nLab operadic (∞,1)-Grothendieck construction

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

$$C \to Cat$$

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

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

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

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

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

Equivalence between $\infty$-Algebras and fibrations

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

This is the central theorem in (Heuts).

Related concepts

• Cartesian fibration of dendroidal sets

References

A construction modeled on dendroidal sets is discussed in

• Gijs Heuts, Algebras over infinity-operads (arXiv:1110.1776)

In section 2.1.3 of

• Jacob Lurie, Higher Algebra

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