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Stream: learning: questions

Topic: Infinity-pretheories vs algebraic patterns

James Deikun (Oct 14 2025 at 07:07):

In section 9 of https://arxiv.org/abs/2106.02706 it is written that algebraic patterns and "polynomial" monads are a special case of their framework of $\mathcal{A}$-pretheories and $\mathcal A$-nervous monads. However, they use $\mathcal{O}^\mathit{in}$ as $\mathcal{A}$ and generally the inclusion of $\mathcal{O}^\mathit{in}$ in $\mathrm{Pr}(\mathcal{O}^\mathit{el})$ is not ff, whereas their framework always assumes $\mathcal A$ is a full subcategory of $\mathcal E$. This shows up pretty obviously in one of the examples, the $E_\infty$ monad on $\mathcal S$, where $\mathcal A$ is the category of finite sets and injections, which is not a full subcategory of $\mathcal S$. Elsewhere in the paper (Theorem 8.7) they show the $E_\infty$ monad is $\mathrm Fin$-nervous, but it's not very clear how to extend this kind of treatment nicely to all algebraic patterns.

On the other hand, one asks, does one even need to do that? Examining the main theorems of the article I was unable to really find any uses of the assumption of fullness. Is it actually fine to just reject this assumption and use non-full dense functors $\mathcal A \to \mathcal E$ to construct pretheories? And if not, what is a good way to obtain a suitable eso functor $\mathcal A \to \mathcal K$ from an algebraic pattern?

Nathanael Arkor (Oct 14 2025 at 11:03):

I can't speak for the higher dimensional setting, but for monad--theory correspondences in the one-dimensional setting, full faithfulness is essentially never necessary. However, given a dense functor $\mathcal A \to \mathcal B$, if we take the fully faithful part of the full image factorisation, it will also be dense (e.g. by Proposition 5.11 of Kelly's Basic Concepts), and theories relative to this functor are equivalent to theories relative to the original functor. I expect the same is true for $(\infty, 1)$-categories.