Fibered Multicategory Theory

TL;DR

This paper introduces fibered multicategories as a new framework extending fibered categories, using a functor-based approach to generalize reindexing of arrows in multicategory contexts.

Contribution

It defines fibered multicategories via functors from a multicategory to a base category, extending the concept of fibered categories to multicategory settings.

Findings

Defines fibered multicategories with a key reindexing axiom.

Introduces cartesian fibered multicategories and studies their properties.

Extends the theory of fibered categories to multicategory contexts.

Abstract

Given a fibration in groupoids d : D -> I, we define a fibered multicategory as a particular functor p : M -> I, where M has the same objects as D, and its arrows a : X -> Y should be thought of as families of arrows in the multicategory, indexed by pY. The key axiom extends the reindexing of objects, given by d, to a reindexing of arrows in M along pullback squares in I. When D is included in M, in an appropriate sense, one gets again fibered categories. In this context, cartesian fibered multicategories are defined and studied in a natural way.