how under further Axioms <ref>, <ref> and <ref> on the 1-cell, the relation of homotopy is an equivalence relation is invariant under composition. Separately, Section <ref> shows how Axioms <ref> and <ref> on $\calC$ and $F_1$ enable the construction of a functor $F:\Delta \to \calC$. Any such functor creates a homology, as shown in Section <ref>. The key connection between homology and homotopy is the homotopy invariance of homology. This is enabled by Axiom <ref>. Section <ref> shows how convexity implies acyclicity, a key component for the connection discussed.

In summary, an arbitrary category $\calC$ can be equipped with notions of homotopy and homology which play the analogous as the case when $\calC= \Topo$. This provides a gateway for all theorems, techniques and numerical methods for Topology <cit.> to be applied to a different category $\calC$. We end the paper with a brief discussion of some of these new avenues. The first idea is that of complexes.

Cell complexes A finite category $J$ will be called a cell-complex diagram if it satisfies the following properties :

* The objects of $J$ can be partitioned into levels $J^{(1)}, \ldots, J^{(L)}$.

* The morphisms in $J$ are generated by arrows that connect an $l$-level object to an $(l+1)$-level object.

A  cellular complex  in $\calC$ is a functor from $\Phi : J \to \calC$ such that

* $\Phi$ maps each $l$-level object into the object $F(l)$.

* Each morphism $f$ in $J$ that connects an $l$-level object to an $(l+1)$-level object, is mapped into a face map $F \paran{ d_{l+1,j} }$ for some $0\leq j \leq l+1$.

Thus a cellular complex is a specific type of a finite diagram in $\calC$. It respects the cells and face maps prescribed by the functor $F$ from (<ref>). By a reuse of terminology, we say that an object $X \in \calC$ is a cellular complex if there is a cellular complex $\Phi : J \to \calC$ such that $X = \colim \Phi$. In this case, the functor $\Phi$ will be called a cellular decomposition of the object $X$. The integer $L$ is to be represented as the dimension of the simplicial complex. The collection of $L$-dimensional $\calC$-simplicial complexes form a category of its own, which we shall denote as $\SqBrack{ L-\text{complex} }$.

Reconstruction The idea of finite dimensional complexes in $\calC$ enables a data-driven study of the category $\calC$. Consider the following commutation diagram :

\begin{equation} \label{eqn:TDA:scheme}
\begin{tikzcd} [row sep = large]
\calC \arrow[d, "\text{Homology}"'] && \SqBrack{ L-\text{simplex} } \arrow[dashed, Holud]{d}[name=n1]{} \arrow[ll, "\colim"'] \\
\AbelCat^{\num} \arrow[rr, "\proj_L"'] && \AbelCat^{L} \arrow[bend right=60, Shobuj]{u}[swap, name=n2]{ \text{Reconstruct} }
\arrow[shorten <=0.1pt, shorten >=4pt, -Bar, to path={(n1) to[out=0,in=180] node[xshift=10pt]{} (n2)} ]{  }
\end{tikzcd}
\end{equation}

The left loop shows the interpretation of $L$ -simplices as $\calC$-objects, the calculation of their Homology, and finally restricting them to the their first $L$ indices. These signature of any $\calC$ object may be directly computable from data. The task of a data-driven reconstruction may be interpreted as the creation of the functor $\text{Reconstruct}"'$ shown in green, which must be the right adjoint to the composite functor shown in dashed yellow. The interpretation and realization of the reconstruction functor is a subject of future work.

