A functor from entry paths to the nerve of simplicial complexes

Fix $n\in \Z_{>0}$ and let $X=\Ran^{\leqslant n}(M)\times \R_{>0}$ for $M$ a compact, connected PL manifold embedded in $\R^N$. Take $\widetilde h\colon X\to (B,\leqslant)$ the conical stratifying map from a previous post (``Conical stratifications via semialgebraic sets," 2018-04-16) compatible with the natural stratification $h\colon X\to SC$. The goal of this post is to construct a functor $F\colon \Sing_B(X) \to N(SC)$ from the $\infty$-category of entry paths that encodes the structure of $X$.

Recall that a simplicial set is a functor, an element of $\text{Fun}(\Delta^{op},\Set)$. A simplicial set $S$ is defined by its collection of $n$-simplices $S_n$, its face maps $s_i:S_{n-1}\to S_n$, and degeneracy maps $d_i:S_{n+1}\to S_k$, for all $i=0,\dots,n$. For the first simplicial set of interest in this post, we have
\begin{align*}
\Sing_B(X)_n & = \Hom_{\Top}^B(|\Delta^n|,X), \\
\left(s_i\colon [n]\to [n-1]\right) & \mapsto \left( \begin{array}{c}
\left(|\Delta^{n-1}|\to X \right) \mapsto \left(|\Delta^n|\to X\right) \\
\text{collapses $i$th with $(i+1)$th vertex, then maps as source}
\end{array}\right)\\
\left(d_i\colon [n]\to [n+1]\right) & \mapsto  \left(\begin{array}{c}
\left(|\Delta^{n+1}|\to X \right) \mapsto \left(|\Delta^n|\to X\right) \\
\text{maps as $i$th face of source map}
\end{array}\right)
\end{align*}
We write $\Hom^B_{\Top}$ for the subset of $\Hom_{\Top}$ that respects the stratification $B$ in the context of entry paths. For the second simplicial set, the nerve, we have
\begin{align*}
N(SC)_n & = \{(S_0\tov{f_1} \cdots \tov{f_n} S_n)\ :\ S_i\in SC,\ f_i\ \text{are simplicial maps}\}, \\
\left(s_i\colon [n]\to [n-1]\right) & \mapsto \left( \left(S_0\tov{f_1}\cdots \tov{f_{n-1}} S_{n-1} \right) \mapsto \left(S_0\tov{f_1} \cdots\tov{f_i} S_i \tov{\id} S_i\tov{f_{i+1}} \cdots \tov{f_{n-1}} S_{n-1}\right)\right),\\
\left(d_i\colon [n]\to [n+1]\right) & \mapsto \left(\begin{array}{r l}
i=0: & \left(S_0\cdots S_{n+1} \right) \mapsto \left(S_1\tov{f_2}\cdots \tov{f_{n+1}} S_{n+1} \right) \\
0<i<n: & \left(S_0 \cdots S_{n+1} \right) \mapsto \left(S_0\tov{f_1} \cdots\tov{f_{i-1}} S_{i-1} \tov{f_{i+1}\circ f_i} S_{i+1} \tov{f_{i+2}} \cdots \tov{f_{n+1}} S_{n+1}\right) \\
i=n: & \left(S_0 \cdots  S_{n+1} \right) \mapsto \left(S_0\tov{f_1}\cdots \tov{f_n} S_n \right)
\end{array} \right). 
\end{align*}
 Define $F$ on $k$-simplices as \[ F\left(\gamma\colon |\Delta^k|\to \Ran^{\leqslant n}(M)\times \R_{>0}\right) = \left(\widetilde h(\gamma(1,0,\dots,0)) \tov{\left(\widetilde h\circ \gamma \circ s_k\circ \cdots \circ s_2\right)\left( |\Delta^1|\right)} \cdots \tov{\left(\widetilde h\circ \gamma \circ s_{k-2}\circ \cdots \circ s_0\right)\left(|\Delta^1|\right)} \widetilde h(\gamma(0,\dots,0,1))\right). \] A morphism in $\Sing_B(X)$ is a composition of face maps $s_i$ and degeneracy maps $d_i$, so $F$ must satisfy the commutative diagrams

for all $s_i$, $d_i$. Since the maps are unwieldy when in coordinates, we opt for heuristic arguments, neglecting to trace out notation-heavy diagrams.

