1,891 research outputs found
Enhanced 2-categories and limits for lax morphisms
We study limits in 2-categories whose objects are categories with extra
structure and whose morphisms are functors preserving the structure only up to
a coherent comparison map, which may or may not be required to be invertible.
This is done using the framework of 2-monads. In order to characterize the
limits which exist in this context, we need to consider also the functors which
do strictly preserve the extra structure. We show how such a 2-category of weak
morphisms which is "enhanced", by specifying which of these weak morphisms are
actually strict, can be thought of as category enriched over a particular base
cartesian closed category F. We give a complete characterization, in terms of
F-enriched category theory, of the limits which exist in such 2-categories of
categories with extra structure.Comment: 77 pages; v2 minor changes only, to appear in Advance
(Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories
Motivated by the challenge of defining twisted quantum field theories in the context of higher categories, we develop a general framework for lax and oplax transformations and their higher analogs between strong (∞,n)-functors. We construct a double (∞,n)-category built out of the target (∞,n)-category governing the desired diagrammatics. We define (op)lax transformations as functors into parts thereof, and an (op)lax twisted field theory to be a symmetric monoidal (op)lax natural transformation between field theories. We verify that lax trivially-twisted relative field theories are the same as absolute field theories. As a second application, we extend the higher Morita category of E d -algebras in a symmetric monoidal (∞,n)-category C to an (∞,n+d)-category using the higher morphisms in C
Semantic Factorization and Descent
Let be a -category with suitable opcomma objects and
pushouts. We give a direct proof that, provided that the codensity monad of a
morphism exists and is preserved by a suitable morphism, the factorization
given by the lax descent object of the higher cokernel of is up to
isomorphism the same as the semantic factorization of , either one existing
if the other does. The result can be seen as a counterpart account to the
celebrated B\'{e}nabou-Roubaud theorem. This leads in particular to a
monadicity theorem, since it characterizes monadicity via descent. It should be
noted that all the conditions on the codensity monad of trivially hold
whenever has a left adjoint and, hence, in this case, we find monadicity to
be a -dimensional exact condition on , namely, to be an effective
faithful morphism of the -category .Comment: Full revision, new diagrams, 48 page
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