§ APPENDIX

§.§ Constructions from categorical products

In this short section we briefly review some commutations created by categorical products. The following commutation can be taken as the definition of categorical products of morphisms :

\begin{equation} \label{eqn:od3l9}
\begin{tikzcd} a \arrow[d, "f"'] \\ b \end{tikzcd} , \;
\begin{tikzcd} a' \arrow[d, "f'"'] \\ b' \end{tikzcd}
\imply
\begin{tikzcd}
& a \arrow[r, "f"] & b \\
a\times a' \arrow[rrr, "f\times f'", dotted] \arrow[ur, "\proj_2", Akashi] \arrow[dr, "\proj_1"', Akashi]  &&& b\times b' \arrow[dl, "\proj_2", Holud] \arrow[ul, "\proj_1"', Holud] \\
& a' \arrow[r, "f'"'] & b'
\end{tikzcd}
\end{equation}

The blue and yellow sub-diagrams are limiting cones, and the central dotted morphism is the unique morphism by which they transform. Equation (<ref>) shows how products extend from objects to morphisms. In fact, this is functorial :

In any category $\calS$ with finite products, for every $n\in \num$, the $n$-fold product is a functor from $\calS^n \to \calS$. Moreover, if $\calS$ has a terminal element $1_{\calS}$, then $1_{\calS} : \calS \to \calS$ is the identity functor.

The next commutation relation that we need involves commutation with the terminal element $1_{\calC}$ of $\calC$ :

\begin{equation} \label{eqn:iojd9}
\begin{tikzcd}
a \arrow[dr, "="'] \arrow[r, "\cong"] & a \times 1_{\calC} \arrow[d, "\proj_1"] \\
& a
\end{tikzcd}
\end{equation}

It is a well known fact <cit.> that products with the terminal object is an identity operation. The second diagonal equality arrow follows from the universality of the limit. We use these two basic commutations to derive some useful results required in our proofs.

§.§ Some technical lemmas

Throughout this section, we assume an object $X\in \calC$ and a morphism $\sigma:F(0)\to X$. Using a combination of (<ref>) and (<ref>) we can draw the following diagram involving $X, F(0), F(1)$ and $\sigma$ :

\[\begin{tikzcd}
&& & & F(1) \\
F(0) \arrow[urrrr, "F\paran{ d_{1,j} }", bend left=20] \arrow[rr, "F\paran{ d_{1,j} }"] && F(1) \arrow[rru, bend left=20, "="] \arrow[r, "="] & F(1) \times F(0) \arrow[dr, "\proj_2"'] \arrow[ur, bend left=10, "\proj_1"] \arrow{rrr}{ \Id_{F(1)} \times \sigma } &&& F(1)\times X \arrow[ull, "\proj_1"'] \arrow[dl, "\proj_2"] \\
&& & & F(0) \arrow[r, "\sigma"] & X
\end{tikzcd}\]

Note that the following two dotted arrows can be added to this diagram :

\[\begin{tikzcd}
&& & & F(1) \\
F(0) \arrow[urrrr, "F\paran{ d_{1,j} }", bend left=20] \arrow[rr, "F\paran{ d_{1,j} }"] && F(1) \arrow[rru, bend left=20, "="] \arrow[r, "="] & F(1) \times F(0) \arrow[dr, "\proj_2"'] \arrow[ur, bend left=10, "\proj_1"] \arrow{rrr}{ \Id_{F(1)} \times \sigma } &&& F(1)\times X \arrow[ull, "\proj_1"'] \arrow[dl, "\proj_2"] \\
&& & & F(0) \arrow[r, "\sigma"] & X \\
&& & & & & F(0) \times X \arrow[ul, "\proj_2"', dotted] \arrow[uullllll, bend left=30, "\proj_1", dotted]
\end{tikzcd}\]

Next, by utilizing the terminal property of $F(0)$, we can added the following dotted diagram :

\[\begin{tikzcd}
&& & & F(1) \\
F(0) \arrow[drrrr, bend right=10, "="', dotted] \arrow[urrrr, "F\paran{ d_{1,j} }", bend left=20] \arrow[rr, "F\paran{ d_{1,j} }"] && F(1) \arrow[rru, bend left=20, "="] \arrow[r, "="] & F(1) \times F(0) \arrow[dr, "\proj_2"'] \arrow[ur, bend left=10, "\proj_1"] \arrow{rrr}{ \Id_{F(1)} \times \sigma } &&& F(1)\times X \arrow[ull, "\proj_1"'] \arrow[dl, "\proj_2"] \\
&& & & F(0) \arrow[r, "\sigma"] & X \\
&& & & & & F(0) \times X \arrow[ul, "\proj_2"'] \arrow[uullllll, bend left=30, "\proj_1"] \arrow[bend right=20]{uu}[swap]{F\paran{ d_{1,j} } \times \Id_{X}}
\end{tikzcd}\]

