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Stream: deprecated: our papers

Topic: Adjoint functor theorems for lax-idempotent pseudomonads

Nathanael Arkor (Jun 21 2023 at 09:50):

@Ivan Di Liberti, @fosco and I have a new preprint on arXiv: Adjoint functor theorems for lax-idempotent pseudomonads.

Here's the abstract:

For each pair of lax-idempotent pseudomonads $R$ and $I$, for which $I$ is locally fully faithful and $R$ distributes over $I$, we establish an adjoint functor theorem, relating $R$-cocontinuity to adjointness relative to $I$. As special cases, we recover variants of the adjoint functor theorem of Freyd, the multiadjoint functor theorem of Diers, and the pluriadjoint functor theorem of Solian–Viswanathan. More generally, we recover enriched $\Phi$-adjoint functor theorems for weakly sound classes of weight $\Phi$.

Let me give a quick overview of the idea behind the paper. The concept of an [[adjoint functor]] is central in category theory, and we encounter examples all over the place. A somewhat less well-known concept is that of a [[multiadjoint functor]], where we ask for a natural isomorphism of the form

$$\mathcal D(L x, y) \cong \coprod_{i \in I} C(x, R_i y)$$

for some set $I$. Despite being less well-known, multiadjoints also appear commonly in category theory: for instance, non-algebraic structures such as fields often have multiadjointness properties; and [[parametric right adjoints]], which appear, for example, in the theory of polynomial functors and familial monads, are examples of multiadjoints. Adjoint functors and multiadjoint functors are both special cases of a more general concept: that of a $\Phi$-adjoint functor, where $\Phi$ is a class of colimits (viz. diagram shapes, or weights). Adjoint functors arise by taking $\Phi = \varnothing$, multiadjoint functors arise by taking $\Phi$ to be the class of discrete colimits, and then there are other examples in the literature that arise from other choices, such as pluriadjoint functors, which arise by taking $\Phi$ to be the class of filtered categories. This notion of adjointness with respect to a class of colimits was first studied by Tholen in Pro-Categories and Multiadjoint Functors.

One of the natural questions that arises when learning about these concepts is: how do these interact with cocontinuity? For instance, we know that left adjoints preserve colimits and, conversely, a cocontinuous functor that satisfies a smallness condition (e.g. the solution-set condition) is left-adjoint. So we might hope for something similar for multiadjoints, and $\Phi$-adjoints more generally. It turns out this is the case for many particular examples. For instance, Diers proved that a functor is left-multiadjoint if and only if it preserves connected colimits and satisfies a smallness condition. Similarly, Solian and Viswanathan proved that a functor is left-pluriadjoint if and only if it preserves finite colimits and satisfies a smallness condition.

The motivation behind our paper is to understand the nature of these $\Phi$-adjoint theorems conceptually, and to prove a general adjointness theorem that captures these examples as special cases. Our main observation is that one can see these adjointness results as relating properties of two classes of colimits that are related by a distributivity property. Formally speaking, this relationship corresponds to a [[pseudodistributive law]] between [[lax-idempotent pseudomonads]], which are pseudomonads that capture the notion of cocompletion in a suitable sense.

While our main theorem is entirely formal (i.e. formulated in an arbitrary 2-category), which allows us, for example, to capture adjointness results for enriched $\Phi$-adjoint functors, the introduction should be readable even for those who are not familiar with 2-category theory. The introduction gives an intuition for the relationship between notions of cocontinuity, adjointness, and smallness. Even for those familiar with the classical [[adjoint functor theorem]], the precise formulation (e.g. the solution set condition) can appear quite mysterious or ad hoc. Our intention is that the understanding of the adjoint functor theorem we present in the introduction shows that that it really is intuitive, once one looks at it in the right way. So, even if you're only interested in ordinary adjoint functors, you may find the paper sheds some light on this topic.

As always, do let us know if you have any questions or comments.

