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coCartesian fibration of (∞,1)-operads

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Definition

Definition

Let 𝒪 \mathcal{O}^\otimes be an (∞,1)-operad. A coCartesian fibration of (∞,1)-operads is

  1. a coCartesian fibration p:𝒞 𝒪 p : \mathcal{C}^\otimes \to \mathcal{O}^\otimes of the underlying quasi-categories;

  2. such that the composite

    𝒞 p𝒪 FinSet */=Comm \mathcal{C}^\otimes \stackrel{p}{\to} \mathcal{O}^\otimes \to FinSet^{*/} = Comm^\otimes

    exhibits 𝒞 \mathcal{C}^\otimes as an (∞,1)-operad.

In this case we say that the underlying (∞,1)-category

𝒞=𝒞 × 𝒪 𝒪 \mathcal{C} = \mathcal{C}^\otimes \times_{\mathcal{O}^\otimes} \mathcal{O}

is equipped by pp with the structure of an 𝒪\mathcal{O}-monoidal (∞,1)-category (see remark below).

This is (Lurie, def. 2.1.2.13).

Remark

For 𝒞 𝒪 \mathcal{C}^\otimes \to \mathcal{O}^\otimes a coCartesian fibration of (.1)(\infty.1)-operads by def. , the underlying map

𝒞𝒪 \mathcal{C} \to \mathcal{O}

is a coCartesian fibration of (∞,1)-categories. Therefore by the (∞,1)-Grothendieck construction it is classified by an (∞,1)-functor

χ:𝒪(,1)Cat. \chi \colon \mathcal{O} \to (\infty,1)Cat \,.

This inherits monoidal structure (with respect to the cartesian monoidal (∞,1)-category structure on (∞,1)Cat) and hence exhibits an 𝒪\mathcal{O}-algebra in (∞,1)Cat. This way coCartesian fibrations of (,1)(\infty,1)-operads over some 𝒪 \mathcal{O}^\otimes are equivalently 𝒪\mathcal{O}-algebras in (∞,1)Cat. Therefore their identification with 𝒪\mathcal{O}-monoidal (∞,1)-categories.

(Lurie, remark 2.1.2.17, 2.4.2.6)

References

Last revised on February 12, 2013 at 05:40:48. See the history of this page for a list of all contributions to it.