Related papers: An introduction to model categories with examples
There are versions of "calculus" in many settings, with various mixtures of algebra and analysis. In these informal notes we consider a few examples that suggest a lot of interesting questions.
We define bicategories internal to 2-categories. When the ambient 2-category is symmetric monoidal categories, this provides a convenient framework for encoding the structures of a symmetric monoidal 3-category. This framework is well…
We define the concept of a regular object with respect to another object in an arbitrary category. We present basic properties of regular objects and we study this concept in the special cases of abelian categories and locally finitely…
We use the theory of approximable triangulated categories to give a condition for a proper DG-category to be reflexive in the sense of Kuznetsov and Shinder. To do this we provide another description of the completion of an approximable…
We introduce $\mu$-Abstract Elementary Classes ($\mu$-AECs) as a broad framework for model theory that includes complete boolean algebras and Dirichlet series, and begin to develop their classification theory. Moreover, we note that…
We develop some aspects of the model theory of additive structures, with particular emphasis on the abelian category of pp-imaginaries.
This paper is more an essay than a report. There is a gentle introduction to some issues in modeling, followed by the use of steepest descent methods to develop a model as contrasted to using such methods to solve one already in hand, as in…
A brief introduction is given to the topic of Smith normal forms of incidence matrices. A general discussion of techniques is illustrated by some classical examples. Some recent advances are described and the limits of our current…
For every functor $\mathcal{F} : \mathcal{K} \to \mathbf{C}$, where $\mathcal{K}$ is a small category and $\mathbf{C}$ is a model category which satisfies some mild hypotheses, we define a model category $\mathbf{C}^m$ of…
These are notes for a graduate-level introductory course on singularity categories.
Are introduced six examples of non-braidable tensor categories which are extensions of the category Comod(H), for H a super-group algebra; and two examples of braided categories where the only possible braiding is the trivial braiding.
We give a brief introduction to the notion of an 'approximate group' and some of its numerous applications.
\emph{Approximation Theory} uses nicely-behaved subcategories to understand entire categories, just as projective modules are used to approximate arbitrary modules in classical homological algebra. We use set-theoretic \emph{elementary…
In this paper we put a cofibrantly generated model category structure on the category of small simplicial categories. The weak equivalences are a simplicial analogue of the notion of equivalence of categories.
In spite of physics terms in the title, this paper is purely mathematical. Its purpose is to introduce triangulated categories related to singularities of algebraic varieties and establish a connection of these categories with D-branes in…
The notion of retrocell in a double category with companions is introduced and its basic properties established. Explicit descriptions in some of the usual double categories are given. Monads in a double category provide an important…
We look at equivalence relations on the set of models of a theory -- MERs, for short -- such that the class of equivalent pairs is itself an elementary class, in a language appropriate for pairs of models. We provide many examples of…
In this paper we relate triangulated category structures to the cohomology of small categories and define initial obstructions to the existence of an algebraic or topological enhancement. We show that these obstructions do not vanish in an…
An n-category is some sort of algebraic structure consisting of objects, morphisms between objects, 2-morphisms between morphisms, and so on up to n-morphisms, together with various ways of composing them. We survey various concepts of…
Given a variety of universal algebras. A method is suggested for describing automorphisms of a category of free algebras of this variety. Applying this general method all automorphisms of such categories are found in two cases: 1) for the…