Commutativity of the diagram on the left is immediate, as considering a simplex $|\Delta^{n-1}|$ as the $i$th face of a larger simplex $|\Delta^n|$ is the same as adding a step that is the identity map in the Hamiltonian path of vertices of $|\Delta^{n-1}|$. Similarly, observing that the image of the shortest path $v_{i-1}\to v_i\to v_{i+1}$ in $|\Delta^{n+1}|$, for $v_i = (0,\dots,0,1,0,\dots,0)$ the $i$th standard basis vector, induced by an element $\gamma\colon |\Delta^{n+1}|\to X$ in $\Sing_B(X)_{n+1}$, is homotopic to the image of the shortest path $v_{i-1} \to v_{i+1}$ shows that the diagram on the right commutes. Since $F$ is a natural transformation between the two functors $\Sing_B(X)$ and $N(SC)$, it is a functor on the functors as simplicial sets.

Remark: The particular choice of $X$ did not seem to play a large role in the arguments above. However, the stratifying map $\widetilde h\colon X\to B$ has image sitting inside $SC$, the nerve of which is the target of $F$, and every morphism in $\Sing_B(X)$ can be interpreted as a relation in $B\subseteq SC$ (both were necessary for the commutativity of the diagrams). Hence it is not unreasonable to expect a similar functor $\Sing_A(X)\to N(A')$ may exist for a stratified space $X\to A\subseteq A'$.

Functorial persistence

The goal of this post is to overcome some hurdles encountered by Bauer and Lesnick. In their approach, some geometric information is lost in passing from persistence modules to matchings. Namely, if an interval ends, we forget if the  $k$-cycle it represents becomes part of another $k$-cycle or goes to 0. Recall:

• $(\R,\leqslant)$ is the category of real numbers and unique morphisms $s\to t$ whenever $s\leqslant t$,
• $\Vect$ ($\BVect$) is the category of (based) finite dimensional vector spaces, and
• $\Set_*$ is the category of pointed sets.

We begin by recalling all the classical notions in the TDA pipeline.

Defintion: A persistence module is a functor $F:(\R,\leqslant)\to \Vect$. The barcode of a persistence module $F$ is a collection of pairs $(I,k)$, where $I\subseteq \R$ is an interval and $k\in \Z_{>0}$ is a positive integer.

Crawley-Boevey describes how to find the decomposition of a persistence module into interval modules. The $k$ for each $I$ is usually 1, but is 2 (and more) if the same interval appears twice (or more) in the decomposition. A barcode contains the same information as a \emph{persistence diagram}, though the former is drawn as horizontal bars and the latter is presented on a pair of axes.

Definition: A matching $\chi$ of barcodes $\{(I_i,k_i)\}_i$ and $\{(J_j,\ell_j)\}_j$ is a bijection $I'\to J'$, for some $I'\subseteq \{(I_i,k_i)\}_i$ and $J'\subseteq \{(J_j,\ell_j)\}_j$.

We write matchings as $\chi\colon \{(I_i,k_i)\}_i \nrightarrow \{(J_j,\ell_j)\}_j$.

Definition: A filtered persistence module is a functor $F:(\R,\leqslant) \to \BVect$ for which $F(s\leqslant t)(e_i) =f_j$ or 0, for every $e_i$ in the basis of $F(s)$ and $f_j$ in the basis of $F(t)$.

The notion of filtered persistence module is used for a stronger geometric connection. Indeed, for every filtered space $X$ the persistence module along this filtration is also filtered (once interval modules have been found), as then inclusions $X_s\hookrightarrow X_t$ will induce isomorphisms in homology onto their image. That is, a pair of homology classes from the source may combine in the target, but if the classes come from interval modules, a class from the source can not be in two non-homologous classes of the target.