We next add an instance of (<ref>) involving $F(0)$ :

\[\begin{tikzcd}
&& & & F(1) \\
F(0) \arrow[drrrr, bend right=10, "="'] \arrow[urrrr, "F\paran{ d_{1,j} }", bend left=20] \arrow[rr, "F\paran{ d_{1,j} }"] && F(1) \arrow[rru, bend left=20, "="] \arrow[r, "="] & F(1) \times F(0) \arrow[dr, "\proj_2"'] \arrow[ur, bend left=10, "\proj_1"] \arrow{rrr}{ \Id_{F(1)} \times \sigma } &&& F(1)\times X \arrow[ull, "\proj_1"'] \arrow[dl, "\proj_2"] \\
&& & & F(0) \arrow[r, "\sigma"] & X \\
&& & & & X \arrow[u, "="] \arrow[r, "="'] & F(0) \times X \arrow[ul, "\proj_2"'] \arrow[uullllll, bend left=30, "\proj_1"] \arrow[bend right=20]{uu}[swap]{F\paran{ d_{1,j} } \times \Id_{X}}
\end{tikzcd}\]

Now note that we can add another instance of $\sigma : F(0) \to X$ to this diagram :

\[\begin{tikzcd}
&& & & F(1) \\
F(0) \arrow[drrrr, bend right=10, "="', Itranga] \arrow[urrrr, "F\paran{ d_{1,j} }", bend left=20] \arrow[rr, "F\paran{ d_{1,j} }", Itranga] && F(1) \arrow[rru, bend left=20, "="] \arrow[r, "=", Itranga] & F(1) \times F(0) \arrow[dr, "\proj_2"'] \arrow[ur, bend left=10, "\proj_1"] \arrow[Itranga]{rrr}{ \Id_{F(1)} \times \sigma } &&& F(1)\times X \arrow[ull, "\proj_1"'] \arrow[dl, "\proj_2"] \\
&& & & F(0) \arrow[r, "\sigma"] \arrow[dr, "\sigma"', Itranga] & X \\
&& & & & X \arrow[u, "="] \arrow[r, "="', Itranga] & F(0) \times X \arrow[ul, "\proj_2"'] \arrow[uullllll, bend left=30, "\proj_1"] \arrow[bend right=20, Itranga]{uu}[swap]{F\paran{ d_{1,j} } \times \Id_{X}}
\end{tikzcd}\]

The final picture (<ref>) has some morphisms traced out in red. Together these create the commutation relations :

\begin{equation} \label{eqn:kubsd9}
\lambda^X_j \circ \sigma = \paran{ F\paran{ d_{1,j} } \times \Id_{X} } \circ \sigma = \paran{ \Id_{F(1)} \times \sigma } \circ F\paran{ d_{1,j} } , \quad j=0,1 .
\end{equation}

One can similarly show that

\begin{equation} \label{eqn:sl5g}
\paran{ \Id_{F(1)} \times \sigma } \circ \lambda^{F(n)}_i = \lambda^X_j \circ \sigma : F(n) \to X\times F(1), \quad \forall n\in \num_0, \; j=0,1.
\end{equation}

These will be key to proving Lemma <ref> later.

