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 at all.

In algebraic topology you (for example) associate every topological space with a group, like or . 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,

Note that is a functor

i.e. to every space we can associate a group . (Of course, replace by integer of your choice.) Recall that:

Thus for a map we can take its **homotopy class** (the equivalence class under this relationship). This has the nice property that and so on.

**Definition 2**

Two spaces and are **homotopic** if there exists a pair of maps and such that and .

In light of this, we can define

**Definition 3**

The category is defined as follows:

- The objects are topological spaces .
- The morphisms are
*homotopy classes*of continuous maps .

**Remark 4**

Composition is well-defined since . Two spaces are isomorphic in if they are homotopic.

Then the big result is that:

**Theorem 6**

The induced map of a map depends only on the homotopy class of . Thus is a functor

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

**Corollary 7**

if and are homotopic.

In particular, the *contractible* spaces are those spaces which are homotopy equivalent to a point. In which case, for all .

## 2. Relative Homology,

In fact, we also defined homology groups

for . We will now show this is functorial too.

**Definition 8**

Let and be subspaces, and consider a map . If we write

We say is a **map of pairs**, between the pairs and .

**Definition 9**

We say that are **pair-homotopic** if they are “homotopic through maps of pairs”.

More formally, a **pair-homotopy** is a map , which we’ll write as , such that is a homotopy of the maps and each is itself a map of pairs.

Thus, we naturally arrive at two categories:

- , the category of
*pairs*of toplogical spaces, and - , the same category except with maps only equivalent up to homotopy.

**Definition 10**

As before, we say pairs and are **pair-homotopy equivalent** if they are isomorphic in . An isomorphism of is a **pair-homotopy equivalence**.

Then, the prism operators now let us derive

**Theorem 11**

We have a functor

The usual corollaries apply.

Now, we want an analog of contractible spaces for our pairs: i.e. pairs of spaces such that for . The correct definition is:

**Definition 12**

Let . We say that is a **deformation retract** of if there is a map of pairs which is a pair homotopy equivalence.

**Example 13** **(Examples of Deformation Retracts)**

- If a single point is a deformation retract of a space , then is contractible, since the retraction (when viewed as a map ) is homotopic to the identity map .
- The punctured disk deformation retracts onto its boundary .
- More generally, deformation retracts onto its boundary .
- Similarly, deformation retracts onto a sphere .

Of course in this situation we have that

## 3. Homotopy,

As a special case of the above, we define

**Definition 14**

The category is defined as follows:

- The objects are pairs of spaces with a distinguished basepoint . We call these
**pointed spaces**. - The morphisms are maps , meaning is continuous and .

Now again we mod out:

**Definition 15**

Two maps 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 .

And lo and behold:

**Theorem 16**

We have a functor

Same corollaries as before.