Homotopy Theory

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Jonathan Beardsley

12:31 AM

@TimCampion Ah nice okay, so every complete simplicially enriched category is also cotensored over simplicial sets.

2 hours later…

Alexander Campbell

2:54 AM

@JonathanBeardsley Are you invoking Kelly Theorem 3.46 here? Note that sSet does not satisfy the hypotheses of that theorem, since the "underlying set" functor sSet --> Set is not conservative.

Jonathan Beardsley

Oops...

yep nevermind.

Alexander Campbell

The category of reduced simplicial sets (i.e. with a single 0-simplex) should be a counterexample.

Jonathan Beardsley

anyway i think it's okay, because i think in my case i have a concrete candidate for the cotensor

Alexander Campbell

What's your case?

Jonathan Beardsley

i wanted an explicit description anyway,.

slice in a simplicial model category

Q: (Co)tensoring of enriched slice categories

In an answer to this question: Enriched slice categories, a description of the enrichment of the slice category in an enriched category is given. I'm interested in going a bit further. If we assume that the original enriched category, say $\mathcal{C}$, is $\mathcal{V}$-enriched but also $\mathca...

Alexander Campbell

2:56 AM

Ah, I see.

So that's not too hard to describe. In your notation, the cotensor of $f : A \to X$ by $K$ is the pullback of $f^K : A^K \to X^K$ along the "diagonal" map $X \to X^K$.

Jonathan Beardsley

I sort of think the cotensor of $g:B\to X$ with $K$ should be the map obtained by pulling back $g^K:B^K\to X^K$ along $X\to X^K$

Alexander Campbell

Yep, exactly that.

Jonathan Beardsley

OMG i got it right all on my own

Haha, I had to think really hard about the case for $A,X$ both spaces and $K=\Delta^1$ to come up with that idea.

@AlexanderCampbell it should be true that $Map(A,-)$ takes pullbacks in your enriched category to pullbacks in $sSet$ right?

I guess this follows from some general functorial way of writing down the functor $Map(-,-)$ as some kind of bifunctor from $C\times C^{op}$ to $sSet$ or something?

Alexander Campbell

That depends on what you mean by pullback. You could mean it is just the pullback in the underlying ordinary category, or that it has the "enriched' universal property of a pullback.

Jonathan Beardsley

oh sorry of course this is true, precisely because $C$ is tensored over $sSet$

Alexander Campbell

3:03 AM

It's only true in general in the latter case.

Jonathan Beardsley

Ah... uhoh...

Alexander Campbell

However, if, as you say, $C$ is tensored over the base, then all "ordinary" limits are automatically "enriched" limits, so you're safe.

Jonathan Beardsley

Phew.

Alexander Campbell