# Contents

## Idea

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

## Properties

### Relation to algebras over an operad

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

###### Theorem
$Alg_P(Cat) \simeq opFib_P$

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

This is (Heuts, theorem 1.6).

### Relation to representable multicategories

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.

## References

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

Revised on February 15, 2012 02:02:47 by Urs Schreiber (82.169.65.155)