Remark: The above dicussion highlights that choosing a basis in the definition of a persistence module already uses the decomposition of persistence modules into interval modules.

It is immediate that a morphism of persistence modules is a natural transformation. Let $\BPVect$ be the full subcategory of $\BVect$ consisting of elements in the image of some filtered persistence module (the objects are the same, we just have a restriction of allowed morphisms).

Definition:  Let $\mathcal B$ be the functor defined by \[ \begin{array}{r c l}
\mathcal B\colon \BPVect & \to & \Set_*, \\
(V,\{e_1,\dots,e_n\}) & \mapsto & \{0,1,\dots,n\}, \\
\left(\varphi:(V,\{e_i\}) \to (W,\{f_j\})\right) & \mapsto & \left(
i \mapsto \begin{cases}
j & \text{ if } \varphi(e_i) = f_j, \\ 0 & \text{ if } \varphi(e_i)=0 \text{ or } i=0.
\end{cases} \right)
\end{array} \]

The basepoint of every set in the image of $\mathcal B$ is 0.

Definition: Let $F,G$ be persistence modules and $\eta$ a morphism $F\to G$.

• The persistence diagram of $F$ is the functor $\mathcal B\circ F$.
• The matching induced by $\eta$ is the natural transformation $\mathcal B(\eta): \mathcal B\circ F\to \mathcal B\circ G$.

Bauer and Lesnick's definition of "matching" allow for more freedom to mix and match barcode intervals, but this also restricts how much information of a persistence module morphism can be tracked.

Example: The following example has a horizontal filtration with the degree 0 homology barcode on the left and the degree 1 homology barcode on the right. Linear maps of based vector spaces have also been shown to indicate how homology classes are born, die (column of zeros), and combine (row with more than one 1).

Example: Bauer and Lesnick present Example 5.6 to show that functoriality does not work in their setting. We reproduce their example and show that functoriality does work in our setting. Note that vertical ordering of the bars does not matter once they are named.

Apply the functor $\mathcal B$ to the whole diagram to get the matchings induced by $\eta$ and $\xi$, as below.

Next we hope to understand how interleavings fit into this setup.

References: Bauer and Lesnick (Induced matchings and the algebraic stability of persistence barcodes), Crawley-Boevey (Decomposition of pointwise finite-dimensional persistence modules)

New directions in TDA

 Conference topic

This post is informal, meant as a collection of (personally) new things from the workshop "Topological data analysis: Developing abstract foundations" at the Banff International Research Station, July 31 - August 4, 2017. New actual questions:

1. Does there exist a constructible sheaf valued in persistence modules over $\Ran^{\leqslant n}(M)$?
   • On the stalks it should be the persistence module of $P\in \Ran^{\leqslant n}(M)$. What about arbitrary open sets?
   • Is there such a thing as a colimit of persistence modules?
   • Uli Bauer suggested something to do with ordering the elements of the sample and taking small open sets.
2. Can framed vector spaces be used to make the TDA pipeline functorial? Does Ezra Miller's work help?
   • Should be a functor from $(\R,\leqslant)$, the reals as a poset, to $\text{Vect}$ or $\text{Vect}_{fr}$, the category of (framed) vector spaces. Filtration function $f:\R^n\to \R$ is assumed to be given.
   • Framed perspective should not be too difficult, just need to find right definitions.
   • Does this give an equivalence of categories (category of persistence modules and category of matchings)? Is that what we want? Do we want to keep only specific properties?
   • Ezra's work is very dense and unpublished. But it seems to have a very precise functoriality (which is not the main thrust of the work, however).
3. Can the Bubenik-de Silva-Scott interleaving categorification be viewed as a (co)limit? Diagrams are suggestive.
   • Reference is 1707.06288 on the arXiv.
   • Probably not a colimit, because that would be very large, though the arrows suggest a colimit.
   • Have to be careful, because the (co)limit should be in the category of posets, not just interleavings.