§.§ Chain maps

\begin{equation} \label{eqn:chain:1}
\begin{tikzcd} A \arrow[d, "f"'] \\ B \end{tikzcd} \imply
\begin{tikzcd}
\Chain (A)_n \arrow{d}[swap]{ \Chain(f)_n } \arrow[r, "\partial^{A}_n"] & \Chain (A)_{n-1} \arrow{d}{ \Chain(f)_{n-1} } \\
\Chain (B)_n \arrow[r, "\partial^{B}_n"'] & \Chain (B)_{n-1}
\end{tikzcd}
\end{equation}

\begin{equation} \label{eqn:chain:2}
\begin{tikzcd}
\Chain (A)_n \arrow[dashed]{rr}{ \Chain (f)_n } && \Chain (B)_n \arrow[d, "="] \\
\braces{ \Hom_{\calC} \paran{ F(n); A } \to \calG } \arrow[u, "="] \arrow{rr}{ f\circ } && \braces{ \Hom_{\calC} \paran{ F(n); B } \to \calG }
\end{tikzcd}
\end{equation}

§.§ Proof of Lemma <ref>

The proof will be by induction :

Inductive hypothesis For every object $X\in \calC$, there are maps $P^{X}_{n} : \Chain(X)_n \to \Chain\paran{ X\times F(1) }_{n+1}$ as in Property <ref> which satisfy the identities (<ref>). Moreover, the following commutations hold :

\begin{equation} \label{eqn:P:5b}
\begin{tikzcd}
\Chain\paran{ F(n) }_n \arrow[rr, "P^{F(n)}_n"] \arrow[d, "\sigma\circ"'] && \Chain\paran{ F(n) \times F(1) }_{n+1} \arrow{d}{ \paran{\sigma\times \Id_F(1)} \circ } \\
\Chain(X)_n \arrow[rr, "P^{X}_n"'] && \Chain\paran{ X \times F(1) }_{n+1}
\end{tikzcd}
, \quad \forall \sigma \in \Hom_{\calC} \paran{ F(n); X } .
\end{equation}

Note that the inductive hypothesis for proving Property <ref> is stronger than Property <ref> itself.

Base case We begin with the case $n=0$. Now define

\begin{equation} \label{eqn:def:PX0}
P^X_0 : \Hom_{\calC} \paran{ F(0); X } \to \Hom_{\calC} \paran{ F(1); X \times F(1) } , \quad \sigma \mapsto
\begin{tikzcd}
F(1) \arrow[dr, dashed] \arrow[r, "\cong"] & 1_{\calC} \times F(1)
\arrow[d, "\Id_{F(1)} \times \sigma"] \\
& X \times F(1)
\end{tikzcd}
\end{equation}

\[\begin{split}
\partial_1 \paran{ P^X_0 \sigma } &= \sum_{i=0}^{1} (-1)^i \paran{ P^X_0 \sigma } \circ F\paran{ d_{1,i} } = \paran{ P^X_0 \sigma } \circ F\paran{ d_{1,0} } - \paran{ P^X_0 \sigma } \circ F\paran{ d_{1,1} } \\
&= , \quad \mbox{by \eqref{eqn:kubsd9}} .
\end{split}\]

When $n=0$, the commutation (<ref>) becomes :

\[\begin{tikzcd}
\Chain\paran{ F(0) }_0 \arrow[rr, "P^{F(0)}_0"] \arrow[d, "\sigma\circ"'] && \Chain\paran{ F(0) \times F(1) }_{n+1} \arrow{d}{ \paran{ \sigma\times \Id_F(1) } \circ } \\
\Chain(X)_n \arrow[rr, "P^{X}_0"'] && \Chain\paran{ X \times F(1) }_{1}
\end{tikzcd} \]

Note that the collection $\Chain\paran{ F(0) }_0$ has only one member – the identity morphism $\Id_{F(0)}$. The counter-clockwise path of the above figure is

\[P^{X}_0 \paran{ \sigma\circ \Id_{F(0)} } = P^{X}_0 \paran{ \sigma } = "\Id_{F(1)} \times \sigma , \]

by (<ref>). The clockwise path is

\[ \paran{ \sigma\times \Id_F(1) } \circ \paran{ P^{F(0)}_0 \paran{ \Id_{F(0)} }  } = \paran{ \sigma\times \Id_F(1) } \circ \paran{ \Id_{F(1)} \times \Id_{F(0)} } = \sigma\times \Id_F(1) .\]

Thus the two paths are the same and the verification of the case $n=0$ has been complete.

