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Stream: learning: questions

Topic: Cohesion for (infinity,1)-categories of cubical objects

Madeleine Birchfield (Apr 09 2025 at 23:59):

Let $\mathcal{E}$ be an $(\infty,1)$-topos. For which categories of cubes $\Box$ is the $(\infty,1)$-category $\mathcal{E}^{\Box^\mathrm{op}}$ of cubical objects in $\mathcal{E}$ cohesive over the base $\mathcal{E}$?

Adrian Clough (Apr 10 2025 at 06:48):

So for $A$ any small $(\infty,1)$-category you get a triple adjunction between $\mathcal{E}^{A^{\mathrm{op}}}$ and $\mathcal{E}$ given by $\mathrm{colim} \dashv \mathrm{const} \dashv \mathrm{lim}$. If $A$ admits a final object $1$ then $\mathrm{lim}$ is just given by evaluating at $1$, thus commutes with colimits, and thus admits a right adjoint by AFT. So, all that you need to check is whether $\mathrm{colim}$ commutes with finite products. This is equivalent to $A$ being sifted. AFAIK all cube categories used in practice are test categories, and in this case being sifted is equivalent to being a strong test category. I don't know which cube categories are strong test categories, but now you can significantly narrow the scope of your search :smile:

Morgan Rogers (he/him) (Apr 10 2025 at 07:27):

You also want const to be fully faithful, no?

Morgan Rogers (he/him) (Apr 10 2025 at 07:28):

@Quentin Schroeder this is a discussion of possible interest for your future project :)

Adrian Clough (Apr 10 2025 at 07:52):

If $A$ has a final object, then full faithfulness is automatic. For any object $X$ in $\mathcal{E}$ evaluation at $1$ of the constant diagram at $X$ just gives you back $X$.

Jonas Frey (Apr 10 2025 at 11:29):

The cube categories are cosifted because they have finite products.

Jonas Frey (Apr 10 2025 at 11:32):

Furthermore I think we get the Nullstellensatz because all objects have points (contrary to eg presheaves on a meet semi lattice where we get the the desired adjoints but no Nullstellensatz)

Jonas Frey (Apr 10 2025 at 11:36):

There are other cohesion axioms like continuity which probably don't hold

Jonas Frey (Apr 10 2025 at 11:36):

(continuity fails in simplicial sets)

Madeleine Birchfield (Apr 10 2025 at 19:14):

Adrian Clough said:

AFAIK all cube categories used in practice are test categories, and in this case being sifted is equivalent to being a strong test category. I don't know which cube categories are strong test categories, but now you can significantly narrow the scope of your search :smile:

I'm personally interested in the Cartesian cube categories with connections, such as the Dedekind cube category, etc. If I remember correctly, Cartesian cube categories with connections are strong test categories, so any such $\mathcal{E}^{\Box^\mathrm{op}}$ should then be cohesive in the above sense.