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Stream: theory: category theory

Topic: arrow category as a monad

James Deikun (Jan 05 2024 at 18:06):

Are these known facts (or known non-facts for that matter):

• Arrow category forms a monad on $\mathsf{Cat}$ with restriction along the inclusion of a diagonal in a square as multiplication and identity morphism as unit.
• $\mathrm{cod}$ is naturally an algebra of the $\overrightarrow{-}$ monad.
• The bar resolution of $\mathrm{cod}$ gives an embedding of $\mathsf{Cat}$ in $\mathsf{Cat}^{\Delta_a^{\mathrm{op}}}$.
• It restricts to give an embedding of the finitely complete categories and pullback-preserving functors in the non-full subcategory of $\mathsf{Cat}^{\Delta_a^{\mathrm{op}}}$ where the augmentation maps are fibrations and the other morphisms all respect that.

Versions of this with higher categorical structure (e.g. 2-monad, Cartesian natural transformation) are even better. I looked for this a bit but it doesn't seem like the kind of thing that gets its own article.

fosco (Jan 05 2024 at 18:19):

https://www.sciencedirect.com/science/article/pii/002240499390171O you might want to read this :-) it was a staple on which I built my PhD thesis, so yes, there is a quasicategorical version of the same theorem -although I wasn't really heavy on the simplicial stuff, and a version in the setting of Grothendieck derivators https://arxiv.org/abs/1705.08565

James Deikun (Jan 05 2024 at 18:51):

Thanks! These cover the first point thoroughly and in higher categorical detail, and dip into the second point a bit. (It seems $\mathrm{cod}$ and $\mathrm{dom}$ are representations of the factorization systems that factor an arrow all to one side or the other.) Anyone have anything that touches on the third or fourth points, or deals with the second in more detail?

John Onstead (Jan 06 2024 at 01:07):

@James Deikun Interesting questions! However, I may need some clarification on something: What is the difference between the arrow category monad on Cat and the Cat endo-hom-functor Hom(2, -) which sends a category C to the hom-category Hom(2, C)? (In this case 2 is the interval category with two objects and one non trivial morphism between them A -> B) Maybe comparing these two functors could help, since hom functors have many nice properties!

James Deikun (Jan 06 2024 at 02:49):

They're actually the same. The arrow category monad is what's called a reader monad on the walking arrow. Since Cat's monoidal closed structure is Cartesian, this is enough information to fully determine the monad. But it even has extra nice stuff because the walking arrow has initial and terminal objects and so the unit of the monad has left and right adjoint retractions. What's more they're both algebras naturally in the carrier.

James Deikun (Jan 06 2024 at 03:00):

I feel like the thing with the bar resolution being an embedding ought to follow by general abstract nonsense from $\mathrm{cod}$ being left adjoint to the unit.

James Deikun (Jan 06 2024 at 03:18):

Basically: I think the bar resolution being a simplicial object in the sense that it respects the full 2-categorical structure of the simplex category, plus being contractible, should be enough to make simplicial maps between them fully determined by a map of the carrier.

James Deikun (Jan 06 2024 at 13:01):

Hm, it turns out the adjoints don't join up into a string as I'd hoped, so that proof strategy won't work. However, there is a string of adjoints $\mathrm{cod} \dashv id = \eta \dashv \mathrm{dom}$ and another $\rho \dashv \mu \dashv \lambda$. Say you have a simplicial map between bar resolutions of $\mathrm{cod}$ given by maps $\overset{\bullet}{f}$, and say you know them up to n (inclusive). Can you figure out what $\overset{n+1}{f}$ is? It should be $T\overset{n}{f}$. This is true in general on the level of the [[bar construction]], where everything lives in the category of algebras and you include extra degeneracies/acyclic structure as part of the data, but it may need a bit more to descend to a simple augmented simplicial object in the base category.

James Deikun (Jan 06 2024 at 13:16):

First of all, every algebra natural in the carrier (natural transformation from $T$ to $\mathsf{Id}$ that is componentwise a $T$-algebra) determines an embedding into the category of $T$-algebras, because the naturality squares prove that each morphism of the base category underlies a unique map of algebras.