Related papers: Path categories and quasi-categories
We generalize the concepts of locally presentable and accessible categories. Our framework includes such categories as small presheaves over large categories and ind-categories. This generalization is intended for applications in the…
Homotopy 3-types can be modelled algebraically by Tamsamani's weak 3-groupoids as well as, in the path-connected case, by cat^2-groups. This paper gives a comparison between the two models in the path-connected case. This leads to two…
We present a new method, involving monads and comonads from category theory, to help establish a certain type of equivalence of subcategories. As a case study we consider the category of topological gradings of $C^*$-algebras over a fixed…
This paper is a generalization of arXiv:0810.0808. We develop the de Rham homotopy theory of not necessarily nilpotent spaces, using closed dg-categories and equivariant dg-algebras. We see these two algebraic objects correspond in a…
This paper is a rather informal guide to some of the basic theory of 2-categories and bicategories, including notions of limit and colimit, 2-dimensional universal algebra, formal category theory, and nerves of bicategories. As is the way…
Many first-order equational theories, such as the theory of groups or boolean algebras, can be presented by a smaller set of axioms than the original one. Recent studies showed that a homological approach to equational theories gives us…
The matched pair theory (of groups) is studied for a class of quasigroups; namely, the $m$-inverse property loops. The theory is upgraded to the Hopf level, and the "$m$-invertible Hopf quasigroups" are introduced.
We introduce an abstract topos-theoretic framework for building Galois-type theories in a variety of different mathematical contexts; such theories are obtained from representations of certain atomic two-valued toposes as toposes of…
Motivated by potential applications to theoretical computer science, in particular those areas where the Curry-Howard correspondence plays an important role, as well as by the ongoing search in pure mathematics for feasible approaches to…
It is an old conjecture, that finite $H$-spaces are homotopy equivalent to manifolds. Here we prove that this conjecture is true for loop spaces. Actually, we show that every quasi finite loop space is equivalent to a stably parallelizable…
This note introduces the theory of quasimaps to GIT quotients with intuition and concrete examples, with the goal of explaining a closed formula for the quasimap $I$-function. Along the way, it emphasizes aspects of this story that…
This is an introduction to type theory, synthetic topology, and homotopy type theory from a category-theoretic and topological point of view, written as a chapter for the book "New Spaces for Mathematics and Physics" (ed. Gabriel Catren and…
This paper defines homology in homotopy type theory, in the process stable homotopy groups are also defined. Previous research in synthetic homotopy theory is relied on, in particular the definition of cohomology. This work lays the…
We discuss the homological algebra of representation theory of finite dimensional algebras and finite groups. We present various methods for the construction and the study of equivalences of derived categories: local group theory, geometry…
We introduce a homotopy theory of digraphs (directed graphs) and prove its basic properties, including the relations to the homology theory of digraphs constructed by the authors in previous papers. In particular, we prove the homotopy…
These notes give a brief introduction to the category of spectra as defined in stable homotopy theory. In particular, Section 5 discusses an extensive list of examples of spectra whose properties have been found to be interesting.
We consider the categorical equivalence between crossed modules over groupoids and double groupoids with thin structures; and by this equivalence, we prove how normality and quotient concepts are related in these two categories and give…
A concise guide to very basic bicategory theory, from the definition of a bicategory to the coherence theorem.
Our aim is to give a fairly complete account on the construction of compatible model structures on exact categories and symmetric monoidal exact categories, in some cases generalizing previously known results. We describe the close…
This document is centered around a main idea: simplicial categories, by which we mean simplicial objects in the category of categories, can be treated as a two-fold categorical structure and their double category theory is homotopically…