Assuming only a very rudimentary knowledge of enriched category theory — V-categories, V-functors, and V-natural transformations for a closed, symmetric monoidal, complete and cocomplete category V — we introduce weighted limits and colimits, the appropriate sort of limits for the enriched setting. This “modern” approach was introduced to the author through talks by Mike Shulman at the Category...

Quasi-categories live at the intersection of homotopy theory with category theory. In particular, they serve as a model for (∞, 1)-categories, that is, weak higher categories with n-cells for each natural number n that are invertible when n > 1. Alternatively, an (∞, 1)-category is a category enriched in ∞-groupoids, e.g., a topological space with points as 0-cells, paths as 1-cells, homotopies...

We define the notions of Azumaya category and Brauer group in category theory enriched over some very general base category V. We prove the equivalence of various definitions, in particular in terms of separable categories or progenerating bimodules. When V is the category of modules over a commutative ring R with unit, we recapture the classical notions of Azumaya algebra and Brauer group and ...

We study 2-monads and their algebras using a Cat-enriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2-categorical points of view. Every 2-category with finite limits and colimits has a canonical model structure in which the weak equivalences are the equivalences; we use these to construct more interesting model structures on 2-categories, includi...

The term Stone-type duality often refers to a dual equivalence between category of lattices or other partially ordered structures on one side and topological the other. This paper is part larger endeavour that aims extend web dualities from metric and, more generally, quantale-enriched categories. In particular, we improve our previous work show how certain results for categories [ 0 , 1 ] -enr...

In this paper we present background results in enriched category theory and model necessary for developing categories of functors suitable doing functor calculus.

Operads were originally defined as V-operads, that is, enriched in a symmetric or braided monoidal category V. The symmetry or braiding in V is required in order to describe the associativity axiom the operads must obey, as well as the associativity that must be a property of the action of an operad on any of its algebras. A sequence of categorical types that filter the category of monoidal cat...

Operads were originally defined as V-operads, that is, enriched in a symmetric or braided monoidal category V. The symmetry or braiding in V is required in order to describe the associativity axiom the operads must obey, as well as the associativity that must be a property of the action of an operad on any of its algebras. A sequence of categorical types that filter the category of monoidal cat...

A prespecified set of genes may be enriched, to varying degrees, for genes that have altered expression levels relative to two or more states of a cell. Knowing the enrichment of gene sets defined by functional categories, such as gene ontology (GO) annotations, is valuable for analyzing the biological signals in microarray expression data. A common approach to measuring enrichment is by crossc...

It is well known that the existence of a braiding in a monoidal category V allows many structures to be built upon that foundation. These include a monoidal 2-category V-Cat of enriched categories and functors over V , a monoidal bicategory V-Mod of enriched categories and modules, a category of operads in V and a 2-fold monoidal category structure on V . We will begin by focusing our expositio...