nLab
operadic (∞,1)-Grothendieck construction

Context

Higher algebra

(,1)(\infty,1)-Category theory

Contents

Idea

The operadic (∞,1)-Grothendieck construction is the generalization of the (∞,1)-Grothendieck construction from (∞,1)-categories to (∞,1)-operads.

Properties

Notice that where in the categorical context we had pseudofunctors

CCat C \to Cat

and then in the (∞,1)-category theoretic context (∞,1)-functors

C(,1)Cat, C \to (\infty,1)Cat \,,

as input if the Grothendieck construction, in the (∞,1)-operadic context such morphisms

P(,1)Cat × P \to (\infty,1)Cat^{\times}

have the interpretation of ∞-algebra over an (∞,1)-operad.

Equivalence between \infty-Algebras and fibrations

For PP an (∞,1)-operad, there is an equivalence of (∞,1)-categories between ∞-algebras over PP in (∞,1)Cat and opCartesian fibrations into PP.

This is the central theorem in (Heuts).

References

A construction modeled on dendroidal sets is discussed in

In section 2.1.3 of

the statement of the above equivalence is essentially taken as a definition.

Created on February 15, 2012 02:31:44 by Urs Schreiber (82.169.65.155)