nLab opposite quasi-category




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.


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 SS underlying the quasi-category, this amounts to reversing the order of the face and degeneracy maps:

S n op:=S n S_n^{op} := S_n
(d i:S n opS n1 op):=(d ni:S nS n1) (d_i : S_n^{op} \to S_{n-1}^{op}) := (d_{n-i} : S_n \to S_{n-1})
(s i:S n opS n+1 op):=(s ni:S nS n+1). (s_i : S_n^{op} \to S_{n+1}^{op}) := (s_{n-i} : S_n \to S_{n+1}) \,.


Secton 1.2.1 in

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