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On internal categories and crossed objects in the category of monoids

Ilia Pirashvili

Abstract

It is a well-known fact that the category $\mathsf{Cat}(\mathbf{C})$ of internal categories in a category $\mathbf{C}$ has a description in terms of crossed modules, when $\mathbf{C}=\mathbf{Gr}$ is the category of groups. The proof of this result heavily uses the fact that any split epimorphism decomposes as a semi-direct product. An equivalent statement does not hold in the category $\mathbf{Mon}$ of monoids. In a previous work on quadratic algebras, I constructed an internal category in the category of monoids, see Section 6. Based on this construction, this paper will introduce the notion of a crossed semi-bimodule and show that it gives rise to an object in $\mathsf{Cat}(\mathbf{Mon})$. I will also relate this new notion to the crossed semi-modules introduced earlier by A. Patchkoria.

On internal categories and crossed objects in the category of monoids

Abstract

It is a well-known fact that the category of internal categories in a category has a description in terms of crossed modules, when is the category of groups. The proof of this result heavily uses the fact that any split epimorphism decomposes as a semi-direct product. An equivalent statement does not hold in the category of monoids. In a previous work on quadratic algebras, I constructed an internal category in the category of monoids, see Section 6. Based on this construction, this paper will introduce the notion of a crossed semi-bimodule and show that it gives rise to an object in . I will also relate this new notion to the crossed semi-modules introduced earlier by A. Patchkoria.
Paper Structure (11 sections, 12 theorems, 79 equations)

This paper contains 11 sections, 12 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Crossed (semi-)modules and crossed semi-bimodules
  3. Internal categories
  4. Crossed modules and internal categories in $\bf{Gr}$
  5. Crossed semi-modules
  6. Crossed semi-bimodules and internal categories in the category of monoids
  7. Definition and first consequences
  8. Crossed semi-bimodules and internal categories in $\bf{Mon}$
  9. Relationship between crossed semi-modules and crossed semi-bimodules
  10. The case when $K$ and $A$ are groups
  11. A construction related to rank two commutative algebras

Key Result

Lemma 3.4

The category of semi-bimodule over a monoid $A$ is equivalent to the full subcategory of crossed semi-bimodules over $A$, for which holds for all $a\in A$ and all $x\in K$.

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • Lemma 3.5
  • proof
  • Proposition 3.6
  • proof
  • ...and 18 more