nLab
opposite (infinity,1)-category

Contents

Idea

The analog of the notion of opposite category for (∞,1)-categories.

Definition

For CC an (∞,1)-category regard it in its incarnation as a ∞-groupoid-enriched category. Then its opposite is the (,1)(\infty,1)-category C opC^{op} with the same objects and with hom-objects given by

C op(X,Y):=C(Y,X) C^{op}(X,Y) := C(Y,X)

with the obvious composition law.

With CC incarnated as a quasi-category, the simplicial set C opC^{op} is that obtained by reversing the order of all the face and degeneracy maps. See opposite quasi-category.

Properties

The operation extends to an automorphic (∞,1)-functor

op:(,1)Cat(,1)Cat op : (\infty,1)Cat \to (\infty,1)Cat

from (∞,1)Cat to itself. Up to equivalence, this is the only nontrivial such automorphism. For more on this see (∞,1)Cat.

Revised on February 24, 2010 18:36:09 by Urs Schreiber (131.211.235.127)