## Homotopy and the Homotopy Category (Pt. III)

**Point of Post: **This is a continuation of this post.

**The Homotopy Category**

The notion of homotopy allows us to define a new category— the homotopy category . The objects of are still topological spaces, but now the arrows in are not continuous maps but instead are homotopy classes of continuous maps –thus, . The composition law in is the obvious–representative wise composition (function composition!) of homotopy classes. In other words, if then we define to be . We claim that this is well-defined:

**Theorem: ***Let and be such that and , then .*

**Proof: **Suppose that is a homotopy from to and is a homotopy from to . Define the map defined by . Clearly is continuous since it is the composition of the maps where is the map defined by (which is continuous since each coordinate function is continuous). Now, it’s easy to see that

and

from where it follows that is a homotopy from to and so the theorem follows.

Now, it’s clear that the identity arrow for each object in is the homotopy class . It totally makes sense, at least with this definition, why the notion of homotopy equivalence was defined as it was. Namely, it’s clear that the notion of a homotopy equivalence is precisely what it means for to be an isomorphism in .

**References:**

[1] Spanier, Edwin Henry. *Algebraic Topology*. New York: McGraw-Hill, 1966. Print.

[2] Hatcher, Allen. *Algebraic Topology*. Cambridge: Cambridge UP, 2002. Print.

[3] Bredon, Glen E. *Topology and Geometry*. New York: Springer-Verlag, 1993. Print.

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