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Stream: theory: category theory

Topic: Example of a localization that needs to be iterated

Brendan Murphy (Apr 22 2024 at 06:04):

In Joyal's notes on quasicategories he discusses a factorization system on small 1-categories (Cat is considered as a 1-category here) with left class "iterated strict localizations" and right class conservative functors. Here a strict localization is a functor isomorphic to something of the form $\mathcal{C} \to \mathcal{C}[W^{-1}]$ for a small category $\mathcal{C}$ and set of maps $W$ in $\mathcal{C}$ and an iterated strict localization is a transfinite composite of these. He says we need to talk about iterated localizations because if we have a functor $F : \mathcal{C} \to \mathcal{D}$ and set $W = \{ f \mid F(f) \text{ is an isomorphism} \}$, giving a factorization of $F$ into a strict localization $P : \mathcal{C} \to \mathcal{C}[W^{-1}]$ followed by a functor $G : \mathcal{C}[W^{-1}] \to \mathcal{D}$, the resulting $G$ might not be conservative. So even after inverting everything inverted by $F$ we may still need to invert yet more maps $\mathcal{V} \subseteq \mathcal{C}[W^{-1}]$, those inverted by $G$. And $P^{-1}(\mathcal{V}) = \mathcal{W}$, so these are genuinely new things created in the localization. What's an example of a functor $F$ where this occurs? An example where we need to iterate transfinitely many times would be appreciated

Rémy Tuyéras (Apr 22 2024 at 10:10):

I am not used to that specific kind of use case (i.e. with quasi-categories), but could it be that the first inversion inverts all $f$ such that $F(f)$ is an iso, but forgets to invert, say:

$f_1 \circ f_2^{-1}$ (where both $f_1$ and $f_2$ belong to $W$)

because $f_1 \circ f_2^{-1}$ needs to wait for $f_2^{-1} \in \mathcal{C}[W^{-1}]$ to be computed?

(I actually doubt the previous example, but it might be a coherence thing where you correct expressions that require some inverses to already exist?)

Brendan Murphy (Apr 22 2024 at 15:47):

No, this isn't possible. That's a composition of two isomorphisms and so is an isomorphism

Brendan Murphy (Apr 22 2024 at 16:16):

(also fwiw this isn't about quasicategories at all, I was just providing a citation for where I saw the result)

Mike Shulman (Apr 22 2024 at 16:49):

I think this is a case where you can just look at the categories freely generated by an example. Let $C = (w \xrightarrow{f} x \xleftarrow{g} y \xrightarrow{h} z)$, let $D$ be the walking section-retraction pair with $s:a\to b$ and $r:b\to a$ and $r s = 1_a$, and let $F:C\to D$ be defined by $F(f) = s$, $F(g) = 1_b$, and $F(h) = r$. Then $W = F^{-1}(\mathrm{Iso})$ contains only $g$ (and identities), so $C[W^{-1}]$ inverts only $g$. But then in $C[W^{-1}]$ we have a new morphism $h g^{-1} f$ which maps to $r 1_b s = 1_a$ in $D$, hence lies in $V = G^{-1}(\mathrm{Iso})$.

Brendan Murphy (Apr 22 2024 at 17:44):

Ah sure, thanks for working it out!

Brendan Murphy (Apr 22 2024 at 17:59):

The reason I was thinking about this is I wanted to know whether any limit is the limit of a conservative functor. But my original framing of the question was wrong, since I thought you could factor through the localization and get something conservative. That said, is the following true?

Let $F : \mathcal{J} \to \mathcal{C}$ be a functor which has a limit. Factor $F = G\circ p$ with $G$ conservative and $p$ an iterated strict localization. Is it true that $G$ has a limit and $\lim G \to \lim F$ is an isomorphism?

If $p$ is a localization I think we can prove this by reducing to $\mathcal{C} = \mathsf{Set}$ with the yoneda embedding and arguing explicitly there (identifying the limits with subsets of a product and showing they're equal by looking at equations). Iterating this finitely many times is fine, but I'm not sure how to extend transfinitely (and my proof for the one step case is pretty sketchy). Does anyone have a reference for this? Or maybe a slick proof using factorization systems?

