Crossed Modules of Monoids I: Relative Categories

Abstract

This is the first part of a series of three strongly related papers in which three equivalent structures are studied:-internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans -crossed modules of monoids relative to this class of spans -simplicial monoids of so-called Moore length 1 relative to this class of spans. The most important examples of monoids that are covered are small categories (treated as monoids in categories of spans) and bimonoids in symmetric monoidal categories (regarded as monoids in categories of comonoids). In this first part the theory of relative pullbacks is worked out leading to the definition of a relative category.Date: March 2018.

Crossed Modules of Monoids II: Relative Crossed Modules

“…We freely use definitions, notation and results from [2]. Throughout, the composition of some morphisms A g G G B and B f G G C in an arbitrary category will be denoted by A f.g G G C .…”

“…[2,Assumption 4.1] asserts that there exist the relative pullbacks of those cospans whose legs are in S in the sense of [2,Definition 2.9]. Under this assumption it was proven in [2,Corollary 4.6] that the spans whose legs are in S (again in the sense of [2, Definition 2.9]) constitute a monoidal category. An S-relative category is defined as a monoid therein, see [2,Definition 4.9].…”

“…Under this assumption it was proven in [2,Corollary 4.6] that the spans whose legs are in S (again in the sense of [2, Definition 2.9]) constitute a monoidal category. An S-relative category is defined as a monoid therein, see [2,Definition 4.9].…”

“…A class of spans in a monoidal category, which is compatible with the monoidal structure -meaning multiplicativity and unitality in a natural sense -was termed monoidal in [2,Definition 2.5]. It is discussed in [2,Example 2.8] that a monoidal admissible class S of spans in a braided monoidal category C induces a monoidal admissible class of spans in the category of monoids in C; and it is shown in [2,Example 4.4] that if S satisfies [2, Assumption 4.1] then so does the induced class in the category of monoids. This allows for the discussion of relative categories in the category of monoids.…”

“…-Categories of comonoids in symmetric monoidal categories whose monoids are bimonoids including Hopf monoids in particular. In the first part [2] of this series of papers we discussed classes of spans satisfying appropriate conditions; and relative pullbacks with respect to them. Assuming that such pullbacks exist -as they do in our key examples -we introduced a monoidal category with monoidal product provided by these pullbacks.…”