Cauchy characterization of enriched categories

Abstract

Soon after the appearance of enriched category theory in the sense of Eilenberg-Kelly 1 , I wondered whether V-categories could be the same as W-categories for non-equivalent monoidal categories V and W. It was not until my four-month sabbatical in Milan at the end of 1981 that I made a serious attempt to properly formulate this question and try to solve it.By this time I was very impressed by the work of Bob Walters [28] showing that sheaves on a site were enriched categories. On sabbatical at Wesleyan University (Middletown) in 1976-77, I had looked at a preprint of Denis Higgs showing that sheaves on a Heyting algebra H could be viewed as some kind of H-valued sets. The latter seemed to be understandable as enriched categories without identities. Walters' deeper explanation was that they were enriched categories (with identities) except that the base was not H but rather a bicategory built from H. A stream of research was initiated in which the base monoidal category for enrichment was replaced, more generally, by a base bicategory.In analysis, Cauchy complete metric spaces are often studied as completions of more readily defined metric spaces. Bill Lawvere [15] had found that Cauchy completeness could be expressed for general enriched categories with metric spaces as a special case. Cauchy sequences became left adjoint modules 2 and convergence became representability. In Walters' work it was the Cauchy complete enriched categories that were the sheaves.It was natural then to ask, rather than my original question, whether Cauchy complete V-categories were the same as Cauchy complete W-categories for appropriate base bicategories V and W. I knew already [20] that the bicategory V-Mod whose morphisms were modules between V-categories could be constructed from the bicategory whose morphisms were V-functors. So the question became: given a base bicategory V, for which This paper characterizes those bicategories M biequivalent to ones of the form W-Mod for a base bicategory W with a small set of objects. The possible W are those biequivalent to full subbicategories of M whose objects form a 'small cauchy generator'. Soon after, it was realized 3 that the bicategory W-Mod of small W-categories and W-modules could be defined whether W were small or not and, if you omitted the small cauchy generator requirement, that the idempotency property Proposition 1 of the present paper brings out how the existence of colimits in the homs of a bicategory is a structural form of additivity. Not only do coproducts become products but all lax colimits become lax limits. These ideas were developed nicely by Richard Wood. 4The 2-category (W-Cat) cc of Cauchy complete W-categories and W-functors can be obtained up to biequivalence from W-Mod by restricting to left-adjoint morphisms. In further work 5 , the question of abstractly characterizing bicategories of the form (W-Cat) cc was addressed.I would like to warmly thank Vacation Scholars James Douglas and Rony Kirollos for retyping the paper using modern techniques, and Michael Bar...

Monoidal Bicategories and Hopf Algebroids

“…Our question now is: when can we recapture M itself from V ? An answer is provided by [St3]. If M admits collages, is locally cocomplete, and the single object I is a cauchy generator for M, then M is biequivalent (as a bicategory) to V-Mod.…”

Formal Hopf algebra theory I: Hopf modules for pseudomonoids

“…The result follows form Theorem 6.4 since the arrows (24) and (25) are the components of the natural transformations γ and ω defined in (13) and (15) respectively.…”

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“…Indeed, if (25) is an isomorphism, by taking c = I and a = d, we deduce that 1 a = (1 a ⊗ g)(f ⊗ 1 a ) for some f : I → a ⊗ x, g : x ⊗ a → I. By an argument due to Paré, the splitting of the idempotent (g ⊗ 1 x )(1 x ⊗ f ) : x → x provides a left dual for a.…”

The Family Approach to Total Cocompleteness and Toposes

“…For categories with small homsets Theorem 14 asserts that familial exactness is equivalent to lex-totality. With a mild size restriction on a category, it is lex-total precisely when it is a Grothendieck topos (the proof of this, basically due to Peter Freyd, appears in [6]). I believe the family approach brings out the essence of topos theory in a way which unifies the divers aspects both logical and geometric.…”