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Stream: theory: category theory

Topic: limit preserving left Kan extension

Matteo Capucci (he/him) (Jan 01 2021 at 12:04):

Let $f: C \to D$ be a limit preserving functor between complete categories.
Example A4.1.10 from the Elephant shows that the induced functor $f^* : [D^{op}, Set] \to [C^{op}, Set]$ has a left exact left adjoint. It does so only assuming $f :C \to D$ is a cartesian functor between cartesian categories, so I was hoping to leverage my stronger assumptions and conclude $\operatorname{Lan}_f$ actually preserves all limits.
I'm not at ease with the kind of arguments employed by Johnstone so I don't really know if they straightforwardly generalize to my case.

Matteo Capucci (he/him) (Jan 01 2021 at 12:05):

Screenshot-from-2021-01-01-13-04-44.png

Reid Barton (Jan 01 2021 at 13:59):

Are $C$ and $D$ large categories? If so I don't see why the left Kan extension should exist in the first place (let alone preserve limits).

Dan Doel (Jan 01 2021 at 17:28):

Isn't the left exact left adjoint $\mathrm{Lan}_f$?

John Baez (Jan 01 2021 at 17:52):

Yes.

Dan Doel (Jan 01 2021 at 18:00):

I suppose the point is that the adjoint may not exist unless $C$ and $D$ are small.

John Baez (Jan 01 2021 at 18:00):

I'm not very good at this stuff, so I'll just say some guesses to keep the conversation rolling. The category of presheaves on $C$ is the free cocompletion of $C$ when $C$ is small, and this is another way of saying that any functor $f: C \to D$ between small categories extends to a cocontinuous functor from $[C^{op},Set]$ to $[D^{op},Set]$, namely $Lan_f: [C^{op},Set] \to [D^{op},Set]$. But this may not work when $C$ is large - I believe the problem is more with $C$ being large than $D$ being large. And as Reid Barton hinted, complete categories are large unless they are preorders. So Matteo's question is problematic unless he's willing to assume $C$ is a preorder.

John Baez (Jan 01 2021 at 18:01):

How about this substitute, then? Instead of assuming $C$ and $D$ have all small limits, assume they are small categories with all $J$-limits where $J$ is any chosen set of diagrams (not class, set).

John Baez (Jan 01 2021 at 18:01):

Suppose $f: C \to D$ preserves all $J$-limits.

Question: does $Lan_f : [C^{op},Set] \to [D^{op},Set]$ preserve $J$-limits?

My guess is yes, but I'm not very good at this stuff.

Dan Doel (Jan 01 2021 at 18:02):

Oh right. Or he's in a constructive setting where that fact about preorders is wrong.

Reid Barton (Jan 01 2021 at 18:35):

I think this is probably related to the notion of a sound doctrine (page itself is not helpful, but has useful links), though I never properly understood this stuff.

Reid Barton (Jan 01 2021 at 18:40):

If $J$ is the collection of all $\kappa$-small diagrams ($\kappa$ a regular cardinal), then the same sort of argument as in Johnstone would work. For $J$ the collection of finite discrete categories you can argue directly: products commute with colimits in each argument so you can reduce to the case of representables, which is true by definition.

Matteo Capucci (he/him) (Jan 01 2021 at 19:15):

$C$ and $D$ are small preorders in my case, yes. EDIT: Oh, I see why you're discussing completeness. I put it in my first post without thinking about Freyd's result.

Matteo Capucci (he/him) (Jan 01 2021 at 19:17):

However my issue is not really with the existence of $\mathrm{Lan}_f$, but with its continuity

Matteo Capucci (he/him) (Jan 01 2021 at 19:17):

I know it preserves finite limits when $f$ does, can one extend this to small limits?

John Baez (Jan 01 2021 at 19:56):

Okay, preorders can be complete and small, so all our worries dissolve. Then my guess is just yes, if $C$ and $D$ are complete small preorders and $f: C \to D$ preserves all small limits then so does $Lan_f$. But my belief is based on my feel for how presheaves work; I'm not good at proving this stuff.

Matteo Capucci (he/him) (Jan 01 2021 at 20:26):

Can you elaborate on that feel? :thinking: Above you mentioned the universal property of small presheaves, but that gives cocontinuity of $\mathrm{Lan}_f$, right? Why shall I expect continuity?

