Higher category theory
higher category theory
Extra properties and structure
The notion of -surjective functor is the continuation of the sequence of notions
from category theory to an infinite sequence of notions in higher category theory.
Roughly, a functor between ∞-categories and is -surjective if for each boundary of a k-morphisms in , each -morphism between the image of that boundary in is in the image of .
Generalization to -categories
For the moment, this here describes the notion for globular models of -categories. See below for the simplicial reformulation.
An -functor between -categories is 0-surjective if is an epimorphism.
For , the functor is -surjective if the universal morphism
to the pullback in
coming from the commutativity of the square
(which commutes due to the functoriality axioms of ) is an epimorphism.
If you interpret and as sets and take ‘epimorphism’ in a strict sense (the sense in Set, a surjection), then you have a strictly -surjective functor. But if you interpret and as -categories or -groupoids and take ‘epimorphism’ in a weak sense (the homotopy sense from -Grpd), then you have an essentially -surjective functor; equivalently, project and to -equivalence-classes before testing surjectivity. A functor is essentially -surjective if and only if it is equivalent to some strictly -surjective functor, so essential -surjectivity is the non-evil notion.
For and categories we have
- is (essentially) -surjective is (essentially) surjective on objects;
- is (essentially) -surjective is full;
- is (essentially) -surjective is faithful;
- is always -surjective.
In terms of lifting diagrams
An -functor is -surjective for precisely if it has the right lifting property with respect to the inclusion of the boundary of the -globe into the -globe.
One recognizes the similarity to situation for geometric definition of higher category. A morphism of simplicial sets is an acyclic fibration with respect to the model structure on simplicial sets if it all diagrams
have a lift
for all , where now is the -simplex.
Weak equivalences, acyclic fibrations and hypercovers
With respect to the folk model structure on -categories an -functor is
All this has close analogs in other models of higher structures, in particular in the context of simplicial sets: an acyclic fibration in the standard model structure on simplicial sets is a morphism for which all diagrams
have a lift
This is precisely in simplicial language the condition formulated above in globular language.
The general idea of -surjectivity is described around definition 4 of
The concrete discussion in the context of strict omega-categories is in
- Yves Lafont, Francois Métayer, Krzysztof Worytkiewicz, A folk model structure on -cat (arXiv).
For the analogous discussion for simplicial sets see
and references given there.