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Stream: learning: questions

Topic: classifying (∞,1)-topos for nontrivial distributive lattices

Madeleine Birchfield (Nov 04 2024 at 04:47):

The classifying (∞,1)-topos for nontrivial bounded total orders is the (∞,1)-topos of simplicial sets. What is the classifying (∞,1)-topos for nontrivial bounded distributive lattices?

Morgan Rogers (he/him) (Nov 06 2024 at 10:11):

I know how to construct an answer in 1-topos land, and since we're dealing with algebraic things maybe it will lift to infinity.
Bounded distributive lattice are models of an algebraic theory; you can use Gabriel-Ulmer duality to deduce that they are classified by Set-valued functors on the category of finitely presented bounded distributive lattices. I think things are sufficiently nice for algebraic theories that you can just replace that with space-valued functors, but I would need someone who has thought about duality theory more recently to confirm that.

daniel gratzer (Nov 06 2024 at 14:27):

This should work, yes. In general, if $C$ is a category with finite limits, the "enveloping $\infty$-\topos" of $\mathrm{Pr}_{\mathsf{Set}}(C)$ is given by just swapping out $\mathsf{Set}$ for $\mathcal{S}$. So $\mathrm{Pr}(\mathsf{BDLat}_{\mathrm{fp}})$ has the right universal property for classifying bounded distributive lattices. For the non-triviality bit, presumably one can pass to an appropriate subtopos or similar

Chaitanya Leena Subramaniam (Nov 06 2024 at 22:06):

Warning: Given a 1-category with finite limits $C$, a lex functor $C\to\mathcal{E}$ to an $\infty$-topos necessarily factors through the 1-topos $\mathcal{E}_0$ of 0-truncated objects (since it must preserve (all iterated) diagonals).

On the other hand if $C$ is a 1-category with finite products, this is no longer true, i.e. a finite-product preserving functor $C\to\mathcal{E}$ is not 0-truncated. Which is the sense in which homotopy theorists [Badzioch, Bergner, Rezk etc] talk about "homotopy algebras of algebraic theories" (i.e. finite-product preserving functors to the $\infty$-topos of spaces).

As a concrete example, a homotopy algebra of the algebraic (aka finite-product) theory of monoids is an $A_\infty$-space, while a lex functor $\text{Monoid}_{fp}^{op}\to\mathcal{S}$ to spaces is the same thing as a 0-truncated monoid in spaces, in other words a plain old (set-based) monoid.

So the answer is: it depends on whether you mean models in an $\infty$-topos of the algebraic theory of distributive lattices, or the essentially algebraic theory of distributive lattices.

daniel gratzer (Nov 07 2024 at 01:10):

Chaitanya Leena Subramaniam said:

As a concrete example, a homotopy algebra of the algebraic (aka finite-product) theory of monoids is an $A_\infty$-space, while a lex functor $\text{Monoid}_{fp}^{op}\to\mathcal{S}$ to spaces is the same thing as a 0-truncated monoid in spaces, in other words a plain old (set-based) monoid.

Very good point! While this is a bit off-topic, is it obvious whether there exists a topos whose points $\mathrm{Fun}(\mathcal{S}, \mathcal{E})$ correspond to (say) $A_\infty$-spaces? The naive approach doesn't work for the reasons you outline above, but perhaps there is a more clever/$\infty$-categorical way to proceed?

Chaitanya Leena Subramaniam (Nov 07 2024 at 02:06):

$A_\infty$-spaces form a locally finitely presentable $\infty$-category, so the exact same "easy Diaconescu" theorem says that $\mathcal{S}^{(A_\infty\text{-spaces})_{fp}}$ is this $\infty$-topos.

Chaitanya Leena Subramaniam (Nov 07 2024 at 02:10):

On second thought, maybe not. Hmmm
I believe we have to be careful to ensure we mean $(A_\infty\text{spaces})_{fp}$ is the category of compact $A_\infty$-spaces, i.e. it is idempotent complete.

Graham Manuell (Nov 07 2024 at 20:11):

Morgan Rogers (he/him) said:

I know how to construct an answer in 1-topos land, and since we're dealing with algebraic things maybe it will lift to infinity.
Bounded distributive lattice are models of an algebraic theory; you can use Gabriel-Ulmer duality to deduce that they are classified by Set-valued functors on the category of finitely presented bounded distributive lattices. I think things are sufficiently nice for algebraic theories that you can just replace that with space-valued functors, but I would need someone who has thought about duality theory more recently to confirm that.

For what it's worth, finitely presented distributive lattices are just finite distributive lattices. The category is also equivalent to the opposite of the category of finite posets. So this is actually quite analogous to presheaves on finite ordinals. I know nothing about the infinity situation though.