nLab
fibration of multicategories

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 PP be a symmetric operad over Set

Theorem

The operadic Grothendieck construction induces an equivalence of 2-categories

Alg P(Cat)opFib P Alg_P(Cat) \simeq opFib_P

between the weak algebras over PP and op-fibrations over PP.

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)