Jens Hemelaer (Jan 01 2021 at 20:30):

I think this will work for posets:
if $f : C \to D$ preserves all small limits (equivalently, all meets), then it has a left adjoint functor.
Further, the functor that sends a category to its presheaf topos, and a functor to its left Kan extension, is a 2-functor, and as such it preserves adjunctions. So the left Kan extension $\mathrm{Lan}_f$ has a left adjoint as well. More precisely, its left adjoint is given by $\mathrm{Lan}_g$, where $g$ is the left adjoint of $f$.

John Baez (Jan 01 2021 at 20:34):

That's a super-nice argument, taking advantage of the "dream world" of preorders, where the adjoint functor theorem loses all its technicalities: a functor between preorders has a left adjoint iff its continuous.

Matteo Capucci (he/him) (Jan 01 2021 at 20:40):

Mmmh it feels like too many adjunctions between the presheaves categories :thinking:

John Baez (Jan 01 2021 at 20:40):

For what little it's worth, @Matteo Capucci, here was my intuition. When you embed a small category C into its category of presheaves you are "freely throwing in colimits". So colimits that used to be colimits in C no longer are, unless there's a damn good reason they have to be ("absolute colimits"). However, limits in C are still limits inside the category of presheaves.

Because we're "freely throwing in colimits" when we take the preseheaf category, we can extend any functor $f: C \to D$ to a colimit-preserving preserving functor between presheaves, and that's $Lan_f$. But because taking presheaves doesn't mess with limits, $Lan_f$ should preserve any limits that $f$ did.

That last sentence is a total hand-wave, not an argument. I guess the rigorous portion of it is that if $f$ preserves limits of some sort, $Lan_f$ must preserve that sort of limit on representables. But why should it do so for all presheaves?

If I had to try to create a proof, I'd try to use the fact that all presheaves are colimits of representables, and do some commuting of limits and colimits... but I don't know if this works.

Someone like @Todd Trimble could settle these questions in his sleep.

John Baez (Jan 01 2021 at 20:41):

But Jens' argument is so slick even I can understand it. It does the job for preorders.

Matteo Capucci (he/him) (Jan 01 2021 at 20:45):

John Baez said:

For what little it's worth, Matteo Capucci, here was my intuition. When you embed a small category C into its category of presheaves you are "freely throwing in colimits". So colimits that used to be colimits in C no longer are, unless there's a damn good reason they have to be ("absolute colimits"). However, limits in C are still limits inside the category of presheaves.

Because we're "freely throwing in colimits" when we take the preseheaf category, we can extend any functor $f: C \to D$ to a colimit-preserving preserving functor between presheaves, and that's $Lan_f$. But because taking presheaves doesn't mess with limits, $Lan_f$ should preserve any limits that $f$ did.

That last sentence is a total hand-wave, not an argument. I guess the rigorous portion of it is that if $f$ preserves limits of some sort, $Lan_f$ must preserve that sort of limit on representables. But why should it do so for all presheaves?

If I had to try to create a proof, I'd try to use the fact that all presheaves are colimits of representables, and do some commuting of limits and colimits... but I don't know if this works.

I see. Then there's should be an easy proof by coend calculus then :thinking: I'll try fiddling around.

Matteo Capucci (he/him) (Jan 01 2021 at 20:46):

Jens Hemelaer said:

I think this will work for posets:
if $f : C \to D$ preserves all small limits (equivalently, all meets), then it has a left adjoint functor.
Further, the functor that sends a category to its presheaf topos, and a functor to its left Kan extension, is a 2-functor, and as such it preserves adjunctions. So the left Kan extension $\mathrm{Lan}_f$ has a left adjoint as well. More precisely, its left adjoint is given by $\mathrm{Lan}_g$, where $g$ is the left adjoint of $f$.

Also I'm now convinced by this. Neat!

Reid Barton (Jan 01 2021 at 20:48):

I don't think this "commuting limits and colimits" approach will work for a general $J$--if you take something like a pullback over a colimit, you can't easily relate that to a colimit of pullbacks.

John Baez (Jan 01 2021 at 20:50):

Yeah, I had a feeling of desperation right around when I suggested that. It was helpful for me to make explicit my intuitions because then I could see the huge hole.