# nLab opposite quasi-category

Contents

### Context

#### $(\infty,1)$-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:

$S_n^{op} := S_n$
$(d_i : S_n^{op} \to S_{n-1}^{op}) := (d_{n-i} : S_n \to S_{n-1})$
$(s_i : S_n^{op} \to S_{n+1}^{op}) := (s_{n-i} : S_n \to S_{n+1}) \,.$

## References

Secton 1.2.1 in

Last revised on January 17, 2011 at 22:14:44. See the history of this page for a list of all contributions to it.