John Baez (Jun 21 2023 at 14:29):

Interesting! This came as a big surprise:

Our main observation is that one can see these adjointness results as relating properties of two classes of colimits that are related by a distributivity property. Formally speaking, this relationship corresponds to a [[pseudodistributive law]] between [[lax-idempotent pseudomonads]], which are pseudomonads that capture the notion of cocompletion in a suitable sense.

You were talking about adjoint functor theorems for multi-adjoints, and all of a sudden you're talking about two lax-idempotent monads. Where do these lax-idempotent monads come from? Is there an easy example where we can see them? I guess I should read the paper.

Jason Erbele (Jun 21 2023 at 14:39):

Minor nitpick – In the abstract, is "adjointess" supposed to be "adjointness"? It also appears the same way in the abstract on arxiv and in the abstract section of the pdf.

Nathanael Arkor (Jun 21 2023 at 14:42):

The link between $\Phi$-adjoints and lax-idempotent pseudomonads is that, for any class of colimits $\Phi$, there is a lax-idempotent pseudomonad $\overline\Phi$ on $\mathbf{CAT}$ whose pseudoalgebras are the locally small categories with $\Phi$-colimits. Furthermore, a functor $f \colon a \to b$ is a left $\Phi$-adjoint if and only if it is "$\overline\Phi$-admissible", which means that $\overline\Phi f \colon \overline\Phi a \to \overline\Phi b$ is left-adjoint. If we take $\Phi$ to be the discrete colimits, then we can recover multiadjointness as an "admissibility" property with respect to the induced lax-idempotent pseudomonad.

Nathanael Arkor (Jun 21 2023 at 14:43):

Lax-idempotent pseudomonads can be thought of as capturing the notions of cocompleteness, cocontinuity, and admissibility/adjointness purely 2-categorically.

Nathanael Arkor (Jun 21 2023 at 14:44):

Jason Erbele said:

Minor nitpick – In the abstract, is "adjointess" supposed to be "adjointness"? It also appears the same way in the abstract on arxiv and in the abstract section of the pdf.

Yes, thank you :sweat:

John Baez (Jun 21 2023 at 14:54):

Nathanael Arkor said:

The link between $\Phi$-adjoints and lax-idempotent pseudomonads is that, for any class of colimits $\Phi$, there is a lax-idempotent pseudomonad $\overline\Phi$ on $\mathbf{Cat}$ such that a functor $f \colon a \to b$ is a left $\Phi$-adjoint if and only if it is "$\overline\Phi$-admissible", which means that $\overline\Phi f \colon \overline\Phi a \to \overline\Phi b$ is left-adjoint. If we take $\Phi$ to be the discrete colimits, then we can recover multiadjointness as an "admissibility" property with respect to the induced lax-idempotent pseudomonad.

Thanks. So if $\Phi$ is the discrete colimits then $\overline{\Phi}$ is the "free cocompletion" pseudomonad on $\mathbf{Cat}$, namely

$\overline{\Phi} C = \mathsf{Set}^{C^{\text{op}}}$ ?

Nathanael Arkor (Jun 21 2023 at 15:00):

Sorry, I should have spelled it out explicitly. When $\Phi$ is the discrete colimits, then $\overline\Phi$ is the [[free coproduct completion]] pseudomonad on $\mathbf{CAT}$ (often called the $\mathbf{Fam}$ completion). A functor is a left multiadjoint if and only if $\mathbf{Fam}(f)$ is left adjoint.

Similarly, the free small-cocompletion pseudomonad arises by taking $\Phi$ to be all small colimits.

Morgan Rogers (he/him) (Jun 21 2023 at 15:01):

@Axel Osmond will be interested in this

Kevin Arlin (Jun 21 2023 at 16:34):

Are free completions colax idempotent?

Nathanael Arkor (Jun 21 2023 at 16:36):

Kevin Arlin said:

Are free completions colax idempotent?

Yes. (And in this way the results of our paper dualise to give adjoint functor theorems for right adjointness, rather than left adjointness.)

John Baez (Jun 21 2023 at 19:45):

Oh, I somehow read "discrete colimits" to mean "small colimits" - I have no idea why, but that's the origin of my wrong guess. Now things sort of make sense.