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)

Unit and counit adjunction

 Lecture topic

 Let $\mathcal F:C\to D$ and $\mathcal G:D\to C$ be adjoint functors. That is, let $\mathcal F$ be left-adjoint to $\mathcal G$, and let $\mathcal G$ be right-adjoint to $\mathcal F$, so that $\Hom_D(\mathcal F(X),Y)\cong\Hom_C(X,\mathcal G(Y))$ for any $X\in\Obj(C)$ and $Y\in \Obj(D)$.

This isomorphism gives natural maps $\eta_X$, from \begin{align*} \Hom_D(\mathcal F(X),\mathcal F(X)) & \cong \Hom_C(X,\mathcal G(\mathcal F(X)),\\ \id_{\mathcal F(X)} & \mapsto \left(X\tov{\eta_X}(\mathcal G\circ \mathcal F)(X)\right), \end{align*} and $\epsilon_Y$, from \begin{align*} \Hom_C(\mathcal G(Y),\mathcal G(Y)) & \cong \Hom_D(\mathcal F(\mathcal G(Y)),Y), \\ \id_{\mathcal G(Y)} & \mapsto \left((\mathcal F\circ \mathcal G)(Y) \tov{\epsilon_Y}Y\right). \end{align*} These may be viewed as natural transformations called the unit $\eta$ and the counit $\epsilon$, \[ \eta:1_C \to \mathcal G\circ \mathcal F \hspace{2cm} \epsilon: \mathcal F\circ \mathcal G \to 1_D. \] They satisfy the triangle identities, that is, the following diagrams commute.