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Stream: community: our work

Topic: Jason Parker

Jason Parker (Sep 13 2022 at 19:37):

Hello, I’m Jason Parker, and I’m just starting my third year as a postdoctoral fellow working with Rory Lucyshyn-Wright at Brandon University in Canada (as I mentioned in my introduction post!). I’d like to describe some of my past, current, and future research in this stream; hopefully these initial posts are not too long!

• My PhD at the University of Ottawa was advised by Pieter Hofstra (who passed away unexpectedly in May) and Philip Scott, and investigated the topic of isotropy in the context of locally finitely presentable categories. To give some quick background on this, George Bergman showed in this paper that the inner automorphisms of group theory can be characterized purely categorically within the category of groups. This categorical characterization of group-theoretic inner automorphisms can then be generalized to an arbitrary category. My thesis was about explicitly uncovering what this categorical characterization of inner automorphisms amounts to in locally finitely presentable categories. In this initial paper we characterized the categorical inner automorphisms in the category of models of any (finitary, single-sorted) algebraic theory (of which the theory of groups is a typical example). In this subsequent paper (which I presented at ACT2021; see these slides), we then characterized the categorical inner automorphisms in the category of models of any (multi-sorted, finitary) essentially algebraic theory, i.e. in any locally finitely presentable category. In this preprint I characterize the categorical inner automorphisms in the category of presheaves of $\mathbb{T}$-models for a large class of essentially algebraic theories $\mathbb{T}$; in particular, I characterize the categorical inner automorphisms in the category of presheaves of groups. These slides provide an overview of most of these papers.

Since working with Rory Lucyshyn-Wright, my research has primarily turned to enriched algebraic theories and enriched monads. Overall, we’ve been working to extend much of the previous literature on these topics to a much more general setting. Here’s the basic setup: you start with a symmetric monoidal closed category $\mathscr{V}$, a $\mathscr{V}$-category $\mathscr{C}$, and a full dense subcategory $\mathscr{J} \hookrightarrow \mathscr{C}$, which is called a subcategory of arities. In the classical context of finitary universal algebra, you have $\mathscr{V} = \mathscr{C} = \mathsf{Set}$ and $\mathscr{J} = \mathsf{FinCard}$, the full subcategory of $\mathsf{Set}$ consisting of the finite cardinals. The objects of $\mathscr{J}$ are your “arities”, which you can use to define enriched algebraic structure on objects of the $\mathscr{V}$-category $\mathscr{C}$. In most of the previous literature on enriched algebra relative to a subcategory of arities, $\mathscr{V}$ has been taken to be a locally (finitely) presentable closed category and $\mathscr{C}$ a locally (finitely) presentable $\mathscr{V}$-category. Here's what we’ve been working on:

• Generalizing this paper by Max Kelly and John Power and this paper by Max Kelly and Steve Lack (see also this paper by Steve Lack), in this preprint we show under fairly modest assumptions on $\mathscr{C}$ and $\mathscr{V}$ that if $\mathscr{J} \hookrightarrow \mathscr{C}$ is a small subcategory of arities that is bounded and eleutheric, then one obtains very nice presentation and algebraic colimit results for $\mathscr{J}$-ary $\mathscr{V}$-monads on $\mathscr{C}$. For an overview, see these slides. Our results apply in particular when $\mathscr{J} \hookrightarrow \mathscr{C}$ is a small and eleutheric subcategory of arities in a locally bounded $\mathscr{V}$-category $\mathscr{C}$ enriched over a locally bounded closed category $\mathscr{V}$; in this paper we develop the general theory of locally bounded enriched categories (see these slides for an overview). For example, any cartesian closed topological category over $\mathsf{Set}$ (e.g. the category of compactly generated (Hausdorff) spaces) is a locally bounded cartesian closed category. I presented this work at CT2020-21.

Jason Parker (Sep 13 2022 at 19:38):

• The results of our preprint are somewhat theoretical, and are not necessarily easily used in practice to explicitly construct enriched monads. In this preprint we use the results of the former preprint as a theoretical underpinning to show that when $\mathscr{J} \hookrightarrow \mathscr{C}$ is a subcategory of arities satisfying the conditions of this preprint, then we obtain very practical and flexible methods for explicitly describing enriched $\mathscr{J}$-ary $\mathscr{V}$-monads on $\mathscr{C}$, using notions of enriched operations and enriched equations akin to those from classical universal algebra. Rory gave a talk about this work at ACT 2022.

