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Stream: community: our work

Topic: The pullback theorem for relative monads

Nathanael Arkor (Apr 02 2024 at 09:33):

@Dylan McDermott and I have a new preprint on arXiv: The pullback theorem for relative monads. Here's the abstract:

A fundamental result in the theory of monads is the characterisation of the category of algebras for a monad in terms of a pullback of the category of presheaves on the category of free algebras: intuitively, this expresses that every algebra is a colimit of free algebras. We establish an analogous result for enriched relative monads with dense roots, and explain how it generalises the nerve theorems for monads with arities and nervous monads. As an application, we derive sufficient conditions for the existence of algebraic colimits of relative monads. More generally, we show that the pullback theorem for relative monads holds in any exact virtual equipment. In doing so, we are led to study the relationship between a $j$-relative monad $T$ and its associated loose-monad $E(j,T)$, and consequently show that the opalgebra object and the algebra object for $T$ may be constructed from certain double categorical limits and colimits associated to $E(j,T)$.

and here are some slides from a talk I gave on this subject recently.

This is the latest paper in a series on the theory of [[relative monads]] (following The formal theory of relative monads and Relative monadicity). It should be noted that one of our intentions with these papers is to demonstrate that the theory of relative monads is useful even for problems that appear only to concern ordinary monads, so, even if you're never come across relative monads before, you may be interested to take a look at the examples we give.

The main theorem is the following. Given a dense functor $j : A \to E$ and a $j$-relative monad $T$, the category of $T$-algebras may be constructed by the following pullback of categories (where $u_T$ is the forgetful functor, $n_j$ is the [[restricted Yoneda embedding]]/nerve along $j$, and $k_T$ is the Kleisli inclusion). This generalises a result due to Linton for ordinary monads.
image.png

If you're familiar with monads with arities, or monad–theory correspondences more generally, you've probably encountered pullbacks that look rather like this before. One of the things we do in the paper is explain how the pullback theorem for relative monads subsumes the "nerve theorems" for monads with arities and nervous monads.

One of our motivations for this paper was to not just to prove this result, but to also give a conceptual explanation for it. For this, we relate the pullback theorem to the theory of [[distributors]] / [[profunctors]] and (co)limits in double categories (or, more precisely, in [[virtual equipments]]). More generally, this allows us to establish the pullback theorem in the general setting of an exact virtual equipment, which gives us a pullback theorem for enriched relative monads for free.

Anyway, that's probably long enough an introduction :) As always, do let us know if you have any questions or comments.

Matteo Capucci (he/him) (Apr 02 2024 at 13:22):

(minor point, would you be bummed if I renamed this thread with you and Dylan's name? that's the informal convention in this stream)

Nathanael Arkor (Apr 02 2024 at 13:24):

Matteo Capucci (he/him) said:

(minor point, would you be bummed if I renamed this thread with you and Dyaln's name? that's the informal convention in this stream)

We should either not impose this convention in #community: our work, or we should reinstate #deprecated: our papers. I'm happy with either, but I want a thread for the paper, not for me.