nLab opposite (infinity,1)-category

Idea

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

Definition

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

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

with the obvious composition law.

With $C$ incarnated as a quasi-category, the simplicial set $C^{op}$ 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

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

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

Related concepts

• opposite model category