Inductive step We now assume that there is an $N\geq 0$ such that (<ref>) and  (<ref>) are true for all $n\leq N$. Our focus will be on the homomorphism

\[ Q_N := \Chain\paran{ \lambda^{F(N)}_1 }_{N} - \Chain\paran{ \lambda^{F(N)}_0 }_{N} - P^{F(N)}_{N-1} \partial_{N} : \Chain \paran{ F(N) }_N \to \Chain \paran{ F(N) \times F(1) }_{N+1} . \]

\[\begin{split}
& \partial_N Q_N = \partial_{N} \circ \paran{ \Chain\paran{ \lambda^{F(N)}_1 }_{N} - \Chain\paran{ \lambda^{F(N)}_0 }_{N} - P^{F(N)}_{N-1} \partial_{N} } \\
& \quad \quad = \partial_{N} \circ \Chain\paran{ \lambda^{F(N)}_1 }_{N} - \partial_{N} \circ \Chain\paran{ \lambda^{F(N)}_0 } - \partial_{N} \circ P^{F(N)}_{N-1} \partial_{N} \\
& \quad \quad = \Chain\paran{ \lambda^{F(N)}_1 }_{N} \circ \partial_{N} - \Chain\paran{ \lambda^{F(N)}_0 }_{N} \circ \partial_{N} - \partial_{N} \circ P^{F(N)}_{N-1} \circ \partial_{N}, \quad \mbox{by \eqref{eqn:chain:2} } .
\end{split}\]

We shall now use our inductive assumption and replace the term $\partial_N P^{F(N)}_{N-1}$ by

\[P^{F(N)}_{N-1} \partial_N = \Chain\paran{ \lambda^{F(N)}_1 }_N - \Chain\paran{ \lambda^{F(N)}_0 }_N + \partial_{N+1} P^{F(N)}_{N} .\]

As a result we get :

\[\begin{split}
& \SqBrack{ \Chain\paran{ \lambda^{F(N)}_1 }_{N} - \Chain\paran{ \lambda^{F(N)}_0 }_{N} - P^{F(N)}_{N-1} \partial_{N} } \circ \partial_{N} \\
& \quad \quad = \Chain\paran{ \lambda^{F(N)}_1 }_{N} \circ \partial_{N} - \Chain\paran{ \lambda^{F(N)}_0 }_{N} \circ \partial_{N} \\
& \quad \quad - \left[ \Chain\paran{ \lambda^{F(N)}_1 }_N - \Chain\paran{ \lambda^{F(N)}_0 }_N + \partial_{N+1} P^{F(N)}_{N} \right] \circ \partial_{N} \\
& \quad \quad = 0.
\end{split}\]

This indicates that the entire image of the homomorphism $Q_N$ are cycles. We shall consider the image under $Q_N$ of the trivial morphism $\Id_{F(N)}$. This image $Q_N \paran{ \Id_{F(N)} }$ is an $N$-cycle of $F(N) \times F(1)$. By Axiom <ref> both $F(1)$ and $F(N)$ are convex. Therefore by Theorem <ref> (i), $F(N) \times F(1)$ is convex too. Then by by Theorem <ref> (ii) $F(N) \times F(1)$ has zero homology groups, i.e., each cycle is also a boundary. Thus there is $\beta_{N+1} \in \Chain \paran{ F(N) \times F(1) }_{N+1} $ such that

\begin{equation} \label{eqn:pdm39}
\partial_{N+1} \beta_{N+1} = Q_N \paran{ \Id_{F(N)} }
\end{equation}

