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Stream: learning: questions

Topic: ✔ Free cocompletion are lax-idempotent 2-monads

Fernando Chu (Jun 04 2025 at 09:47):

The nlab states (paraphrasing)

The cocompletion under a class of certain diagrams gives a lax-idempotent 2-monad $T:\text{Cat} \to \text{Cat}$, since then an algebra $\alpha:TC \to C$ is a category $C$ with all such colimits, which implies $T$ is lax-idempotent.

I am having trouble seeing the intermediate step, i.e. that algebras must have all such colimits. Am I missing something obvious? Can this be shown using only the universal property of the cocompletion, without reference to a particular presentation?

Writing $\eta$ for the unit of $T$, if $\alpha \dashv \eta_C$ (as any algebra for a lax-idempotent 2-monads has), then for a diagram $K:J \to C$ we have

$$C(\alpha(\text{Colim}\ \eta_C K), X) \cong TC(\text{Colim}\ \eta_C K, \eta_C X) \cong [J,TC](\eta_CK, \Delta_{\eta_CX}) \cong \int_j TC(\eta_CKj, \eta_CX) \cong \int_j C(Kj,X) \cong [J,C](K, \Delta_X)$$

So if we can show that $T$ is indeed lax-idempotent, then the missing step holds; so I'd also be happy with such a proof. However, the nlab used the missing step to prove that $T$ is lax-idempotent, so they probably had in mind some direct proof.

Nathanael Arkor (Jun 04 2025 at 10:02):

There are a few ways to prove this, but perhaps the most intuitive is to show that a category $C$ admits $\Phi$-colimits (for some class of shapes $\Phi$) if and only if the inclusion $C \hookrightarrow \Phi(C)$ into the free $\Phi$-cocompletion admits a left adjoint (which then tells you that the algebras for $T$, which are by definition the $\Phi$-cocomplete categories, are characterised by the adjointness condition that defines a lax-idempotent pseudomonad).

Nathanael Arkor (Jun 04 2025 at 10:03):

I think it's a good exercise to try to show this yourself, but if you get stuck, there's a proof in Proposition 3.17 of Notes on enriched categories with colimits of some class.

Fernando Chu (Jun 04 2025 at 10:11):

Nathanael Arkor said:

...the algebras for $T$, which are by definition the $\Phi$-cocomplete categories...

I'm probably missing something extremely obvious. Why is this the case? An algebra for $T$ should be a category $C$ equipped with a (not necessarily $\Phi$-colimits-preserving) functor $\alpha:TC\to C$ such that some diagrams commute, right?

John Baez (Jun 04 2025 at 10:33):

Intiutively, $\alpha$ preserving colimits of some sort is what says $C$ has them.

It may help to think about a decategorified analogue like the monad $T$ for commutative monoids, though here "having finite sums" is a structure, not a property, so this monad is not idempotent. In this case the map $\alpha : TS \to S$ preserves finite sums when $S$ is a commutative monoid and thus has them.

Nathanael Arkor (Jun 04 2025 at 11:21):

Fernando Chu said:

I'm probably missing something extremely obvious. Why is this the case? An algebra for $T$ should be a category $C$ equipped with a (not necessarily $\Phi$-colimits-preserving) functor $\alpha:TC\to C$ such that some diagrams commute, right?

By definition, a "free $\Phi$-cocompletion" provides a left pseudoadjoint to the forgetful 2-functor from the 2-category of $\Phi$-cocomplete categories and $\Phi$-cocontinuous functors to the 2-category of categories.

Nathanael Arkor (Jun 04 2025 at 11:23):

This forgetful 2-functor is furthermore pseudomonadic. This is perhaps the point that's not obvious to you? It can be verified by checking a 2-dimensional version of the monadicity theorem, for instance.

Nathanael Arkor (Jun 04 2025 at 11:25):

However, if you'd like a really concrete proof that the algebras are precisely the $\Phi$-cocomplete categories (i.e. directly using the definition of (pseudo)algebra), without using any abstract machinery, @Philip Saville, Andrew Slattery and I give such a proof in §7 of Bicategories of algebras for relative pseudomonads.

Fernando Chu (Jun 04 2025 at 11:26):

Yes! The forgetful 2-functor being pseudomonadic was the step I was missing. Thanks a lot!

Notification Bot (Jun 04 2025 at 11:26):

Fernando Chu has marked this topic as resolved.

Nathanael Arkor (Jun 04 2025 at 11:28):

Another paper you might find helpful is Kelly–Lack's On the monadicity of categories with chosen colimits. The focus is a little different, because they assume pseudomonadicity from the outset, and then deduce 2-monadicity, but the introduction of the paper may provide useful context.