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Stream: learning: questions

Topic: small products in model categories

Elif Uskuplu (Mar 28 2023 at 19:21):

Hi,
I don't know much about model categories, but as I've been working from Hovey's book, I'd like to ask a question. As in the attached file, Screenshot-from-2023-03-28-12-00-14.png we have the homotopy category of a model category has all small products.

From this, I can talk about countable products in particular. Well, can we say the following?
If the maps $f_n,g_n:B\rightarrow X_n$ are left homotopic for each $n\in\mathbb{N}$, and $f=\prod f_n$ and $g=\prod g_n$, then $f,g:B\rightarrow \prod X_n$ are left homotopic. It seems right to me, but I'm not sure about the details or extra assumptions, if needed. Any explanation or correction would be great.

Mike Shulman (Mar 29 2023 at 04:12):

This is true if $B$ is cofibrant and each $X_n$ is fibrant. In that case, there's a lemma saying that if $f_n \sim g_n$ then that left homotopy can be witnessed by any desired cylinder object for $B$. So if you pick a single cylinder object $\tilde{B}$ for $B$ and use it for all the homotopies $f_n \sim g_n$, then you can induce a map $\tilde{B} \to \prod_n X_n$ and deduce $f\sim g$.

Elif Uskuplu (Mar 29 2023 at 06:47):

Got it, thanks.

Reid Barton (Mar 29 2023 at 06:59):

And, it's not true in general. For instance in simplicial sets, if $B$ is two points and $X_n$ is a path/string of 1-simplices of length $n$, and $f_n$ picks the first vertex of the each path and $g_n$ the last vertex. Then $f_n \sim g_n$ for each $n$ but in the product $f$ and $g$ are actually points in two different connected components, basically because any path in the product has finite length so it's too short to connect the endpoints after projecting to $X_n$ for large $n$.

Reid Barton (Mar 29 2023 at 07:01):

So you have to do a fibrant replacement before taking infinite products, to get a homotopy invariant construction.
In sSet you don't have to do fibrant replacement to form "finite homotopy products", but in a general model category you could have to.