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Session S27 - Categories and Topology

Friday, July 16, 14:30 ~ 15:00 UTC-3

Locally bounded enriched categories

Rory Lucyshyn-Wright

Brandon University, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak93c50c7a163bfba0750a4b161ff8ee6c').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy93c50c7a163bfba0750a4b161ff8ee6c = 'l&#117;cyshyn-wr&#105;ghtr' + '&#64;'; addy93c50c7a163bfba0750a4b161ff8ee6c = addy93c50c7a163bfba0750a4b161ff8ee6c + 'br&#97;nd&#111;n&#117;' + '&#46;' + 'c&#97;'; var addy_text93c50c7a163bfba0750a4b161ff8ee6c = 'l&#117;cyshyn-wr&#105;ghtr' + '&#64;' + 'br&#97;nd&#111;n&#117;' + '&#46;' + 'c&#97;';document.getElementById('cloak93c50c7a163bfba0750a4b161ff8ee6c').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy93c50c7a163bfba0750a4b161ff8ee6c + '\'>'+addy_text93c50c7a163bfba0750a4b161ff8ee6c+'<\/a>';

Locally bounded categories [3] have some of the convenient features of locally presentable categories, such as reflectivity results for orthogonal subcategories [1,2] as well as results on the existence of free monads, colimits of monads, and colimits in categories of algebras [2]. The axioms for locally bounded categories are formulated in terms of factorization systems and are general enough to admit a vast array of categories that are not locally presentable but provide important backgrounds for topology and analysis. In particular, there are various categories of topological structures and many quasitoposes that are locally bounded but not locally presentable. Based on ideas of Freyd and Kelly [1], the notion of locally bounded category was introduced by Kelly [3], who employed a given locally bounded closed category $\mathcal{V}$ as the basis for a general treatment of enriched limit theories. The latter theories generalize Kelly's enriched finite limit theories, which Kelly introduced in [4] along with notions of locally presentable closed category and locally presentable $\mathcal{V}$-category. However, notably absent from the literature is a notion of locally bounded $\mathcal{V}$-category that would complete the parallel between the locally presentable and locally bounded settings.

In this talk on joint work with Jason Parker, we introduce a notion of locally bounded $\mathcal{V}$-category enriched over a locally bounded closed category $\mathcal{V}$. Locally bounded $\mathcal{V}$-categories are examples of the more basic notion of $\mathcal{V}$-factegory, by which we mean a $\mathcal{V}$-category equipped with an enriched proper factorization system $(\mathcal{E},\mathcal{M})$ that is suitably compatible with the given factorization system carried by $\mathcal{V}$, and we say that a $\mathcal{V}$-factegory $\mathcal{C}$ is cocomplete if it is cocomplete as a $\mathcal{V}$-category and has arbitrary cointersections of $\mathcal{E}$-morphisms. We then say that a cocomplete $\mathcal{V}$-factegory $\mathcal{C}$ is locally bounded if it has a small $(\mathcal{E},\mathcal{M})$-generator $\mathcal{G}$ consisting of objects $G$ that are $\alpha$-bounded for some regular cardinal $\alpha$, meaning that $\mathcal{C}(G,-):\mathcal{C} \rightarrow \mathcal{V}$ preserves $\alpha$-filtered unions of $\mathcal{M}$-subobjects.

We show that locally bounded $\mathcal{V}$-categories support several adjoint functor theorems and a reflectivity result for enriched orthogonal subcategories, as well as results on the existence of algebraically free $\mathcal{V}$-monads, colimits of $\mathcal{V}$-monads, and enriched colimits in $\mathcal{V}$-categories of algebras. As a technique for constructing examples of locally bounded $\mathcal{V}$-categories, we show that a cocomplete $\mathcal{V}$-factegory $\mathcal{C}$ is locally bounded as soon as there exists a right adjoint $\mathcal{V}$-functor $G:\mathcal{C} \rightarrow \mathcal{B}$ that is valued in some locally bounded $\mathcal{V}$-category $\mathcal{B}$ and satisfies certain axioms; we call such a $\mathcal{V}$-functor $G$ a bounding right adjoint for $\mathcal{C}$. We characterize locally bounded $\mathcal{V}$-categories as precisely those cocomplete $\mathcal{V}$-factegories $\mathcal{C}$ that admit a bounding right adjoint valued in a presheaf $\mathcal{V}$-category.

We show that bounding right adjoints give rise to various classes of examples of locally bounded $\mathcal{V}$-categories, including certain orthogonal subcategories of locally bounded $\mathcal{V}$-categories as well as certain $\mathcal{V}$-categories of algebras/models for monads/theories on locally bounded $\mathcal{V}$-categories.

By analogy with Kelly's notion of finite weighted limit enriched in a locally finitely presentable closed category $\mathcal{V}$, we define a notion of $\alpha$-small weighted limit enriched in a locally $\alpha$-bounded closed category $\mathcal{V}$. We show that the $\mathcal{V}$-category of models for an $\alpha$-small-limit sketch or theory is locally $\alpha$-bounded as soon as $\mathcal{V}$ is locally $\alpha$-bounded and $\mathcal{E}$-cowellpowered.

We also discuss applications of locally bounded $\mathcal{V}$-categories to the study of enriched monads for subcategories of arities.

[1] P. J. Freyd and G. M. Kelly, Categories of continuous functors, Journal of Pure and Applied Algebra 2 (1972) 169-191.

[2] 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 22 (1980) 1-83.

[3] 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].

[4] G. M. Kelly, Structures defined by finite limits in the enriched context, I, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 23 (1982), 3-42.

Joint work with Jason Parker (Brandon University, Canada).