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### 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. document.getElementById('cloakebd27d7ff8405bb00a836b0cdd02fcf2').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyebd27d7ff8405bb00a836b0cdd02fcf2 = 'p&#97;rk&#101;rj' + '&#64;'; addyebd27d7ff8405bb00a836b0cdd02fcf2 = addyebd27d7ff8405bb00a836b0cdd02fcf2 + 'br&#97;nd&#111;n&#117;' + '&#46;' + 'c&#97;'; var addy_textebd27d7ff8405bb00a836b0cdd02fcf2 = 'p&#97;rk&#101;rj' + '&#64;' + 'br&#97;nd&#111;n&#117;' + '&#46;' + 'c&#97;';document.getElementById('cloakebd27d7ff8405bb00a836b0cdd02fcf2').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyebd27d7ff8405bb00a836b0cdd02fcf2 + '\'>'+addy_textebd27d7ff8405bb00a836b0cdd02fcf2+'<\/a>';

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).