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Stream: deprecated: our papers

Topic: Relative monadicity

Nathanael Arkor (May 18 2023 at 09:47):

@Dylan McDermott and I have a new preprint on arXiv: Relative monadicity.

It builds upon the theory we developed in our earlier paper regarding [[relative monads]] in a [[virtual equipment]]. Here's the abstract:

We establish a relative monadicity theorem for relative monads with dense roots in a virtual equipment, specialising to a relative monadicity theorem for enriched relative monads. In particular, for a dense $\mathbb V$-functor $j \colon A \to E$, a $\mathbb V$-functor $r \colon D \to E$ is $j$-monadic if and only if $r$ has a left $j$-relative adjoint and creates $j$-absolute colimits. We also establish a pasting law for relative adjunctions, and examine its interaction with relative monadicity. As a consequence, we derive necessary and sufficient conditions for the composite of a $\mathbb V$-functor with a $j$-monadic $\mathbb V$-functor to itself be $j$-monadic.

The quick summary is that we generalise the classical [[monadicity theorem]] to relative monads. Relative monadicity is typically a stronger condition than monadicity, and is able to capture algebraic structures whose operations have arities that are constrained in some way: for instance, if we know that a category is monadic over $\mathbf{Set}$, then we know it is presented by some infinitary algebraic theory. However, if we know that it is monadic relative to the inclusion $(\mathbf{FinSet} \hookrightarrow \mathbf{Set})$, then we furthermore know that it is presented by some finitary algebraic theory.

If you're not interested in the formal aspects of the theorem, you can skip from the introduction directly to §4, where we spell out the relative monadicity theorem in the context of (enriched) categories, and give a number of examples. In §5, we show that relative monadicity is useful to consider even if we're only interested in classical (non-relative) monadicity. In particular, we use relative monadicity to give necessary and sufficient conditions for the composite of a functor with a monadic functor to itself be monadic.

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

John Baez (May 18 2023 at 15:11):

Sounds cool! Do you find any interesting new examples of composites of monadic functors that are monadic?

Nathanael Arkor (May 18 2023 at 19:08):

Good question; this isn't something we've thought about yet. Our primary motivation for considering the monadicity of composites was a little more abstract: we were interested in understanding why algebraic functors (i.e. those functors induced by morphisms of algebraic theories) were monadic (a topic that was discussed on Zulip very recently). Since algebraic theories can be seen naturally as certain relative monads, we expected this fact ought to be an instance of a more general result about functors between categories of algebras for relative monads, and this led us to the pasting law that describes the (relative) monadicity of composites (Theorem 5.5 in the paper). I would definitely be interested to see if some new concrete examples could be found this way!