nLab
family of (infinity,1)-operads

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Contents

Definition

For the following we model (∞,1)-operads 𝒪\mathcal{O} specifically by their (∞,1)-categories of operators 𝒪 \mathcal{O}^\otimes and these specifically as quasi-categories.

Let 𝒞\mathcal{C} be an (∞,1)-category. A 𝒞\mathcal{C}-family of (∞,1)-operads, is a fibration in the model structure for quasi-categories

p:𝒪 𝒞×FinSet * p \colon \mathcal{O}^{\otimes} \to \mathcal{C} \times FinSet_*

such that

  • For C𝒞C \in \mathcal{C} any object, for X𝒪 C X \in \mathcal{O}^\otimes_C over nFinSet *\langle n\rangle \in FinSet_*, and for α:nk\alpha \colon \langle n\rangle \to \langle k\rangle any inert morphism, then there exists a pp-coCartesian morphism α¯:XY\overline \alpha \colon X \to Y in 𝒪 C \mathcal{O}^\otimes_C.

(Lurie, 2.3.2.10)

Properties

Relation to generalized (,1)(\infty,1)-operads

For 𝒞\mathcal{C} an (∞,1)-category, a fibration 𝒪 𝒞×FinSet *\mathcal{O}^\otimes \to \mathcal{C} \times FinSet_* in the model structure for quasi-categories is a 𝒞\mathcal{C}-family of (,1)(\infty,1)-operads precisely if it is a fibration of generalized (∞,1)-operads such that the underlying map 𝒪 0 𝒞\mathcal{O}^\otimes_{\langle 0\rangle} \to \mathcal{C} is an acyclic Kan fibration.

References

Created on February 11, 2013 at 19:06:37. See the history of this page for a list of all contributions to it.