Related papers: Fibered Multicategory Theory

Notions of `operad' and `multicategory' abound. This work provides a single framework in which many of these various notions can be expressed. Explicitly: given a monad * on a category S, we define the term `(S,*)-multicategory', subject to…

Multimodal normal incestual systems are investigated in terms of multiple categories. The different sorted composition of operators are exhibited as 2-cells in multiple categories built up from 2-categories giving rise to different axioms.…

Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades,…

This paper studies the Euler characteristic of a bicategory based on the concept of magnitudes introduced by Leinster. We focus on its invariance with respect to biequivalence and on the product formula for Buckley's fibered bicategories.

We prove that the arrow category of a monoidal model category, equipped with the pushout product monoidal structure and the projective model structure, is a monoidal model category. This answers a question posed by Mark Hovey, and has the…

The filtered derived category of an abelian category has played a useful role in subjects including geometric representation theory, mixed Hodge modules, and the theory of motives. We develop a natural generalization using current methods…

A graded tensor category over a group $G$ will be called a strongly $G$-graded tensor category if every homogeneous component has at least one multiplicativily invertible object. Our main result is a description of the module categories…

Let $\mathcal C$ be a category with finite colimits, writing its coproduct $+$, and let $(\mathcal D, \otimes)$ be a braided monoidal category. We describe a method of producing a symmetric monoidal category from a lax braided monoidal…

We introduce and investigate the category of factorization of a multiplicative, commutative, cancellative, pre-ordered monoid $A$, which we denote $\mathcal{F}(A)$. The objects of $\mathcal{F}(A)$ are factorizations of elements of $A$, and…

We introduce a relative version of the spherical objects of Seidel and Thomas. Define an object E in the derived category D(Z x X) to be spherical over Z if the corresponding functor from D(Z) to D(X) gives rise to autoequivalences of D(Z)…

Let $\Delta$ be a numerical semigroup and let $d\ge 2$ be an integer. We study the fiber of the quotient map \(S\mapsto S/d\) over $\Delta$. We describe its elements as semigroups of the form $\langle X\rangle+d\Delta$, for suitable finite…

We study polynomial functors over locally cartesian closed categories. After setting up the basic theory, we show how polynomial functors assemble into a double category, in fact a framed bicategory. We show that the free monad on a…

The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful;…

This paper introduces the concept of gluing in a general category, enabling us to define categories that admit glued-up objects. To achieve this, we introduce the notion of a gluing index category. Subsequently, we provide an entirely…

Many structures of interest in two-dimensional category theory have aspects that are inherently strict. This strictness is not a limitation, but rather plays a fundamental role in the theory of such structures. For instance, a monoidal…

Multisorted modules, equivalently representations of quivers, equivalently additive functors on preadditive categories, encompass a wide variety of additive structures. In addition, every module has a natural and useful multisorted…

Given an appropriate diagram of left Quillen functors between model categories, one can define a notion of homotopy fiber product, but one might ask if it is really the correct one. Here, we show that this homotopy pullback is well-behaved…

We develop a notion of iterated monoidal category and show that this notion corresponds in a precise way to the notion of iterated loop space. Specifically the group completion of the nerve of such a category is an iterated loop space and…

In this work, we study the notion of cofinal functor of $\infty$-bicategories with respect to the theory of partially lax colimits. The main result of this paper is a characterization of cofinal functors of $\infty$-bicategories via…

This paper supplements [17], showing that categorically the layered theory is the same as the theory of ordered monoids (e.g. the max-plus algebra) used in tropical mathematics. A layered theory is developed in the context of categories,…