I started following a course on Algebraic Topology without having any background in Category Theory, hence I am finding some difficulties in the exercises.
Let $G:$ Top $\to$ hTop be the usual functor.
I have the following exercise.
Let F : Top → C be a functor. TFAE:
(i) F is homotopy invariant, i.e. $f ≃ g$ implies $F(f) = F(g)$.
(ii) F sends homotopy equivalences to isomorphisms.
(iii) For every $X ∈$ Top, $F$ sends the projection $X × I → X$ to an isomorphism.
(iv) $F$ factors through the homotopy category hTop, i.e., there exists a functor $\overline{F}$ : hTop →
C such that $\overline{F} \circ G = F$.
Now, I know how to prove (i) $\Rightarrow$(ii), (ii) $\Rightarrow$ (iii) (I showed that the projection is a homotopy equivalence) and (iv) $\Rightarrow$ (i), but I am stuck on the implication (iii) $\Rightarrow$ (iv). It seems more natural to prove (i) $\Rightarrow$ (iv), but then I wouldn't know how to close the series of implications.
Any help would be highly appreciated. Thanks!
Best Answer
I think it is fairly obvious that (i) and (iv) are equivalent. Thus it suffices to prove (iii) $\implies$ (i).
Let $f, g : X \to Y$ be homotopic maps and $H : X \times I \to Y$ be a homotopy from $f$ to $g$. That is, if we define $i_t : X \to X \times I, i_t(x) = (x,t)$, we have $Hi_0=f$ and $Hi_1=g$.
Obviuosly $pi_t = id$ for all $t$, thus $F(pi_t) = F(p)F(i_t) = id$ which implies $F(i_t) = F(p)^{-1}$. We get $$F(f) = F(Hi_0) = F(H)F(i_0) = F(H)F(p^{-1}) = F(H)F(i_1) = F(Hi_1) = F(g) .$$