The map $P^{X}_{N+1}$ can now be defined as :

\begin{equation} \label{eqn:def:PXN}
P^{X}_{N+1} : \Chain(X)_{N+1} \to \Chain\paran{ X\times F(1) }_{N+2} , \quad \sigma \mapsto \paran{ \sigma \times \Id_{F(1)} } \circ \beta_{N+1} .
\end{equation}

\[\begin{split}
\partial_{N+1} P^{X}_{N+1} (\sigma) &= \partial_{N+1} \paran{ \sigma \times \Id_{F(1)} } \circ \beta_{N+1}, \quad \mbox{by \eqref{eqn:def:PXN}}, \\
&= \partial_{N+1} \Chain \paran{ \sigma \times \Id_{F(1)} }_{N+2} \paran{ \beta_{N+1} } , \quad \mbox{by \eqref{eqn:chain:2} }, \\
&= \Chain \paran{ \sigma \times \Id_{F(1)} }_{N+2} \partial_{N+1} \paran{ \beta_{N+1} } , \quad \mbox{by \eqref{eqn:chain:1} } ,\\
&= \Chain \paran{ \sigma \times \Id_{F(1)} }_{N+2} Q_N \paran{ \Id_{F(N)} }, \quad \mbox{by \eqref{eqn:pdm39} }
\end{split}\]

Next we utilize the definition of $Q_N$ to get

\[\begin{split}
\partial_{N+1} P^{X}_{N+1} (\sigma) &= \paran{ \sigma \times \Id_{F(1)} } \circ \SqBrack{ \Chain\paran{ \lambda^{F(N)}_1 }_{N} - \Chain\paran{ \lambda^{F(N)}_0 }_{N} - P^{F(N)}_{N-1} \partial_{N} } \paran{ \Id_{F(N)} } , \\
&= \paran{ \sigma \times \Id_{F(1)} } \circ \lambda^{F(N)}_1 - \paran{ \sigma \times \Id_{F(1)} } \circ \lambda^{F(N)}_0 - \paran{ \sigma \times \Id_{F(1)} } \circ  P^{F(N)}_{N-1} \partial_{N} \paran{ \Id_{F(N)} } ,\\
&= \paran{ \sigma \times \Id_{F(1)} } \circ \lambda^{F(N)}_1 - \paran{ \sigma \times \Id_{F(1)} } \circ \lambda^{F(N)}_0 - P^{F(N)}_{N-1} \circ \sigma \circ  \partial_{N} \paran{ \Id_{F(N)} },
\end{split}\]

where the last equality follows from the inductive assumption (<ref>) for $n=N$. We now proceed by simplifying the first two terms on the LHS using (<ref>) :

\[\begin{split}
\partial_{N+1} P^{X}_{N+1} (\sigma) &= \lambda^{X}_1 \circ \sigma  - \lambda^{X}_0 \circ \sigma - P^{F(N)}_{N-1} \circ \sigma \circ  \partial_{N} \paran{ \Id_{F(N)} } \\
&= \lambda^{X}_1 \circ \sigma  - \lambda^{X}_0 \circ \sigma - P^{F(N)}_{N-1} \circ \partial_{N} \sigma \circ \paran{ \Id_{F(N)} }, \quad \mbox{by \eqref{eqn:chain:2} } \\
&= \SqBrack{ \lambda^{X}_1 - \lambda^{X}_0 - P^{F(N)}_{N-1} } \paran{ \sigma } ,
\end{split}\]

and thus (<ref>) is verified for $n=N+1$. It remains to verify (<ref>) for $n=N+1$. That involves testing the commutation loop on an $N$-simplex $\tau : F(n) \to F(n)$. The CW path gives

\[ \paran{\sigma \times \Id_{F(1)} } \circ P^{F(N)}_N (\tau) = \paran{\sigma \times \Id_{F(1)} } \circ \paran{ \tau \times \Id_{F(1)} } \circ \beta_{N+1} = \paran{ \sigma \tau \times \Id_{F(1)} } \circ \beta_{N+1} .\]

The CCW path gives

\[ P^{X}_N \paran{ \sigma\circ \tau  } = \paran{ \sigma \tau \times \Id_{F(1)} } \circ \beta_{N+1} . \]

Thus the two paths are equal, and (<ref>) stands verified for $n=N+1$. This completes the inductive reasoning, and completes the proof of Lemma <ref>.