Exact completion of path categories and algebraic set theory: Part I: Exact completion of path categories

We introduce the notion of a “category with path objects”, as a slight strengthening of Kenneth Brown's classical notion of a “category of fibrant objects”. We develop the basic properties of such a category and its associated homotopy category. Subsequently, we show how the exact completion of this homotopy category can be obtained as the homotopy category associated to a larger category with path objects, obtained by freely adjoining certain homotopy quotients. In a second part of this paper, we will present an application to models of constructive set theory. Although our work is partly motivated by recent developments in homotopy type theory, this paper is written purely in the language of homotopy theory and category theory, and we do not presuppose any familiarity with type theory on the side of the reader.     

Idempotent functors and localizations in categories of modules and Abelian groups

The present paper contains various results on idempotent functors and localizations in categories of modules and Abelian groups.     

Idempotent splittings,colimit completion,and weak aspects of the theory of monads

We show that some recent constructions in the literature, named ‘weak’ generalizations, can be systematically treated by passing from 2-categories to categories enriched in the Cartesian monoidal category of Cauchy complete categories.     

Group completion in the K-theory and Grothendieck–Witt theory of proto-exact categories

We study the algebraic K-theory and Grothendieck–Witt theory of proto-exact categories, with a particular focus on classes of examples of ${\mathbb{F}}_1$-linear nature. Our main results are analogues of theorems of Quillen and Schlichting, relating the K-theory or Grothendieck–Witt theory spaces of proto-exact categories defined using the (hermitian) Q-construction and group completion.     

Idempotent uninorms

Uninorms are an important generalization of t-norms and t-conorms, having a neutral element lying anywhere in the unit interval. Two broad classes of idempotent uninorms are fully characterized: the class of left-continuous ones and the class of right-continuous ones. In particular, the important subclasses of conjunctive left-continuous idempotent uninorms and of disjunctive right-continuous idempotent uninorms are characterized by means of super-involutive and sub-involutive decreasing unary operators. As a consequence, it is shown that any involutive negator gives rise to a conjunctive left-continuous idempotent uninorm and to a disjunctive right-continuous idempotent uninorm.     

Fibration of Idempotent Measures

Ukrainian Mathematical Journal - We prove that the idempotent barycenter map restricted to the points with nontrivial fibers is a trivial fibration with Hilbert cube fibers whenever it is open.     

Idempotent bounded C-semigroups

A semigroup with zero isidempotent bounded (IB) if it is the 0-direct union of idempotent generated principal left ideals and the 0-direct union of idempotent generated principal right ideals. Notable examples are completely 0-simple semigroups and the wider class of primitive abundant semigroups. Significant to the structure of these semigroups is that they are all categorical at zero. In this paper we describe IB semigroups that are categorical at zero in terms ofdouble blocked Rees matrix semigroups. This generalises Fountain's characterisation of primitive abundant semigroups via blocked Rees matrix semigroups [1], which in turn yields the Rees theorem for completely 0-simple semigroups.     

Globals of Idempotent Semigroups

If S is a semigroup, the global (or the power semigroup) of S is the set P(S) of all nonempty subsets of S equipped with a naturally defined multiplication. A class K of semigroups is globally determined if any two semigroups of K with isomorphic globals are themselves isomorphic. We study properties of globals of idempotent semigroups and show, in particular, that the class of normal bands is globally determined.     

Idempotent factorization of matrices

Necessary and sufficient conditions are presented for a square matrix over an arbitrary field to be a product of k ≥ 1 idempotent matrices of prescribed nullities.     

Idempotent probabilities on semigroups

Summary In this paper, idempotent probability measures have been considered on semigroups which are locally compact or metric and satisfy: (*) A ^–1 B and Ax ^–1 are compact whenever A and B are so, for every x in the semigroup. Such semigroups are more general than compact semigroups which do admit of such measures. On such semigroups we can construct such measures by the usual process if there is a compact sub-semigroup. It is shown in this paper that if such a measure exists in such semigroups, then it must be such an extension measure. Some related results concerning the conditions (*) are also discussed here.     

Idempotent Completion of n-Angulated Categories

Applied Categorical Structures - Let $${\mathcal {C}}$$ be an n-angulated category. We prove that its idempotent completion $$\widetilde{{\mathcal {C}}}$$ admits a unique n-angulated structure such...