nLab
Cartesian fibration of dendroidal sets

Idea

The notion of Cartesian fibration of dendroidal sets is the generalization from simplicial sets to dendroidal sets of the notion of Cartesian fibration. Accordingly, it is models the notion of Grothendieck fibration for (∞,1)-operads. Its 1-operadic analog is the notion of fibration of multicategories.

Definition

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Properties

Relation to $\mathrm{\infty }$-algebra

Let $P$ be an (∞,1)-operad, incarnated as a dendroidal set. For instance the homotopy coherent dendroidal nerve of a topological operad/simplicial operad.

Then coCartesian fibrations over $P$ are equivalent to ∞-algebras over $P$ in (∞,1)Cat:

$${\mathrm{coCart}}_P\simeq {\mathrm{Alg}}_P({\mathrm{Cat}}_{(\mathrm{\infty },1)}).$$coCart_P \simeq Alg_P(Cat_{(\infty,1)}) \,.

This is (Heuts, theorem 0.1).

Related concepts

• model structure for dendroidal Cartesian fibrations

References

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

• Gijs Heuts, An infinite loop space machine for infinity-operads (arXiv:1112.0625)