This will be old news to anyone who does algebraic topology, but oddly enough I can’t seem to find it all written in one place anywhere, and in particular I can’t find the bit about $\mathsf{hPairTop}$ at all.

In algebraic topology you (for example) associate every topological space $X$ with a group, like $\pi_1(X, x_0)$ or $H_5(X)$. All of these operations turn out to be functors. This isn’t surprising, because as far as I’m concerned the definition of a functor is “any time you take one type of object and naturally make another object”.

The surprise is that these objects also respect homotopy in a nice way; proving this is a fair amount of the “setup” work in algebraic topology.

1. Homology, $H_n : \mathsf{hTop} \rightarrow \mathsf{Grp}$

Note that $H_5$ is a functor $$H_5 : \mathsf{Top} \rightarrow \mathsf{Grp}$$ i.e. to every space $X$ we can associate a group $H_5(X)$. (Of course, replace $5$ by integer of your choice.) Recall that:

Definition 1. Two maps $f, g : X \rightarrow Y$ are homotopy equivalent if there exists a homotopy between them.

Thus for a map we can take its homotopy class $[f]$ (the equivalence class under this relationship). This has the nice property that $[f \circ g] = [f] \circ [g]$ and so on.

Definition 2. Two spaces $X$ and $Y$ are homotopic if there exists a pair of maps $f : X \rightarrow Y$ and $g : Y \rightarrow X$ such that $[f \circ g] = [\mathrm{id}_X]$ and $[g \circ f] = [\mathrm{id}_Y]$.

In light of this, we can define

Definition 3. The category $\mathsf{hTop}$ is defined as follows:

• The objects are topological spaces $X$.
• The morphisms $X \rightarrow Y$ are homotopy classes of continuous maps $X \rightarrow Y$.

Remark 4. Composition is well-defined since $[f \circ g] = [f] \circ [g]$. Two spaces are isomorphic in $\mathsf{hTop}$ if they are homotopic.

Remark 5. As you might guess this “quotient” construction is called a quotient category.

Then the big result is that:

Theorem 6. The induced map $f_\sharp = H_n(f)$ of a map $f: X \rightarrow Y$ depends only on the homotopy class of $f$. Thus $H_n$ is a functor $$H_n : \mathsf{hTop} \rightarrow \mathsf{Grp}.$$

The proof of this is geometric, using the so-called prism operators. In any case, as with all functors we deduce

Corollary 7. $H_n(X) \cong H_n(Y)$ if $X$ and $Y$ are homotopic.

In particular, the contractable spaces are those spaces $X$ which are homotopy equivalent to a point. In which case, $H_n(X) = 0$ for all $n \ge 1$.

2. Relative Homology, $H_n : \mathsf{hPairTop} \rightarrow \mathsf{Grp}$

In fact, we also defined homology groups $$H_n(X,A)$$ for $A \subseteq X$. We will now show this is functorial too.

Definition 8. Let $\varnothing \neq A \subset X$ and $\varnothing \neq B \subset X$ be subspaces, and consider a map $f : X \rightarrow Y$. If $f(A) \subseteq B$ we write $$f : (X,A) \rightarrow (Y,B).$$ We say $f$ is a map of pairs, between the pairs $(X,A)$ and $(Y,B)$.

Definition 9. We say that $f,g : (X,A) \rightarrow (Y,B)$ are pair-homotopic if they are “homotopic through maps of pairs”.

More formally, a pair-homotopy $f, g : (X,A) \rightarrow (Y,B)$ is a map $F : [0,1] \times X \rightarrow Y$, which we’ll write as $F_t(X)$, such that $F$ is a homotopy of the maps $f,g : X \rightarrow Y$ and each $F_t$ is itself a map of pairs.

Thus, we naturally arrive at two categories:

• $\mathsf{PairTop}$, the category of pairs of topological spaces, and
• $\mathsf{hPairTop}$, the same category except with maps only equivalent up to homotopy.

Definition 10. As before, we say pairs $(X,A)$ and $(Y,B)$ are pair-homotopy equivalent if they are isomorphic in $\mathsf{hPairTop}$. An isomorphism of $\mathsf{hPairTop}$ is a pair-homotopy equivalence.

Then, the prism operators now let us derive

Theorem 11. We have a functor $$H_n : \mathsf{hPairTop} \rightarrow \mathsf{Grp}.$$ The usual corollaries apply.

Now, we want an analog of contractable spaces for our pairs: i.e. pairs of spaces $(X,A)$ such that $H_n(X,A) = 0$ for $n \ge 1$. The correct definition is:

Definition 12. Let $A \subset X$. We say that $A$ is a deformation retract of $X$ if there is a map of pairs $r : (X, A) \rightarrow (A, A)$ which is a pair homotopy equivalence.

Example 13 (Examples of Deformation Retracts)

1. If a single point $p$ is a deformation retract of a space $X$, then $X$ is contractable, since the retraction $r : X \rightarrow \{*\}$ (when viewed as a map $X \rightarrow X$) is homotopic to the identity map $\mathrm{id}_X : X \rightarrow X$.
2. The punctured disk $D^2 \setminus \{0\}$ deformation retracts onto its boundary $S^1$.
3. More generally, $D^{n} \setminus \{0\}$ deformation retracts onto its boundary $S^{n-1}$.
4. Similarly, $\mathbb R^n \setminus \{0\}$ deformation retracts onto a sphere $S^{n-1}$.

Of course in this situation we have that $$H_n(X,A) \cong H_n(A,A) = 0.$$

3. Homotopy, $\pi_1 : \mathsf{hTop}_\ast \rightarrow \mathsf{Grp}$

As a special case of the above, we define

Definition 14. The category $\mathsf{Top}_\ast$ is defined as follows:

• The objects are pairs $(X, x_0)$ of spaces $X$ with a distinguished basepoint $x_0$. We call these pointed spaces.
• The morphisms are maps $f : (X, x_0) \rightarrow (Y, y_0)$, meaning $f$ is continuous and $f(x_0) = y_0$.

Now again we mod out:

Definition 15. Two maps $f , g : (X, x_0) \rightarrow (Y, y_0)$ of pointed spaces are homotopic if there is a homotopy between them which also fixes the basepoints. We can then, in the same way as before, define the quotient category $\mathsf{hTop}_\ast$.

And lo and behold:

Theorem 16. We have a functor $$\pi_1 : \mathsf{hTop}_\ast \rightarrow \mathsf{Grp}.$$ Same corollaries as before.