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 degenracy maps:

$$(d_i:S_n^{\mathrm{op}}\to S_{n-1}^{\mathrm{op}})=(d_{n-i}:S_n\to S_{n-1})$$(d_i : S_n^{op} \to S_{n-1}^{op}) = (d_{n-i} : S_n \to S_{n-1})

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