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Stream: theory: category theory

Topic: Categories of (co)sieves

Josh Chen (Jan 08 2024 at 05:59):

Does anyone know of references about categories of (co)sieves?

The nLab page on cosieves gives a brief definition organizing them into a category coSv(c) (which I think is equivalent to the full subcategory, in $[C, Set]^{\rightarrow}$, on subfunctors of corepresentables), and I can show that e.g. the forgetful functor coSv(C) -> C sending cosieves to apexes has left adjoint, while I don't think it's a Grothendieck fibration. But in general, I'm not sure know how to intuit the idea behind the definition of coSv(C) in that page.

Patrick Nicodemus (Jan 08 2024 at 07:19):

Well, in topos theory, for a small category $C$ one studies the presheaf $\Omega$ on $C$ where $\Omega(c)$ is the set of all sieves with codomain $c$. This presheaf is important because it is the subobject classifier for the topos of presheaves, and then Grothendieck topologies can be defined in terms of Lawvere-Tierney topologies on $\Omega$. I would imagine that, all things being dualized, this category coSv(C) is just the category of elements of $\Omega$.

I'm not sure why the category of elements is of particular interest rather than the (co)presheaf itself though.

Patrick Nicodemus (Jan 08 2024 at 07:21):

Toby Bartels, the last person to edit that page, is active here, so you might ping him. They might know why it's important.

Morgan Rogers (he/him) (Jan 08 2024 at 07:45):

Sometimes it can be more natural to work with copresheaves. For example, the classifying topos of an essentially algebraic theory is the category of copresheaves on its category of finitely presentable models. Subtoposes of this topos correspond to quotients of the theory (theories obtained by adding axioms in geometric logic to the original theory) and the corresponding Grothendieck topologies may be more intuitively expressed in terms of cosieves in the category of fp-models rather than sieves in the opposite category.

Josh Chen (Jan 08 2024 at 07:59):

Patrick Nicodemus said:

I would imagine that, all things being dualized, this category coSv(C) is just the category of elements of $\Omega$.

I'm not sure why the category of elements is of particular interest rather than the (co)presheaf itself though.

If I'm not mistaken, I think the category of elements of the copresheaf gives almost but not quite the nLab's definition of coSv(C); the nLab asks for (R' ⚬ f) to be a subset of R, where the category of elements would require equality. But that's a very nice observation, given that I was in fact wanting to use the latter version instead of the nLab's. Thank you! Though now I wonder even more about the former...

Josh Chen (Jan 08 2024 at 08:16):

Interestingly, I think the adjunction I mentioned earlier also no longer holds for the category of elements version, though I'm a bit jetlagged so I'll have to doublecheck everything again.