# nLab opposite quasi-category

### Context

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

(∞,1)-category theory

# Contents

## Idea

The general notion of opposite (∞,1)-category leads to a notion of opposite of a quasi-category , when (∞,1)-categories are incarnated as quasi-categories.

So the notion of opposite of a quasi-category generalizes the notion of opposite category from category theory.

## Definition

Under the relation between quasi-categories and simplicial categories the opposite quasi-category is that corresponding to the obvious opposite SSet-enriched category. Concretely in terms of the simplicial set $S$ underlying the quasi-category, this amounts to reversing the order of the face and degenracy maps:

${S}_{n}^{\mathrm{op}}:={S}_{n}$S_n^{op} := S_n
$\left({d}_{i}:{S}_{n}^{\mathrm{op}}\to {S}_{n-1}^{\mathrm{op}}\right):=\left({d}_{n-i}:{S}_{n}\to {S}_{n-1}\right)$(d_i : S_n^{op} \to S_{n-1}^{op}) := (d_{n-i} : S_n \to S_{n-1})
$\left({s}_{i}:{S}_{n}^{\mathrm{op}}\to {S}_{n+1}^{\mathrm{op}}\right):=\left({s}_{n-i}:{S}_{n}\to {S}_{n+1}\right)\phantom{\rule{thinmathspace}{0ex}}.$(s_i : S_n^{op} \to S_{n+1}^{op}) := (s_{n-i} : S_n \to S_{n+1}) \,.

## References

Secton 1.2.1 in

Revised on January 17, 2011 22:14:44 by Urs Schreiber (131.211.233.246)