nLab coCartesian fibration of (∞,1)-operads

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Higher algebra

higher algebra

universal algebra

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Definition

Definition

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

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

2. such that the composite

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

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

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

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

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

This is (Lurie, def. 2.1.2.13).

Remark

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

$\mathcal{C} \to \mathcal{O}$

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

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

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

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.