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We show that the construction due to Leinster and Weber of a generalized Lawvere theory for a familially representable monad on a (co)presheaf category, and the associated ``nerve'' functor from monad algebras to (co)presheaves, have an…

The aim of this paper is to extend the definition of motivic homotopy theory from schemes to a large class of algebraic stacks and establish a six functor formalism. The class of algebraic stacks that we consider includes many interesting…

Categories can be identified -- up to isomorphism -- with polynomial comonads on Set. The left Kan extension of a functor along itself is always a comonad -- called the density comonad -- so it defines a category when its carrier is…

Mackey functors provide the coefficient systems for equivariant cohomology theories. More generally, enriched presheaf categories provide a classification and organization for many stable model categories of interest. Changing enrichments…

We give a natural-deduction-style type theory for symmetric monoidal categories whose judgmental structure directly represents morphisms with tensor products in their codomain as well as their domain. The syntax is inspired by Sweedler…

We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various…

Exploiting the symmetry topological field theory/topological order correspondence (SymTFT/TO), together with the higher-categorical structure of 6D N =(2,0) SCFTs, we prove that the total quantum dimension of the relative gaugeable algebra…

We prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let $\mathscr{X}$ denote an additive category with finite direct sums and split…

Algebraic datatypes, and among them lists and trees, have attracted a lot of interest in automated reasoning and Satisfiability Modulo Theories (SMT). Since its latest stable version, the SMT-LIB standard defines a theory of algebraic…

Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a general representation theorem for a wide class of…

A model category is called combinatorial if it is cofibrantly generated and its underlying category is locally presentable. As shown in recent years, homotopy categories of combinatorial model categories share useful properties, such as…

This paper introduces an inherently strict presentation of categories with products, coproducts, or symmetric monoidal products that is inspired by file systems and directories. Rather than using nested binary tuples to combine objects or…

Treewidth is a well-known graph invariant with multiple interesting applications in combinatorics. On the practical side, many NP-complete problems are polynomial-time (sometimes even linear-time) solvable on graphs of bounded treewidth. On…

Representations of a group $G$ in vector spaces over a field $K$ form a category. One can reconstruct the given group $G$ from its representations to vector spaces as the full group of monoidal automorphisms of the underlying functor. This…

The purpose of this note is to point out that simplicial methods and the well-known Dold-Kan construction in simplicial homotopy theory can be fruitfully applied to convert link homology theories into homotopy theories. Dold and Kan prove…

Convergence spaces are a generalization of topological spaces. The category of convergence spaces is well-suited for Algebraic Topology, one of the reasons is the existence of exponential objects provided by continuous convergence. In this…

In this short mostly expository note, we sketch a program for gauging fully extended topological field theories in 3 dimensions. One begins with the spherical fusion category with which one wants to do Levin-Wen or Turaev-Viro. One then…

It is informally understood that the purpose of modal type constructors in programming calculi is to control the flow of information between types. In order to lend rigorous support to this idea, we study the category of classified sets, a…

Tambara functors are an equivariant generalization of rings that appear as the homotopy groups of genuine equivariant commutative ring spectra. In recent work, Blumberg and Hill have studied the corresponding algebraic structures, called…

Derivators, introduced independently by Grothendieck and Heller in the 1980s, provide a categorical framework for studying homotopy theory. They are based on the idea that, while the homotopy 1-category of a single model category or…