nLab opposite quasi-category

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 degeneracy 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

• Jacob Lurie, Higher Topos Theory