New things to learn about:

1. Algebraic geometry / homotopy theory: the etale space of a sheaf, Kan extensions, model categories, symmetric monoidal categories.
2. TDA related: Gromov-Hausdorff distance, the universal distance (Michael Lesnick's thesis and papers), merge trees, Reeb graphs, Mapper (the program).

Categories and the TDA pipeline

 Conference topic

This post contains topics and ideas from ACAT at HIM, April 2017, as presented by Professor Ulrich Bauer (see slide 11 of his presentation, online at ulrich-bauer.org/persistence-bonn-talk.pdf). The central theme is to assign categories and functors to analyze the process
\[
\text{filtration}\ \longrightarrow\ \text{(co)homology}\ \longrightarrow\ \text{barcode.}
\hspace{3cm}(\text{pipe}) \] Remark: The categories we will use are below. For filtrations, we have the ordered reals (though any poset $P$ would work) and topological spaces:
\begin{align*}
R\ :\ & \Obj(R) = \R,  & \Top\ :\ & \Obj(\Top) = \{\text{topological spaces}\}, \\[5pt]
& \Hom(r,s) = \begin{cases}
\{r \mapsto s\}, & \text{ if } r\leqslant s, \\ \emptyset, & \text{ else,}
\end{cases} && \Hom(X,Y) = \{\text{functions }f:X\to Y\}.
\end{align*}
For (co)homology groups, we have the category of (framed) vector spaces. We write $V^n$ for $V^{\oplus n} = V\oplus V\oplus \cdots \oplus V$, and $e_n$ for a frame of $V^n$ (see below).
\begin{align*}
\Vect\ :\ & \Obj(\Vect) = \{V^{\oplus n}\ :\ 0\leqslant n< \infty\},\\
& \Hom(V^n,V^m) = \{\text{homomorphisms }f:V^n\to V^m\}, \\[5pt]
\Vect^{fr}\ :\ & \Obj(\Vect^{fr}) = \{V^n\times e^n\ :\ 0\leqslant n<\infty\}, \\
& \Hom(V^n\times e^n,V^m\times e^m) = \{\text{hom. }f:V^n\to V^m,\ g:e^n\to e^m,\ g\in \Mat(n,m)\}.
\end{align*}
Finally for barcodes, we have $\Delta$, the category of finite ordered sets, and its variants. A partial injective function, or matching $f:A\nrightarrow B$ is a bijection $A'\to B'$ for some $A'\subseteq A$, $B'\subseteq B$.
\begin{align*}
\Delta\ :\ & \Obj(\Delta) = \{[n]=(0,1,\dots,n)\ :\ 0\leqslant n<\infty\},\\
& \Hom([n],[m]) = \{ \text{order-preserving functions }f:[n]\to [m]\}, \\[5pt]
\Delta'\ :\
& \Obj(\Delta')= \{a=(a_0<a_1<\cdots<a_n)\ :\ a_i\in \Z_{\geqslant 0}, 0\leqslant n<\infty\},\\ & \Hom(a,b) = \{\text{order-preserving functions }f:a\to b\}, \\[5pt]
\Delta''\ :\
& \Obj(\Delta'')= \{a=(a_0<a_1<\cdots<a_n)\ :\ a_i\in \Z_{\geqslant 0}, 0\leqslant n<\infty\},\\ & \Hom(a,b) = \{\text{order-preserving partial injective functions }f:a\nrightarrow b\}.
\end{align*}

Definition: A frame $e$ of a vector space $V^n$ is equivalently:

• an ordered basis of $V^n$,
• a linear isomorphism $V^n\to V^n$, or
• an element in the fiber of the principal rank $n$ frame bundle over a point.

Frames (of possibly different sizes) are related by full rank elements of $\Mat(n,m)$, which contains all $n\times m$ matrices over a given field.

