### Context

#### Higher algebra

higher algebra

universal algebra

## Theorems

#### $\left(\infty ,1\right)$-Category theory

(∞,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

$C\to \mathrm{Cat}$C \to Cat

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

$C\to \left(\infty ,1\right)\mathrm{Cat}\phantom{\rule{thinmathspace}{0ex}},$C \to (\infty,1)Cat \,,

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

$P\to \left(\infty ,1\right){\mathrm{Cat}}^{×}$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).

## 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)