homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
An (∞,n)-category is said to have 1-adjoints if in its homotopy 2-category every 1-morphism is part of an adjunction. By recursion, for and an (∞,n)-category has -adjoints if for every pair of objects the hom (∞,n-1)-category has adjoints for -morphisms.
An -category has all adjoints (or just has adjoints, for short) if it has adjoints for -morphisms for .
If in addtition every object in is a fully dualizable object, then is called an (∞,n)-category with duals.
The internal language of -categories with duals seems plausibly to be axiomatizable in opetopic type theory.
The notion appears first in section 2.3 of
A model for -categories with all adjoints in terms of (∞,1)-sheaves on a site of a variant of -dimensional manifolds with embeddings between them is discussed in
David Ayala, Nick Rozenblyum, Weak -categories are sheaves on iterated submersions of -manifolds (in preparation)
David Ayala, Nick Rozenblyum, Weak -categories with adjoints are sheaves on -manifolds (in preparation)
previewed in
David Ayala, Higher categories are sheaves on manifolds, talk at FRG Conference on Topology and Field Theories, U. Notre Dame (2012) (video)
Abstract Chiral/factorization homology gives a procedure for constructing a topological field theory from the data of an En-algebra. I’ll explain a multi-object version of this construction which produces a topological field theory from the data of an -category with adjoints. This construction is a consequence of a more primitive result which asserts an equivalence between n-categories with adjoints and “transversality sheaves” on framed -manifolds - of which there is an abundance of examples.
Nick Rozenblyum, Manifolds, Higher Categories and Topological Field Theories, talk Northwestern University (2012) (pdf slides)
Last revised on November 6, 2014 at 18:27:56. See the history of this page for a list of all contributions to it.