Definition: Let $(P,\leqslant)$ be a poset. A (indexed topological) filtration is a functor $F:P\to \Top$, with
\[
\Hom(F(r),F(s)) = \begin{cases}
\{\iota:F(r) \hookrightarrow F(s)\}, & \text{ if }r\leqslant s, \\ \emptyset, & \text{ else,}
\end{cases}
\]
where $\iota$ is the inclusion map. That is, we require $F(r)\subseteq F(s)$ whenever $r\leqslant s$.

Definition: A persistence module is the composition of functors $M_i:P \tov{F} \Top \tov{H_i} \Vect$.

Homology will be taken over some field $k$. A framed persistence module is the same composition as above, but mapping into $\Vect^{fr}$ instead. The framing is chosen to describe how many different vector spaces have already been encountered in the filtration.

Definition: A barcode is a collection of intervals of $\R$. It may also be viewed as the composition of functors $B_i:P\tov{F}\Top\tov{H_i}\Vect \tov{\dim}\Delta$.

Similarly as above, we may talk about a framed barcode by instead mapping into $\Vect^{fr}$ and then to $\Delta''$, keeping track of which vector spaces we have already encountered. This allows us to interpret the process $(\text{pipe})$ in two different ways. First we have the unframed approach
\[
\begin{array}{r c c c l}
\Top & \to & \Vect & \to & \Delta, \\
X_t & \mapsto & H_i(X_t;k) & \mapsto & [\dim(H_i(X_t;k))].
\end{array}
\]
The problem here is interpreting the inclusion $X_t\hookrightarrow X_{t'}$ as a map in $\Delta$, for instance, in the case when $H_i(X_t;k)\cong H_i(X_{t'};k)$, but $H_i(X_t\hookrightarrow X_{t'}) \neq \id$. To fix this, we have the framed interpretation of $(\text{pipe})$
\[
\begin{array}{r c c c l}
\Top & \to & \Vect^{fr} & \to & \Delta'', \\
X_t & \mapsto & H_i(X_t;k)\times e & \mapsto & [e].
\end{array}
\]
The first map produces a frame $e$ of size $n$, where $n$ is the total number of different vector spaces encountered over all $t'\leqslant t$, by setting the first $\dim(H_i(X_t;k))$ coordinates to be the appropriate ones, and then the rest. This is done with the second map to $\Delta''$ in mind, as the size of $[e]$ is $\dim(H_i(X_t;k))$, with only the first $\dim(H_i(X_t;k))$ basis vectors taken from $e$. As usual, these maps are best understood by example.

Example: Given the closed curve $X$ in $\R^2$ below, let $\varphi:X\to \R$ be the height map from the line 0, with $X_i=\varphi^{-1}(-\infty,i]$, for $i=r,s,t,u,v$. Let $e_i$ be the standard $i$th basis vector in $\R^N$.

Remark: This seems to make $(\text{pipe})$ functorial, as the maps $X_t\hookrightarrow X_{t'}$ may be naturally viewed as partial injective functions in $\Delta''$, to account for the problem mentioned with the unframed interpretation. However, we have traded locality for functoriality, as the image of $X_t$ in $\Delta''$ can not be calculated without having calculated $X_{t'}$ for all $t'<t$.

References: Bauer (Algebraic perspectives of persistence), Bauer and Lesnick (Induced matchings and the algebraic stability of persistence barcodes)

Exactness and derived functors

 Lecture topic

Let $0\to X\to Y\to Z\to 0$ be a short exact sequence of objects in a category $A$. Let $\mathcal F:A\to B$ be a covariant functor.

Definition: The functor $\mathcal F$ is right-exact if $\mathcal F(X)\to\mathcal F(Y)\to \mathcal F(Z)\to 0$ is an exact sequence. It is left-exact if $0\to \mathcal F(X)\to\mathcal F(Y)\to \mathcal F(Z)$ is an exact sequence. It is exact if it is both left- and right-exact.

