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Stream: community: our work

Topic: Bicategories of algebras for relative pseudomonads

Nathanael Arkor (Jan 23 2025 at 04:49):

@Philip Saville, Andrew Slattery and I have a new preprint on arXiv: Bicategories of algebras for relative pseudomonads. Here's the abstract:

We introduce pseudoalgebras for relative pseudomonads and develop their theory. For each relative pseudomonad $T$, we construct a free–forgetful relative pseudoadjunction that exhibits the bicategory of $T$-pseudoalgebras as terminal among resolutions of $T$. The Kleisli bicategory for $T$ thus embeds into the bicategory of pseudoalgebras as the sub-bicategory of free pseudoalgebras. We consequently obtain a coherence theorem that implies, for instance, that the bicategory of distributors is biequivalent to the 2-category of presheaf categories. In doing so, we extend several aspects of the theory of pseudomonads to relative pseudomonads, including doctrinal adjunction, transport of structure, and lax-idempotence. As an application of our general theory, we prove that, for each class of colimits $\Phi$, there is a correspondence between monads relative to free $\Phi$-cocompletions, and $\Phi$-cocontinuous monads on free $\Phi$-cocompletions.

I'll give some context for what we did here, since our work builds upon a few different ideas in the literature.

Nathanael Arkor (Jan 23 2025 at 04:53):

A [[relative pseudomonad]] is a joint generalisation of a [[relative monad]] – which is a generalisation of a monad to functors which may not be endofunctors – and a [[pseudomonad]] – which is a generalisation of a monad to situations in which functoriality and the monad laws only hold up to isomorphism. A motivating example of a relative pseudomonad is the [[presheaf construction]], which sends a small category $A$ to the locally small category $[A^{\text{op}}, \text{Set}]$. It is relative because it takes small things and returns large things, so is not an endofunctor; and is pseudo because it is not strictly functorial (since the functor structure is defined in terms of left extensions, i.e. colimits, which are only unique up to isomorphism).

Nathanael Arkor (Jan 23 2025 at 04:57):

Relative pseudomonads were introduced in a paper Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures by Fiore, Gambino, Hyland, and Winskel. There are two fundamental constructions associated to every monad: the Kleisli category and the category of algebras. In [FGHW], the authors showed that relative pseudomonads admit a Kleisli construction: for the presheaf relative pseudomonad, the Kleisli bicategory is precisely the bicategory of [[distributors]] / [[profunctors]].

Nathanael Arkor (Jan 23 2025 at 04:59):

However, [FGHW] did not consider any notion of algebra for relative pseudomonads. This is the contribution of our preprint. We show that, in analogy with the one-dimensional setting, every relative pseudomonad admits a bicategory of pseudoalgebras (in fact, several, depending on the notion of morphism between algebras). We then study various properties of this bicategory and of pseudoalgebras generally. This fills in the half of the story that was missing from [FGHW].

Nathanael Arkor (Jan 23 2025 at 05:01):

One of our main results is that the 2-category of pseudoalgebras for the presheaf construction is the 2-category of cocomplete categories and cocontinuous functors – this is what one would expect, but proving it turns out to be slightly subtle (we end up proving a significantly stronger result in the process).

Nathanael Arkor (Jan 23 2025 at 05:03):

As a fun application of our results, we apply the theory of relative pseudomonads to the theory of relative monads, and give a characterisation of the monads relative to free cocompletions.

Nathanael Arkor (Jan 23 2025 at 05:05):

Finally, I'll note that several of the results we develop appear to be new even for ordinary (i.e. non-relative) pseudomonads, as 2-categorical literature often focuses on 2-monads rather than the more general pseudomonads. So we expect the results of our paper to be useful even in the non-relative setting.

Niles Johnson (Jan 23 2025 at 13:12):

looks cool; thanks for the summary/intro!