Yoneda structures on 2-categories

Street, Ross; Walters, R. F. C.

doi:10.1016/0021-8693(78)90160-6

Cited by 101 publications

(138 citation statements)

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“…Other basic well known facts we shall routinely use below concern the "composability" and "cancellability" of left Kan extensions [28]. That is, suppose that one has…”

Section: Comma Squaresmentioning

confidence: 99%

Algebraic Kan extensions along morphisms of internal algebra classifiers

Weber

2016

Tbilisi Math. J.

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An "algebraic left Kan extension" is a left Kan extension which interacts well with the algebraic structure present in the given situation, and these appear in various subjects such as the homotopy theory of operads and in the study of conformal field theories. In the most interesting examples, the functor along which we left Kan extend goes between categories that enjoy universal properties which express the meaning of the calculation we are trying to understand. These universal properties say that the categories in question are universal examples of some categorical structure possessing some kind of internal structure, and so fall within the theory of "internal algebra classifiers" described in earlier work of the author. In this article conditions of a monad-theoretic nature are identified which give rise to morphisms between such universal objects, which satisfy the key condition of Guitart-exactness, which guarantees the algebraicness of left Kan extending along them. The resulting setting explains the algebraicness of the left Kan extensions arising in operad theory, for instance from the theory of "Feynman categories" of Kaufmann and Ward, generalisations thereof, and also includes the situations considered by Batanin and Berger in their work on the homotopy theory of algebras of polynomial monads.

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“…Other basic well known facts we shall routinely use below concern the "composability" and "cancellability" of left Kan extensions [28]. That is, suppose that one has…”

Section: Comma Squaresmentioning

confidence: 99%

Algebraic Kan extensions along morphisms of internal algebra classifiers

Weber

2016

Tbilisi Math. J.

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“…Smallness of A and local smallness of B also ensure that JI7F has a right adjoint, which we denote by VF. In particular, for small A the Yoneda embedding YK: A -» J7A yields 3(YA) H J7(YK) H V(TA): J7A -J7J7A, and it is shown in [8] that V(TA) is isomorphic to YjrA. We can apply these considerations to A = 0, the empty category, which is the initial object of CAT.…”

Section: Introductionmentioning

confidence: 99%

“…if A is small and B is locally small, then F is Kan [8], but neither condition is necessary; if, say, we have L-\ F, then J7L H JI7F and 3F = JtL. Smallness of A and local smallness of B also ensure that JI7F has a right adjoint, which we denote by VF.…”

Section: Introductionmentioning

confidence: 99%

“…The 2-category of category objects in SET, which we write CAT, is cartesian closed, and set is an object of CAT. Thus, for C a category in CAT, CAT(Cop, set) is also an object of CAT, and we abbreviate it by J7C (it was written ^C in [8]). Substitution gives a 2-functor J7: CAT0005 -» CAT, where CAT000" is the dual which reverses both arrows of CAT (functors) and 2-cells (natural transformations).…”

Section: Introductionmentioning

confidence: 99%

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An Adjoint Characterization of the Category of Sets

Rosebrugh¹,

Wood²

1994

Proceedings of the American Mathematical Society

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“…Previous works [19], [25], [20] have taken the 2-category Cat of categories and functors as primitive and have considered the problem of constructing the bicategory Mod of categories and modules. For categories A, B, a module r from A to B consists of sets r(b, a) indexed by the objects a, b of A, B together with actions:…”

mentioning

confidence: 99%

Cauchy characterization of enriched categories

Street

1981

Seminario Mat. e. Fis. di Milano

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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...

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Yoneda structures on 2-categories

Cited by 101 publications

References 8 publications

Algebraic Kan extensions along morphisms of internal algebra classifiers

Algebraic Kan extensions along morphisms of internal algebra classifiers

An Adjoint Characterization of the Category of Sets

Cauchy characterization of enriched categories

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