Example: These are some examples of left- and right-exact functors:
    $\Hom_A(X,-)$ is covariant left-exact
    $\Hom_A(-,X)$ is contravariant left-exact
    $-\otimes_R X$ is covariant right-exact, for $X$ a left $R$-module

Recall that $X\otimes_R Y$ is naturally isomorphic to $Y\otimes_RX$.

Definition: An object $X\in \Obj(A)$ is projective if $\Hom_A(X,-)$ is an exact functor. Similarly, $X$ is injective if $\Hom_A(-,X)$ is an exact functor.

Recall that a projective resolution of an object $X$ is a sequence of projective objects $\cdots\to P_2\to P_1\to P_0$ that may or may not terminate on the left. The homology of the sequence in degree 0 is $X$, and trivial in other degrees. Similarly, an injective resolution of $X$ is a sequence of injective objects $I_0\to I_1\to I_2\to\cdots$ that may or may not terminate on the right. The cohomology is also concentrated in degree 0, and is $X$ there. A free resolution is a projective resolution where all the objects are free (whatever that means in the context).

These types of resolutions may not exist. A category "has enough injectives (projectives)" means we can always construct injective (projective) resolutions.

Definition: Let $\mathcal F:A\to B$ be a covariant right-exact functor and $\mathcal G:A\to B$ a covariant left-exact functor. Let $X\in \Obj(A)$ with $P_\bullet$ a projective resolution of $X$ and $I_\bullet$ an injective resolution of $X$. The $i$th left-derived functor of $\mathcal F$ is $L_i\mathcal F(X) = H_i(\mathcal F(P_\bullet))$. The $i$th right-derived functor of $\mathcal G$ is $R^i\mathcal G(X) = H^i(\mathcal G(I_\bullet))$.

These objects of $B$ are well-defined up to natural isomorphism. Note that $\mathcal F^{op}:A^{op}\to B^{op}$ is a contravariant right-exact functor. Moreover, if $\mathcal F$ was contravariant right-exact and $\mathcal G$ was contravariant left-exact, then $L_i\mathcal F(X)=H_i(\mathcal F(I_\bullet))$ and $R^i\mathcal G(X)=H^i(\mathcal G(P_\bullet))$.

Example: Let $R$ be a ring with $X$ and $Y$ both $R$-bimodules. Then
\begin{align*}
\Tor_i^R(Y,X) & =  L_i(-\otimes_RX)(Y) &
\Ext^i_R(X,Y) & = R^i(\Hom_R(X,-))(Y) \\
& = L_i(Y\otimes_R - )(X),
&& = R^i(\Hom_R(-,Y))(X).
\end{align*}
Recall that $\Tor_i^R(Y,X)$ is canonically isomorphic to $\Tor_i^R(X,Y)$, but it is not true for $\Ext$. Also note that $\Hom_R(X,-)$ is covariant and $\Hom_R(-,Y)$ is contravariant, while $-\otimes_R X$ and $Y\otimes_R -$ are both covariant functors.
References: Weibel (An introduction to homological algebra, Chapter 2)

The Eilenberg-Steenrod axioms

The category $\text{Top}$ of topological spaces may be generalized to the category $\text{Top}_*$ of pointed topological spaces. This in turn may be generalized to the category $\text{Top}_{rel}$ of pairs $(X,A)$, where $X\in\Obj(\text{Top})$ and $A$ is a subspace of $X$. The morphisms of $\text{Top}_{rel}$ on $(X,A)$ are the morphisms of $\text{Top}$ on $X$ paired with their restrictions to $A$. We write $(X)$ for $(X,\emptyset)$.

