nLab opfibration of multicategories

Idea

Opfibrations of multicategories are the generalization of Grothendieck opfibrations from categories to multicategories.

Properties

Relation to algebras over an operad

For a multicategory regarded as a (non-symmetric) operad, discrete opfibrations 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

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).

Relation to representable multicategories

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

Related concepts

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

References

On opfibrations of planar multicategories:

• 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:

• Gijs Heuts, section 1 of: Algebras over infinity-operads [arXiv:1110.1776]