7,420 research outputs found
Frobenius and the derived centers of algebraic theories
We show that the derived center of the category of simplicial algebras over
every algebraic theory is homotopically discrete, with the abelian monoid of
components isomorphic to the center of the category of discrete algebras. For
example, in the case of commutative algebras in characteristic , this center
is freely generated by Frobenius. Our proof involves the calculation of
homotopy coherent centers of categories of simplicial presheaves as well as of
Bousfield localizations. Numerous other classes of examples are discussed.Comment: 40 page
(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
Natural Transformations of Organismic Structures
The mathematical structures underlying the theories of organismic sets, (M, R)-systems and molecular sets are shown to be transformed naturally within the theory of categories and functors. Their natural transformations allow the comparison of distinct entities, as well as the modelling of dynamics in “organismic” structures
Double Homotopy (Co)Limits for Relative Categories
We answer the question to what extent homotopy (co)limits in categories with
weak equivalences allow for a Fubini-type interchange law. The main obstacle is
that we do not assume our categories with weak equivalences to come equipped
with a calculus for homotopy (co)limits, such as a derivator.Comment: 34 page
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