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Stream: learning: questions

Topic: describing presheaves on Lawvere theories

John Baez (Apr 10 2021 at 21:12):

@Christian Williams and I are trying to figure out how describe all the presheaves on a Lawvere theory T in an intuitive, tractable way. We came up with an idea and I'm wondering if someone can help polish it up.

I'll illustrate this idea with the Lawvere theory for monoids, though you can replace the word "monoid" in what follows by any other concept described by a Lawvere theory. (I like the method of "pretending to be doing an example when you're actually doing something general" - it's psychologically helpful, since it reduces the burden of abstraction.)

So, let $T$ be the Lawvere theory of monoids. We're trying to describe presheaves on $T$.

We think of these as functors $F: T^{\text{op}} \to \mathsf{Set}$. Using a famous result of Lawvere, $T^{\text{op}}$ is a skeleton of the category of free finitely generated monoids. So, there's a full and faithful functor

$i: T^{\text{op}} \to \mathsf{Mon}$

So, we can get a presheaf $F$ on $T$ from any functor

$G : \mathsf{Mon} \to \mathsf{Set}$

simply by composing with $i$:

$F = G \circ i$

I claim that any presheaf on $T$ arises this way.

Next, how do we get lots of functors

$G : \mathsf{Mon} \to \mathsf{Set} ?$

Here's an example:

$G(X) = \displaystyle{ \frac{\{(x,y) \in X^2 : x^2 = y^3\}}{(x,y) \sim (y,1)} }$

Here I'm building a set from a monoid $X$ in a functorial way. This example generalizes: any method of building a set $G(X)$ from (the underlying set of) a monoid $X$ using the monoid operations, and limits, and colimits, gives me a functor $G$.

I think it's pretty easy to show that every presheaf on $T$ arises this way - in fact, just using finite products and arbitrary colimits. Every presheaf is a colimit of representables. I believe the representable presheaves on $T$ come from the functors

$G_n : \mathsf{Mon} \to \mathsf{Set}$

given by

$G_n(X) = \textrm{ underlying set of } X^n$

So, we can start with just the one functor

$G_1(X) = \textrm{ underlying set of } X$

and look at the presheaf on $T$ that this gives:

$G_1 \circ i$

If we then take finite products of this presheaf, and then arbitrary colimits of those products, we'll get all presheaves on $T$.

My question is: what sort of language I can use to conveniently "write down a formula" for enough $G$'s to get all presheaves on $T$?

If I only use finite limits and colimits, then it seems I can write down a kind of formula for $G$ using the Mitchell-Benabou language of the presheaf category on $T$. Right? Is there some generalization of this to infinite limits and colimits?

Nathanael Arkor (Apr 10 2021 at 21:26):

This may not be quite what you're asking, but the models of a Lawvere theory $L$ is given by the sifted cocompletion of $L°$, i.e. $\mathrm{Sind}(L°)$. Hence, the presheaves on $L$ are given by precisely the $\Phi$-colimits of models of $L$, where $\Phi$ is the class of colimits orthogonal to sifted colimits. If one could establish what colimits are given by $\Phi$, you would have a very concrete description of presheaves on Lawvere theories.

John Baez (Apr 10 2021 at 21:28):

I'm sorry, but could you remind me what "sifted" colimits are? Are they filtered colimits where all the arrows in the diagram are monos?

John Baez (Apr 10 2021 at 21:29):

I didn't know that the models of a Lawvere theory form precisely the sifted cocompletion of that theory. I should learn about this...

Fawzi Hreiki (Apr 10 2021 at 21:30):

They're colimits which commute with finite products in Set

John Baez (Apr 10 2021 at 21:31):

Okay, thanks Fawzi - but there should also be some characterization of them, no?

Fawzi Hreiki (Apr 10 2021 at 21:35):

The original definition is the one I gave above but I think this was later found to be equivalent to the diagonal functor (for the diagram category) being final

Fawzi Hreiki (Apr 10 2021 at 21:37):

I think the idea is that what makes filtered colimits useful is that they commute with finite limits in Set and so sifted colimits are the finite product analogue (even though they don't have as nice a characterisation as filtered colimits)

Morgan Rogers (he/him) (Apr 10 2021 at 21:38):

Fawzi Hreiki said:

The original definition is the one I gave above but I think this was later found to be equivalent to the diagonal functor (for the diagram category) being final

Or that for any pair of objects in the diagram category, the category of spans between those two objects is (non-empty and) connected.

Morgan Rogers (he/him) (Apr 10 2021 at 21:39):

I'll come back to this thread tomorrow, I might be able to assist.

Christian Williams (Apr 10 2021 at 21:45):

sifted colimits are basically, filtered colimits plus reflexive coequalizers.

Christian Williams (Apr 10 2021 at 21:46):

a filtered colimit is one where the diagram category has the property that every diagram in it has a cocone.

Christian Williams (Apr 10 2021 at 21:47):

the classic example is a chain of inclusions $A_0 \to A_1\to A_2\to \dots$, and I think in general filtered colimits are certain "gluings" of these chains.

Christian Williams (Apr 10 2021 at 21:48):

so, each layer of this class of colimits should have a fairly concrete description.

John Baez (Apr 10 2021 at 21:51):

I like the characterization of filtered categories as those in which every (ahem) FINITE diagram has a cocone. This often makes it easy to see which colimits are filtered, and it gives a nice intuition for them.

John Baez (Apr 10 2021 at 21:56):

I think there's a similar easily intuited characterization of sifted categories, and I guess Morgan said it: "the category of spans between those two objects is (non-empty and) connected." But I need to mentally process this a bit before I can just look at a diagram and see if it's sifted.

Alexis Hazell (Apr 11 2021 at 01:50):

It's likely I'm misunderstanding something, but shouldn't:

simply by composing with $i$:

$G = F ∘ i$

actually be:

$F = G ∘ i$

?

John Baez (Apr 11 2021 at 07:11):

Yes, that was a typo. I fixed it. Thanks!

Morgan Rogers (he/him) (Apr 11 2021 at 08:24):

Christian Williams said:

a filtered colimit is one where the diagram category has the property that every diagram in it has a cocone.

*finite diagram

Todd Trimble (Apr 11 2021 at 13:08):

Nathanael Arkor said:

This may not be quite what you're asking, but the models of a Lawvere theory $L$ is given by the sifted cocompletion of $L$, i.e. $\mathrm{Sind}(L)$. Hence, the presheaves on $L$ are given by precisely the $\Phi$-colimits of models of $L$, where $\Phi$ is the class of colimits orthogonal to sifted colimits. If one could establish what colimits are given by $\Phi$, you would have a very concrete description of presheaves on Lawvere theories.

I don't think this was stated exactly right: the opposite of a Lawvere theory is the category of finitely generated free algebras, and any algebra can be presented as a reflexive coequalizer of a diagram of free algebras, and free algebras can be constructed as filtered colimits of finitely generated free algebras. So the category of algebras will be the sifted cocompletion of the opposite of the Lawvere theory.

Roman Kniazev (Apr 11 2021 at 19:49):

I'm not sure that I can help, but rather add more (hopefully related) questions: about a year ago or so I was wondering what kind of theory a presheaf topos on a Lawvere theory classifies. I found this post and I suppose that the current topic is not unrelated to it. So my question is whether there is any progress on say, existence of finite axiomatization of theories classified by aforementioned toposes?