### Session S27 - Categories and Topology

Friday, July 16, 15:00 ~ 15:30 UTC-3

## Presentations and Algebraic Colimits of Enriched Monads

### Jason Parker

#### Brandon University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

Signatures and presentations for monads and theories in (enriched) universal algebra have been previously studied mainly in the context of locally presentable enriched categories over a locally presentable symmetric monoidal closed category $\mathscr{V}$; see e.g. [6] and its successor [7], as well as the recent [2]. In this talk, summarizing joint work with Rory Lucyshyn-Wright, we develop a framework for studying such phenomena that subsumes a much wider class of enriched categories, including the locally bounded categories of [5, Chapter 6] and the symmetric monoidal closed $\pi$-categories of [1]. Locally bounded categories in particular provide a vast generalization of locally presentable categories and include many quasitoposes and categories of topological spaces as examples. Moreover, our framework is sufficiently general to encompass the Lawvere $\Phi$-theories of [8] even when the base $\mathscr{V}$ is not locally presentable.

Given a symmetric monoidal closed category $\mathscr{V}$ and a $\mathscr{V}$-category $\mathscr{C}$, a subcategory of arities in $\mathscr{C}$ is a dense sub-$\mathscr{V}$-category $j : \mathscr{J} \hookrightarrow \mathscr{C}$, which is said to be eleutheric if it satisfies a certain exactness condition (which in particular guarantees that arbitrary $\mathscr{V}$-functors $\mathscr{J} \to \mathscr{C}$ are equivalent to $\mathscr{J}$-ary $\mathscr{V}$-endofunctors on $\mathscr{C}$ as defined below; cf. [9, 7.1]). Examples of eleutheric subcategories of arities abound:

-The subcategory of $\alpha$-presentable objects $\mathscr{C}_{\alpha} \hookrightarrow \mathscr{C}$ in any locally $\alpha$-presentable $\mathscr{V}$-category $\mathscr{C}$;

-More generally, the subcategory $\mathscr{C}_\Phi \hookrightarrow \mathscr{C}$ of $\Phi$-presentable objects in any locally $\Phi$-presentable $\mathscr{V}$-category $\mathscr{C}$ for a class of locally small weights $\Phi$ satisfying Axiom A of [8];

-Even more generally, any free $\Psi$-cocompletion $j : \mathscr{J} \hookrightarrow \mathscr{C}$ for a class of weights $\Psi$ and a $\mathscr{V}$-category $\mathscr{J}$;

-The unrestricted subcategory of arities $1_{\mathscr{C}}: \mathscr{C} \to \mathscr{C}$ in an arbitrary $\mathscr{V}$-category $\mathscr{C}$;

-The subcategory of arities $\left\{I\right\} \hookrightarrow \mathscr{V}$ in $\mathscr{V}$ consisting of just the unit object;

-The subcategory $\left\{ n \cdot I \colon n \in \mathbb{N}\right\} \hookrightarrow \mathscr{V}$ consisting of finite copowers of the unit object in any symmetric monoidal closed $\pi$-category $\mathscr{V}$ (cf. [1]), which need not be locally presentable.

Given an eleutheric subcategory of arities $j : \mathscr{J} \hookrightarrow \mathscr{C}$ in a $\mathscr{V}$-category $\mathscr{C}$, a $\mathscr{V}$-endofunctor $T : \mathscr{C} \to \mathscr{C}$ is then said to be $\mathscr{J}$-ary if it preserves left Kan extensions along $j$, and a $\mathscr{V}$-monad $\mathbb{T} = (T, \eta, \mu)$ on $\mathscr{C}$ is said to be $\mathscr{J}$-ary if its underlying $\mathscr{V}$-endofunctor $T$ is so. For example:

-The $\mathscr{J}$-ary monads on a locally $\alpha$-presentable $\mathscr{V}$-category $\mathscr{C}$ are precisely the $\alpha$-ary monads, i.e. the monads that preserve $\alpha$-filtered colimits, and correspond to the enriched Lawvere theories of [10] when $\alpha = \aleph_0$;

-More generally, the $\mathscr{J}$-ary monads on a locally $\Phi$-presentable $\mathscr{V}$-category $\mathscr{C}$ are precisely the $\Phi$-accessible monads, i.e. the monads that preserve $\Phi$-flat colimits, and correspond to the Lawvere $\Phi$-theories of [8];

