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It seems that there is a general sense in which $\pi_0$ preserves finite products, whether $\pi_0$ is acting on topological spaces or simplicial sets or categories or groupoids. I believe taking the skeleton of a category or a preorder also preserves finite products. Is there a common reason for all this?
Tangentially related, does $Ho$ preserve finite products, considered as a functor from relative categories (in the sense of Barwick and Kan [2012]) to categories? If it did then we would have a quintuple adjoint string $Ho \dashv min \dashv und \dashv max \dashv weq : \mathbf{RelCat} \to \mathbf{Cat}$ with the leftmost adjoint preserving finite products – and so we would have something that looks very much like a cohesion structure.
Let’s see: $\pi_0$ for spaces is a reflexive coequalizer, starting with an obvious coreflexive pair involving the interval $I$:
$Top(I, X) \stackrel{\to}{\to} Top(1, X) \to \pi_0(X)$and of course reflexive coequalizers commute with finite limits. Suppose we replace the interval category $\mathbf{2}$ with the codiscrete groupoid $K(2)$ on two objects? It seems to me the corresponding reflexive coequalizer of
$C^{K(2)} \stackrel{\to}{\to} C^1$should be the skeleton of $C$. (I’d never thought of it this way, though.)
Edit: possibly not. I just woke up.
Taking the coequaliser of $C^{K(2)} \rightrightarrows C^1$ kills all the automorphisms, though. But this certainly explains why taking the set of isomorphism classes in a category preserves products. Thanks!
The functor $Ho : \mathbf{RelCat} \to \mathbf{Cat}$ does preserve finite products. This is an exercise in adjointness using the fact that both $\mathbf{RelCat}$ and $\mathbf{Cat}$ are cartesian closed.
Yeah, that was what I was worried about. But it works for preorders.
@Karol: I’m afraid I don’t follow. I can see that $min$ is a cartesian closed functor, which shows that $Ho$ satisfies various Frobenius reciprocity conditions, e.g.
$und ([Y, min Z]_{rel}) \cong [Ho Y, Z]$ $X \times Ho Y \cong Ho (min X \times Y)$and also that $\operatorname{Ho} 1 \cong 1$, but don’t we need to know a little bit more before we can conclude that $Ho$ preserves binary products?
Postscript. Ah, I see now. We need to use the fact that $[Y, min Z]_{rel} \cong min([Ho Y, Z])$ as relative categories. What’s the abstract nonsense way of saying this?
I’m not sure I follow your arguments and I’m also not sure I understand your notation. Is $min$ supposed to stand for “minimal relative structure”? If so then $Ho$ is not a left adjoint of $min$. Minimal relative structure has just identities as weak equivalences and $Ho$ is a left adjoint of a functor $iso$ that sends a category to a relative category with isomorphisms as weak equivalences.
Now, the exercise in adjointness I had in mind is the following. Fix a relative category $A$ and consider a square
$\array{ \mathbf{RelCat} & \stackrel{- \times A}{\to} & \mathbf{RelCat} \\ \downarrow^{Ho} & & \downarrow^{Ho} \\ \mathbf{Cat} & \stackrel{- \times Ho A}{\to} & \mathbf{Cat} \\ }$which can be filled with a natural transformation coming from the universal property of products. We want to show that it is an isomorphism. Since all the functors here are left adjoints it will suffice to verify that the corresponding transformation filling the square of their right adjoints is an isomorphism and this follows by a direct inspection.
I’m not sure I follow your arguments and I’m also not sure I understand your notation. Is min supposed to stand for “minimal relative structure”? If so then Ho is not a left adjoint of min. Minimal relative structure has just identities as weak equivalences and Ho is a left adjoint of a functor iso that sends a category to a relative category with isomorphisms as weak equivalences.
Ah, right, yes. I was assuming that every isomorphism is automatically a weak equivalence. Perhaps I was secretly thinking of homotopical categories rather than relative categories.
After some further thought, there is actually just one reason why the leftmost adjoints in all of these strings of adjoints preserves finite products:
$\pi_0 \dashv disc \dashv ob \dashv codisc : \mathbf{Set} \to \mathbf{Grpd}$ $\mathbf{I} \dashv und \dashv iso : \mathbf{Grpd} \to \mathbf{Cat}$ $\tau_1 \dashv N : \mathbf{Cat} \to \mathbf{SSet}$ $\pi_1 \dashv N : \mathbf{Grpd} \to \mathbf{SSet}$ $\pi_0 \dashv disc \dashv (-)_0 \dashv cosk_0 : \mathbf{Set} \to \mathbf{SSet}$ $Ho \dashv min^+ : \mathbf{Cat} \to \mathbf{RelCat}$Namely, because each one is the reflector of an exponential ideal. Admittedly, it seems to be easier to prove directly that $\tau_1 : \mathbf{SSet} \to \mathbf{Cat}$ preserves finite products, but everything else follows by nonsense. This also explains why $skel = \pi_0 iso : \mathbf{Cat} \to \mathbf{Set}$ preserves finite products.
Nice!
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