Preliminary fact.
Let $d:\sf I\to Set$ be a diagram of sets ($\sf I$ is a small category). Suppose that (1) for any object $i$ in $\sf I$, is given an equivalence relation $\sim_i$ on the set $d(i)$, and that (2) for any arrow $l:i\to i'$ in $\sf I$, if $x,x'\in d(i)$ satisfy $x\sim_ix'$, then $d(l)(x)\sim_{i'} d(l)(x')$. Let $t(i):=d(i)/\sim _i$, and let $t(l):t(i)\to t(i')$ be the map induced by $d(l)$; denote by $t:\sf I\to Set$ the diagram obtained.
Observe that the relations $\sim _i$ yield an equivalence relation $\sim$ on $\prod_{i\in ob(\mathsf I)}d(i)$, and moreover $\prod_{i\in ob(\mathsf I)}d(i)/\sim$ is equal to $\prod_{i\in ob(\mathsf I)}t(i)$.
Now consider $\lim d$, which is a subset of $ \prod_{i\in ob(\mathsf I)}d(i)$. $\sim$ induces an equivalence relation on $\lim d$, which we denote again $\sim$, but it isn't true, in general, that $\lim d/\sim $ is equal to $\lim t$.
Finally let $S$ be a set and $f:S\to \prod_{i\in ob(\mathsf I)}d(i)$ be a map. Suppose that there is an equivalence relation $\sim_S$ on $S$, and that, if $s,s'\in S$ satisfy $s\sim _S s'$, then $f(s)\sim f(s')$. If $f$ induces a bijection $S\to \lim d$, it induces a bijection $S/\sim_S\to \lim d/\sim$, and in case that $\sf I$ is discrete, this is a bijection $S/\sim_S\to \lim t$.
Question.
Let $\sf Top$ be the category of topological spaces. Recall that two continuous maps $f,g\in \mathsf {Top}(X,Y)$ are said homotopic if there is a continuous map $H:X\times I\to Y$ such that $H(x,0)=f(x)$ and $H(x,1)=g(x)$, for all $x\in X$. Homotopy is an equivalence relation, which respects composition: then let $h\sf Top$ be the quotient category of $\sf Top$ by the homotopy relation.
Let $\{X_i\}_{i\in I}$ be a family of topological spaces, and consider the product $X:=\prod_{i\in I}X_i$, with the canonical maps $p_i:X\to X_i$. There is a natural bijection $\mathsf{Top}(-,X)\Rightarrow \prod_{i\in I}\mathsf{Top}(-,X_i)$, defined by composition with the $p_i$. The last paragraph of the preliminary fact shows that:
- it is induced a bijection $h\mathsf{Top}(Y,X)\to \prod_{i\in I}h\mathsf{Top}(Y,X_i)$ at all $Y$ (hence a natural bijection $h\mathsf{Top}(-,X)\Rightarrow \prod_{i\in I}h\mathsf{Top}(-,X_i)$, i.e. the projection $\mathsf {Top}\to h\sf Top$ preserves products);
- the point above may not be true replacing the product with another limit.
If instead of $\sf Top$ we work with the category of pointed spaces, and use pointed homotopy, and want to prove that $\pi:\mathsf{Top}_*\to h\sf Top_*$ preserves products, I don't see what could go differently from above. In Tammo tom Diecks' Algebraic Topology, page 31, is suggested to show that $\pi$ preserves products using this result (Proposition 2.1.6 on the book):
If $p:X\to Y$ is a quotient map of topological spaces, and $H:Y\times I\to Z$ is a map of sets such that $H\circ (p\times \mathrm{id}): X\times I\to Z$ is continuous, then $H$ is continuous.
Do you think that my proof is ok? I fear not, because I wouldn't see how to link the indented result to $\pi$ preserving products, and I don't think it is used implicitly in my proof somewhere. Maybe my argument works in $\sf Top$, but not in $\sf Top_*$? But as I said, I can't see where it fails. Thanks for any clarify.
You do not need 2.1.6 for the (pointed and unpointed) product case nor for the unpointed sum case, but you need it to prove that $\pi:\mathsf{Top}_*\to h\sf Top_*$ preserves sums. This is what tom Dieck says:
The problem is this:
Each family of pointed homotopies $H_j : (X_j,*) \times I \to (Y,*)$ induces a function $H : (\bigvee_ j(X_j,*)) \times I \to (Y,*)$. Show that $H$ is continuous.
The wedge $\bigvee_ j(X_j,*)$ is the quotient of the disjoint sum $\coprod_j X_j$ obtained by identifying all basepoints of the $X_j$ to a single point. Clearly the $H_j$ induce a continuous $\tilde H : (\coprod_j X_j) \times I \to Y$, and now we need 2.1.6 to see that $H$ is continuous.