Definition 1: Let $X,Y\in\Obj(\text{Top}_*)$. Then $f\in\Hom_{\text{Top}_*}(X,Y)$ is an $n$-equivalence if the induced map on homotopy groups $f_*:\pi_k(X,x)\to \pi_k(Y,f(x))$ is an isomorphism for $k<n$ and an epimorphism for $k=n$. Further, $f$ is a weak equivalence if it is an $n$-equivalence for all $n\geqslant 1$. Similarly, $f\in \Hom_{\text{Top}_{rel}}((X,A),(Y,B))$ is a weak equivalence if $f\in \Hom_{\text{Top}_*}(X,Y)$ and $f|_A\in \Hom_{\text{Top}_*}(A,B)$ are weak equivalences.

Definition 2: Let $C,D$ be two categories. A functor $\mathcal F:C\to D$ is an assignment $\mathcal F(X)\in \Obj(D)$ for every $X\in \Obj(C)$, and $\mathcal F(f)\in \Hom_D(\mathcal F(X),\mathcal F(Y))$ for every $f\in\Hom_C(X,Y)$. This assignment satisfies the following relations:
          $\mathcal F(g\circ f) = \mathcal F(g)\circ \mathcal F(f)$ for every $f\in \Hom_C(X,Y)$ and $g\in \Hom_C(Y,Z)$
          $\mathcal F(\id_X) = \id_{\mathcal F(X)}$ for every $X\in\Obj(C)$

Definition 3: Let $C$ be any category and $\mathcal F:\text{Top}\to C$ a functor. Then $\mathcal F$ is homotopy invariant if $f\simeq g$ in $\text{Top}$ implies $\mathcal F(f)=\mathcal F(g)$ in $C$, where $\simeq$ is the homotopy of maps.

Definition 4: A (relative) homology theory of topological spaces is a collection of homotopy-invariant functors $H_n:\text{Top}_{rel}\to \text{Ab}$ and a collection of natural transformations $d_n:H_n(X,A) \to H_{n-1}(A)$.

The Eilenberg-Steenrod axioms are properties a relative homology theory may satisfy. The number of axioms depends on how general a view of homology theories one would like. Eilenberg and Steenrod (7), May (4), Aguilar, Gitler, and Prieto (4), Wikipedia (5), and other sources (6,8) have all different numbers of axioms. The order of the axioms below is alphabetical.

For any $(X,A)\in\Obj(\text{Top}_{rel})$ and all $n$:

Axiom 1: Additivity. If $(X,A)=\bigoplus_i(X_i,A_i)$, then $H_n(X,A) \cong \bigoplus_iH_n(X_i,A_i),$ where the isomorphism is induced by the inclusions $(X_i,A_i)\hookrightarrow (X,A)$.

Axiom 2: Exactness. There is a long exact sequence
\[ \cdots \to H_{n+1}(X,A)\tov{d_{n+1}}H_n(A)\tov{\ \ }H_n(X)\tov{\ \ }H_n(X,A)\tov{d_n}H_{n-1}(A)\tov{\ \ }\cdots \]
where $H_n(A)\to H_n(X)$ and $H_n(X)\to H_n(X,A)$ are induced by the inclusions $(A)\hookrightarrow (X)$ and $(X)\hookrightarrow (X,A)$, respectively.

Axiom 3: Excision. If there exists a subset $U$ of $X$ with $\text{cl}(U)\subset \text{int}(A)$, then there is an isomorphism $H_n(X\setminus U,A\setminus U)\cong H_n(X,A)$ induced by the inclusion $(X\setminus U,A\setminus U)\hookrightarrow (X,A)$.

Axiom 4: Dimension. $H_n(*)=0$ for all $n\neq 0$.

Axiom 5: Weak equivalence. If $f\in\Hom_{\text{Top}_{rel}}((X,A),(Y,B))$ is a weak equivalence, then the induced map on homology $f_*:H_n(X,A)\to H_n(Y,B)$ is an isomorphism.

Singular homology is a homology theory that satisfies all the axioms above. $K$-theory is a homology theory that does not satisfy the dimension axiom.

References: May (A Concise course in Algebraic Topology, Chapter 13.1), Aguilar, Gitler, and Prieto (Algebraic Topology from a Homotopical Viewpoint, Chapter 5.3)