-Even more generally, the $\mathscr{J}$-ary monads on a $\mathscr{V}$-category $\mathscr{C}$ which is the free $\Psi$-cocompletion of a subcategory $\mathscr{J}$ for a class of weights $\Psi$ are precisely the $\Psi$-cocontinuous monads, and correspond to the $\mathscr{J}$-theories of [9] when $\mathscr{C} = \mathscr{V}$;

-The $\mathscr{J}$-ary monads for the unrestricted subcategory of arities are just arbitrary $\mathscr{V}$-monads, and correspond to the $\mathscr{V}$-theories of [3] when $\mathscr{C} = \mathscr{V}$;

-The $\mathscr{J}$-ary monads for the subcategory of arities on the unit object of $\mathscr{V}$ are the tensor-preserving monads on $\mathscr{V}$, and correspond to monoids in $\mathscr{V}$ (cf. [9, 4.2.5]);

-The $\mathscr{J}$-ary monads for the subcategory of arities on the finite copowers of the unit object in any $\pi$-category $\mathscr{V}$ correspond to the Borceux-Day enriched algebraic theories of [1].

Finally, given a subcategory of arities $j : \mathscr{J} \hookrightarrow \mathscr{C}$ in a $\mathscr{V}$-category $\mathscr{C}$, one can define the notion of a $\mathscr{J}$-signature in $\mathscr{C}$ as an $\mathsf{ob}\mathscr{J}$-indexed family of objects in $\mathscr{C}$, analogously to the definition of a traditional signature in universal algebra as a family of sets indexed by the finite cardinals, as well as the notion of a $\mathscr{J}$-presentation for $\mathscr{J}$-ary monads.

In this talk, we will explore the relationship between $\mathscr{J}$-ary monads on the one hand and $\mathscr{J}$-signatures and presentations on the other for a small eleutheric subcategory of arities $j : \mathscr{J} \hookrightarrow \mathscr{C}$ in a (cocomplete and cotensored) $\mathscr{V}$-category $\mathscr{C}$ that satisfies a mild boundedness condition defined in terms of factorization systems and notions from [4]. It turns out that any small and eleutheric subcategory of arities whatsoever in a locally bounded enriched category satisfies this boundedness condition, which thereby enables us to develop the following central results in a much broader context than has been traditionally studied:

-The forgetful functor from $\mathscr{J}$-ary monads on $\mathscr{C}$ to $\mathscr{J}$-signatures in $\mathscr{C}$ is monadic;

-Any small diagram of $\mathscr{J}$-ary monads on $\mathscr{C}$ has a colimit which is both $\mathscr{J}$-ary and algebraic, in the sense that its $\mathscr{V}$-category of algebras is isomorphic to the limit of the $\mathscr{V}$-categories of algebras of the monads in the diagram;

-As a consequence, any $\mathscr{J}$-presentation $P$ generates a $\mathscr{J}$-ary monad, whose $\mathscr{V}$-category of algebras is isomorphic to the (suitably defined) $\mathscr{V}$-category of $P$-algebras for the presentation. Moreover, any $\mathscr{J}$-ary monad has such a presentation.

[1] F. Borceux and B. Day. Universal algebra in a closed category. Journal of Pure and Applied Algebra Vol. 16 No. 2 (1980) 133-147.

[2] J. Bourke and R. Garner. Monads and theories. Advances in Mathematics Vol. 351 (2019) 1024-1071.

[3] E.J. Dubuc. Enriched semantics-structure (meta) adjointness. Revista de la Union Matematica Argentina Vol. 25 (1970) 5-26.

[4] G.M. Kelly. A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on. Bulletin of the Australian Mathematical Society Vol. 22 (1980) 1-83.

[5] G.M. Kelly. Basic concepts of enriched category theory. Repr. Theory and Applications of Categories, No. 10, 2005, Reprint of the 1982 original [Cambridge University Press].

[6] G.M. Kelly and A.J. Power. Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads. Journal of Pure and Applied Algebra Vol. 89 (1993) 163-179.

[7] S. Lack. On the monadicity of finitary monads. Journal of Pure and Applied Algebra Vol. 140 (1999) 65-73.

[8] S. Lack and J. Rosicky. Notions of Lawvere theory. Applied Categorical Structures 19 (2011) 363-391.

[9] R.B.B. Lucyshyn-Wright. Enriched algebraic theories and monads for a system of arities. Theory and Applications of Categories Vol. 31 No. 5 (2016) 101-137.

[10] K. Nishizawa and J. Power. Lawvere theories enriched over a general base. Journal of Pure and Applied Algebra Vol. 213 Issue 3 (2009) 377-386.

Joint work with Rory Lucyshyn-Wright (Brandon University, Manitoba, Canada).