Yoneda structures from 2-toposes. (English) Zbl 1125.18001
While higher category theory was developed with an eye on algebraic topology, it was soon seen as applicable to computer science and was more recently led to naturally in the field theory of physics. The techniques used to define and study higher categories were largely 1-categorical. The present paper gathers 2-categorical techniques to carry that study further. The compelling justification for this, explained in the Introduction, is that higher categories (notably those which are trivial in some low dimensions) tend to be algebras for 2-monads that are not mere liftings of monads on categories.
After elegantly reviewing limits, fibrations, extensions and limits in a 2-category, the paper exposes the theory of weighted colimits for a Yoneda structure on a 2-category in the sense of R. Walters and the reviewer [J. Algebra 50, 350–379 (1978; Zbl 0401.18004)] with a couple of extra axioms. The author defines a 2-topos to be a finitely complete, cartesian closed 2-category with a duality involution and what he calls a classifying discrete opfibration into an object denoted by \(\Omega \). Examples relevant to M. A. Batanin’s globular approach to higher categories [Adv. Math. 136, No. 1, 39–103 (1998; Zbl 0912.18006)] are explained. Each 2-topos supports a Yoneda structure for which special features are explored.
After elegantly reviewing limits, fibrations, extensions and limits in a 2-category, the paper exposes the theory of weighted colimits for a Yoneda structure on a 2-category in the sense of R. Walters and the reviewer [J. Algebra 50, 350–379 (1978; Zbl 0401.18004)] with a couple of extra axioms. The author defines a 2-topos to be a finitely complete, cartesian closed 2-category with a duality involution and what he calls a classifying discrete opfibration into an object denoted by \(\Omega \). Examples relevant to M. A. Batanin’s globular approach to higher categories [Adv. Math. 136, No. 1, 39–103 (1998; Zbl 0912.18006)] are explained. Each 2-topos supports a Yoneda structure for which special features are explored.
Reviewer: Ross H. Street (North Ryde)
MSC:
| 18A05 | Definitions and generalizations in theory of categories |
| 18A15 | Foundations, relations to logic and deductive systems |
| 18B25 | Topoi |
| 18D05 | Double categories, \(2\)-categories, bicategories and generalizations (MSC2010) |
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