# Abstract Nonsense

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

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

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The Homotopy Category

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The notion of homotopy allows us to define a new category— the homotopy category $\mathbf{hTop}$. The objects of $\mathbf{hTop}$ are still topological spaces, but now the arrows $X\to Y$ in $\mathbf{hTop}$ are not continuous maps $X\to Y$ but instead are homotopy classes of continuous maps $X\to Y$–thus, $\text{Hom}_{\mathbf{hTop}}(X,Y)=[X,Y]$. The composition law in $\mathbf{hTop}$ is the obvious–representative wise composition (function composition!) of homotopy classes. In other words, if $X\xrightarrow{[f]}Y\xrightarrow{[g]}Z$ then we define $[g]\circ[f]$ to be $[g\circ f]$. We claim that this is well-defined:

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Theorem: Let $f,g:X\to Y$ and $h,k:Y\to Z$ be such that $f\simeq g$ and $h\simeq k$, then $h\circ f\simeq k\circ g$.

Proof: Suppose that $H$ is a homotopy from $f$ to $g$ and $F$ is a homotopy from $h$ to $k$. Define the map $G:X\times[0,1]\to Z$ defined by $G(x,t)=F(H(x,t),t)$. Clearly $G$ is continuous since it is the composition of the maps $F\circ R$ where $R$ is the map $X\times[0,1]\to Y\times[0,1]$ defined by $R(x,t)=(H(x,t),t)$ (which is continuous since each coordinate function is continuous). Now, it’s easy to see that

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$G(x,0)=F(H(x,0),0)=F(f(x),0)=h(f(x))=(h\circ f)(x)$

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and

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$G(x,1)=F(H(x,1),1)=F(g(x),1)=k(g(x))=(k\circ g)(x)$

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from where it follows that $G$ is a homotopy from $h\circ f$ to $k\circ g$ and so the theorem follows. $\blacksquare$

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Now, it’s clear that the identity arrow for each object $X$ in $\mathbf{hTop}$ is the homotopy class $[\text{id}_X]\in[X,X]$. 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 $f:X\to Y$ is precisely what it means for $[f]\in [X,Y]$ to be an isomorphism in $\mathbf{hTop}$.

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References:

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[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.