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I am about to create an entry called locally algebra-ed topos in the spirit of the section for local algebras at classifying topos.
I tend to think this terminology is better than the undescriptive “structured topos”, but please let me know what you think.
I would like to amplify the following fact:
if we agree to say (which is reasonable) that
an algebra is a model of some essentially algebraic theory, hence a lex functor out of a finite-limite category;
a local algebra with respect to a coverage on the category is such a lex functor that preserves covers.
then the statement is:
I agree that “structured topos” is not very descriptive. But the generality of the notion of “essentially algebraic theory” makes me a bit uneasy about calling any model of such a theory an “algebra”. Moreover, although covers in sites are sometimes about “topology” and hence “locality”, the condition for an “algebra” to be “local” in this sense is related instead to the use of “local” in “local ring”, which I have even more trouble seeing as generalizing naturally to any essentially algebraic theory (especially since one must specify a coverage in order to get a notion of “local algebra”). Any model of any theory in a topos is always automatically “local” in the sense that its structure and properties happen “stalkwise”, so I think there is potential for confusion there. Can you say anything about what exactly is “local” about a “local algebra”, in general?
In fact, geometric theories are so general that I’m not sure that “structured topos” is so bad – since any finitary first-order theory has a classically-equivalent coherent theory, for very many “structures” one might want to consider internal to a topos, there is a corresponding geometric theory. Maybe “geometrically structured topos”?
Okay, thanks for the thoughts. Yes, maybe the terminology needs more tuning.
Can you say anything about what exactly is “local” about a “local algebra”, in general?
I mean this in the sense of local rings: for $\mathcal{C}_{\mathbb{T}}$ a finitely complete category, $\mathcal{E}$ a sheaf topos, a lex functor $A : \mathcal{C} \to \mathcal{E}$ is of course stalkwise a $\mathbb{T}$-algebra. But if we pick in addition a coverage $J$ on $\mathcal{C}_{\mathbb{T}}$ and ask $A$ to respect covers, then this makes it a $J$-local $\mathbb{T}$-algebra.
But I am aware that what is nice here is also a problem: this notion is vastly general.
The key point missing currently, which might be just the ingredient to cut down on the generality and yield the intended geometric meaning, is really what I have been addressing in the other thread (and which we talked about before): the extra structure that constrains the morphisms of “$J$-local $\mathbb{T}$-algebras” (that which involves the “admissibility” structure).
Maybe “geometrically structured topos”?
Yes, maybe that’s good.
In Rosenberg’s reconstruction theorem he assign to every abelian category (viewed as a category of quasicoherent sheaves) a spectrum which is a topological space and a stack of local abelian categories. The center (that is the ring of endomorphisms of the identity functor) of a local abelian category is a local commutative ring. So in the case when the abelian category to start is the category of qcoh sheaves over a commutative scheme, then the center construction, followed by the sheafification, gives the usual structure sheaf of local rings. So in a sense, the notion of local abelian category is sort of categorification of a local ring, It would be interesting if this notion can be derived somehow from the above, also categorified, machinery.
Good point. So I think it is important to notice that there is a 2-dimensional $(n,r)$-category lattice of possible categorifications here:
what happens in Structured Spaces and what I am discussing here is local $(\infty,1)$-algebras. What happens in the setup that you mention (we have something on that at Tannaka duality for algebraic stacks) is local 2-algebras (meaning $(2,2)$-algebras).
While it is pretty clear what 2-algebras are in concrete cases (suitable monoidal categories) I am not aware of much genuine “2-algebraic theory” (as $(2,2)$-algebraic theory) beyond the better understood $(2,1)$-algebraic theory. So I guess there is a lot still to be done here.
Well, if we are in the language of monads, then for 2-monads one can look at pseudoalgebras but also at (co)lax algebras. This is not satisfactory for (2,2) ?
The definition of local abelian category is in terms of certain preorder which has a prominent role in Rosenberg’s spectral theory. This is suitable for the stuff related to qcoh sheaves, what is not a topos but an abelian setup. It would be interesting to see if it boils down to the same definition eventually.
Well, if we are in the language of monads, then for 2-monads one can look at pseudoalgebras but also at (co)lax algebras. This is not satisfactory for (2,2) ?
Right, but what I meant is: is there the full-blown theory of 2-theories as there is for 1-theories? Do we understand “geometric 2-theories”, “classifying 2-toposes”, etc.?
I think that the work of Bunge-Hermida on 2-Diaconescu’s theorem is one of the deepest results related to classifying 2-toposes, cf. Marta’s pdf slides from Calais 2007. Maybe Igor could tell us more.
Okay, good. Is there anything beyond Diaconescu’s theorem? That gives classifying toposes for essentially algebraic 2-theories. For local algebra we need “geometric 2-theories”. did anyone go towards that goal already?
I don’t have the time right now, but somebody should then start an entry on the 2-Diaconescu theorem. At least with some references.
Just getting around to responding to this: my goal at internal logic of a 2-category (michaelshulman) was precisely this, to describe a logic in which we could define “geometric 2-theories” and prove that they have classifying 2-toposes. But I got sidetracked by other things and haven’t gotten back to it for a while. I believe Dan Licata’s thesis has since dealt more exhaustively with the type theory (minus the logic).
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