The analogue of the notion of *Grothendieck fibration* generalized from categories to multicategories.

For a multicategory regarded as a (non-symmetric) operad, discrete fibrations over it are equivalent to algebras over that operad (Hermida, proposition 5.1).

For symmetric multicategories we have the following. Let $P$ be a symmetric operad over Set

The operadic Grothendieck construction induces an equivalence of 2-categories

$Alg_P(Cat)
\simeq
opFib_P$

between the weak algebras over $P$ and op-fibrations over $P$.

This is (Heuts, theorem 1.6).

Fibrations over the terminal multicategory are equivalently representable multicategories (Hermida, corollary 4.3).

The generalization to the context of (∞,1)-operads is given by the notion of Cartesian fibration of dendroidal sets.

Fibrations of planar multicategories are discussed in

- Claudio Hermida,
*Fibrations for abstract multicategories*, Fields Institute Communications (pdf)

For symmetric multicategories a discussion of (op)fibrations and of the operadic Grothendieck construction is in section 1 of

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

Last revised on February 15, 2012 at 02:02:47. See the history of this page for a list of all contributions to it.