Mike Shulman (Apr 22 2024 at 18:29):

Interesting question! Here's an attempt at a slick proof using factorization systems. As you say, suppose $F:J\to C$ has a limit, call it $\ell$, and that $F = GP$ with $P$ left orthogonal to all conservative functors. (In fact I don't think we need that $G$ is conservative.) Then we know that the map $\mathrm{Cone}(x,F) \to C(x,\ell)$ is an isomorphism, and we want to show that the map $\mathrm{Cone}(x,G) \to C(x,\ell)$ is an isomorphism. By 3-for-2, it suffices to show that the map $\mathrm{Cone}(x,G) \to \mathrm{Cone}(x,F)$ is an isomorphism, and now we don't need any more to mention the fact that $F$ has a limit. So we can try to prove that for all functors $F:J\to C$ with a factorization $F = GP$ where $P$ is left orthogonal to conservative functors, we have that the map $\mathrm{Cone}(x,G) \to \mathrm{Cone}(x,F)$ is an isomorphism.

Next observe that to give a functor $F:J\to C$ and a cone over it with vertex $x$ is equivalent to giving a functor $F' : J \to x/C$ to the co-slice under $x$. And therefore, given $F:J\to C$, to give a cone over it with vertex $x$ is to give a lifting of $F$ along the projection $U_x : x/C \to C$. But $U_x$ is conservative! Thus, given $F$ and a cone over it, we have a commutative square $G \circ P = U_x \circ F'$, where $U_x$ is conservative and $P$ is left orthogonal to conservative functors. Therefore, there is a unique lifting, mapping the codomain of $P$ to $x/C$, i.e. a unique extension of our given cone over $F$ to a cone over $G$ with vertex $x$. This is what we wanted.

Brendan Murphy (Apr 22 2024 at 18:45):

Oh that's very nice!!!

Brendan Murphy (Apr 22 2024 at 18:48):

And it makes sense that we wouldn't need G to be conservative, since my original argument (where we just localize once) was in that situation

Kevin Carlson (Apr 23 2024 at 17:35):

I almost have an easier argument: (iterated strict) localizations are both initial and final, because discrete fibrations and opfibrations are conservative. So the loc-cons factorization system is comparable to the comprehensive ones in the order on factorization systems. Therefore as long as you know $G$ has a limit it will be the same as the limit of $G\circ p.$ Maybe you need Mike’s argument in the case where you don’t know a priori that $G$ has a limit at all, though.

Kevin Carlson (Apr 23 2024 at 17:37):

I’ve just never thought about the question of whether it’s possible to have an initial functor $f$ such that some $D\circ f$ has a limit while $D$ does not!

Brendan Murphy (Apr 23 2024 at 17:40):

I think we can use the yoneda embedding to assume the limit is taken in a presheaf category, then argue that the limits of D and D°f are the same by initiality so D has a representable limit in the presheaf category and thus a limit in the original category

Kevin Carlson (Apr 23 2024 at 17:58):

Yeah, I think that's right.

Brendan Murphy (Apr 23 2024 at 18:20):

Your argument also made me realize that in the problem I'm working on I might as well replace my functor with a discrete opfibration instead of just a conservative functor

Brendan Murphy (Apr 23 2024 at 18:26):

Although I'm less certain the things I want to be preserved are preserved

Mike Shulman (Apr 23 2024 at 18:48):

Thanks Kevin! Since my $U_x : x/C\to C$ is also a discrete opfibration, my argument can also be regarded as a re-proof of the fact that functors that are left orthogonal to discrete opfibrations are final.

Kevin Carlson (Apr 23 2024 at 18:51):

Discrete opfibrations are pretty great, you should use them if you can.

Brendan Murphy (Apr 23 2024 at 19:05):

Well the problem I'm thinking about is what limits exist in the augmented simplex category. I conjectured that if a diagram with inhabited indexing category has a limit then some projection from the limit to an object in the diagram must be injective. I was able to wlog to the case where the projections are all surjective, the functor defining the diagram is conservative, and the diagram category is skeletal (this implies the domain category is both direct and inverse, or "finite length"). To get this reduction to go through I used the fact that iterated strict localizations are the identity on objects, so the objects in the diagram & the projections to them stay the same (except when throwing away isomorphic copies when we take skeleta). But if we use the initial/discrete opfibration factorization system it's not so clear to me how to translate information back to the original diagram