  

• There have been many correspondences established in the literature between enriched monads and enriched algebraic theories of some kind. We have the classical Lawvere-Linton equivalence between Lawvere theories and finitary monads on $\mathsf{Set}$, which was subsequently generalized and enriched by Power and Nishizawa (see here and here) to an equivalence between enriched Lawvere theories and finitary $\mathscr{V}$-monads on a locally finitely presentable $\mathscr{V}$-category $\mathscr{C}$ enriched over a locally finitely presentable closed category $\mathscr{V}$. Rory Lucyshyn-Wright subsequently showed in this paper that when $\mathscr{J} \hookrightarrow \mathscr{V}$ is an eleutheric system of arities in an essentially arbitrary symmetric monoidal closed category $\mathscr{V}$, then one obtains an equivalence between $\mathscr{J}$-theories and $\mathscr{J}$-ary $\mathscr{V}$-monads on $\mathscr{V}$. Recently, Bourke and Garner showed in this paper that when $\mathscr{J} \hookrightarrow \mathscr{C}$ is an arbitrary small subcategory of arities in a locally presentable $\mathscr{V}$-category $\mathscr{C}$ enriched over a locally presentable closed category $\mathscr{V}$, then one obtains an equivalence between $\mathscr{J}$-theories and $\mathscr{J}$-nervous $\mathscr{V}$-monads on $\mathscr{C}$. Rory and I are currently finishing a paper, to be titled "Enriched structure-semantics adjunctions and monad-theory equivalences for subcategories of arities", in which we show that all of these enriched monad-theory correspondences can be obtained as particular instances of a general result, which also yields new examples. Specifically, we have shown that if $\mathscr{V}$ is an essentially arbitrary symmetric monoidal closed category and $\mathscr{J} \hookrightarrow \mathscr{C}$ is an amenable subcategory of arities in a $\mathscr{V}$-category $\mathscr{C}$, which basically means that all $\mathscr{J}$-theories admit free algebras, then there is an equivalence between $\mathscr{J}$-theories and $\mathscr{J}$-nervous $\mathscr{V}$-monads on $\mathscr{C}$. This result specializes to recover all of the enriched monad-theory equivalences mentioned above (and more). Moreover, we obtain the new result that when $\mathscr{J} \hookrightarrow \mathscr{C}$ is an arbitrary small subcategory of arities in a $\mathscr{V}$-sketchable $\mathscr{V}$-category $\mathscr{C}$ enriched over a locally bounded closed category $\mathscr{V}$, then $\mathscr{J}$ is automatically amenable. In particular, we may take $\mathscr{C} = \mathscr{V}$, and obtain an enriched monad-theory equivalence for any small subcategory of arities in a locally bounded closed category $\mathscr{V}$, including many convenient categories of topological spaces. I gave a talk on this material at ACT 2022.

• We are also working on extending most of the main results and techniques of this paper and this paper to the more general context that I just discussed, which will in particular allow us to easily present $\mathscr{J}$-nervous $\mathscr{V}$-monads on locally bounded closed categories $\mathscr{V}$ for arbitrary small subcategories of arities $\mathscr{J} \hookrightarrow \mathscr{V}$.

We have a few other projects/papers in the pipeline as well, but these are the main ones that we’ve been working on.

• I’ve recently become (very) interested in applications of category theory and multi-sorted equational logic to database theory (see this paper and this paper), and I’m currently also working on extending the work in these papers to a more general enriched setting, which would hopefully provide explicit methods for migrating/integrating certain kinds of “enriched” data. But this is still in fairly early stages, although I’m hoping to finish my first preprint on it late this year or early next year.

Jason Parker (Dec 09 2022 at 18:14):

My paper “Presentations and algebraic colimits of enriched monads for a subcategory of arities” with Rory Lucyshyn-Wright was published in TAC this week. As I discuss in a bit more detail above, in this paper we show that many results about free monads, presentations of monads, and colimits of monads established by Kelly, Power, and Lack (and other authors) can be recovered as instances of a general axiomatic setting, namely that of a bounded and eleutheric subcategory of arities. Here are some slides (from Category Theory Octoberfest 2021) that summarize the paper.

In the follow-up preprint “Diagrammatic presentations of enriched monads and varieties for a subcategory of arities”, we work in the same setting and show how to provide extremely explicit and user-friendly presentations of enriched monads.

Jason Parker (Aug 10 2023 at 16:31):

I have a new preprint on the arXiv, titled "Free algebras of topologically enriched multi-sorted equational theories". The quick summary is that I develop a topologically enriched generalization of classical multi-sorted equational theories, and I establish explicit and concrete descriptions of free algebras for these enriched equational theories. This work is closely connected to recent work by myself, Lucyshyn-Wright, and many others on enriched algebraic theories and their presentations. But here's the full abstract:

Classical multi-sorted equational theories and their free algebras have been fundamental in mathematics and computer science. In this paper, we present a generalization of multi-sorted equational theories from the classical (Set-enriched) context to the context of enrichment in a symmetric monoidal category V that is topological over Set. Prominent examples of such categories include: various categories of topological and measurable spaces; the categories of models of relational Horn theories without equality, including the categories of preordered sets and (extended) pseudo-metric spaces; and the categories of quasispaces (a.k.a. concrete sheaves) on concrete sites, which have recently attracted interest in the study of programming language semantics.

Given such a category V, we define a notion of V-enriched multi-sorted equational theory. We show that every V-enriched multi-sorted equational theory T has an underlying classical multi-sorted equational theory |T|, and that free T-algebras may be obtained as suitable liftings of free |T|-algebras. We establish explicit and concrete descriptions of free T-algebras, which have a convenient inductive character when V is cartesian closed. We provide several examples of V-enriched multi-sorted equational theories, and we also discuss the close connection between these theories and the presentations of V-enriched algebraic theories and monads studied in recent papers by the author and Lucyshyn-Wright.

I've given a few conference talks on this work this summer, including at ACT; here are the slides for that one. Comments are very welcome!