#
nLab
opposite (infinity,1)-category

Contents
### Context

#### $(\infty,1)$-Category theory

**(∞,1)-category theory**

**Background**

**Basic concepts**

**Universal constructions**

**Local presentation**

**Theorems**

**Extra stuff, structure, properties**

**Models**

# Contents

## 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.

Last revised on September 4, 2024 at 10:01:39.
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