Related papers: Majority categories
Constellations are partial algebras that are one-sided generalisations of categories. It has previously been shown that the category of inductive constellations is isomorphic to the category of left restriction semigroups. Here we consider…
The theory of abelian categories proved very useful, providing an axiomatic framework for homology and cohomology of modules over a ring and, in particular, of abelian groups. For many years, a similar categorical framework has been lacking…
We explore a hierarchy of notions in categorical algebra: Mal'tsev categories (where every reflexive relation is symmetric); naturally Mal'tsev categories (where every reflexive graph underlies a unique internal groupoid structure, also…
Bicommutant categories are higher categorical analogs of von Neumann algebras that were recently introduced by the first author. In this article, we prove that every unitary fusion category gives an example of a bicommutant category. This…
The term ``Boolean category'' should be used for describing an object that is to categories what a Boolean algebra is to posets. More specifically, a Boolean category should provide the abstract algebraic structure underlying the proofs in…
After explaining the importance of model categories in abstract homotopy theory, we provide concrete examples demonstrating that various categories of manifolds do not have all finite colimits, and hence cannot be model categories. We then…
We continue the theory of $\tT$-systems from the work of the second author, describing both ground systems and module systems over a ground system (paralleling the theory of modules over an algebra). The theory, summarized categorically at…
We classify the propositional modal validities arising from the category of sets under its natural classes of morphisms. The resulting validities depend on the morphism class, the size of the world, and the permitted substitution instances.…
It is proved that equalities between arrows assumed for cartesian categories are maximal in the sense that extending them with any new equality in the language of free cartesian categories collapses a cartesian category into a preorder. An…
We call a finitely complete category algebraically coherent when the change-of-base functors of its fibration of points are coherent, which means that they preserve finite limits and jointly strongly epimorphic pairs of arrows. We give…
There are many ways to present model categories, each with a different point of view. Here we'd like to treat model categories as a way to build and control resolutions. This an historical approach, as in his original and spectacular…
Constructing and manipulating homotopy types from categorical input data has been an important theme in algebraic topology for decades. Every category gives rise to a `classifying space', the geometric realization of the nerve. Up to weak…
Algebraic structures in which the property of commutativity is substituted by the mediality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or…
We study extensively the homotopy theory of coalgebras. By coalgebras, we mean the full theory of coalgebras: with counits and not necessarily locally conilpotent. For example $\mathcal E_\infty$-coalgebras, $\mathcal A_\infty$-coalgebras,…
Universal measuring coalgebras provide an enrichment of the category of algebras over the category of coalgebras. By considering the special case of the tensor algebra on a vector space V, the category of linear spaces itself becomes…
Matrix conditions extend linear Mal'tsev conditions from Universal Algebra to exactness properties in Category Theory. Some can be stated in the finitely complete context while, in general, they can only be stated for regular categories. We…
An explicit categorical equivalence is defined between a proper subvariety of the class of $PMV$-algebras, as defined by Di Nola and Dvure$\check{c}$enskij, to be called $PMV_f$-algebras, and the category of semi-low $f_u$-rings. This…
We firstly introduce some key concepts in category theory, such as quotient category, completion of limits, $\mathrm{Mor}$ category, and so on; then give the concept of topology algebras and sheaves, and discuss how to restore the structue…
This article introduces pre-Hilbert $*$-categories: an abstraction of categories exhibiting "algebraic" aspects of Hilbert-space theory. Notably, finite biproducts in pre-Hilbert $*$-categories can be orthogonalised using the Gram-Schmidt…
Presentations of categories are a well-known algebraic tool to provide descriptions of categories by means of generators, for objects and morphisms, and relations on morphisms. We generalize here this notion, in order to consider situations…