Extensions, crossed modules and internal categories in categories of groups with operations

Abstract

In [6] Brown and Spencer noted that internal categories within the category of groups are equivalent to crossed modules. As they remarked, this result was known to various others before them, but it had not until then appeared in print. That paper led me to investigate the question of which algebraic categories, C, were such that a similar result held i.e. internal categories in C are equivalent to crossed modules of the appropriate type. The resulting work was written up in 1980 but was not submitted for publication.

Coverings of internal groupoids and crossed modules in the category of groups with operations

“…The idea of the definition of categories of groups with operations comes from Higgins [14] and Orzech [23]; and the definition below is from Porter [25] and Datuashvili [13, p. 21], which is adapted from Orzech [23].…”

“…The internal category in C is defined in [25] as follows. We follow the notations of Section 1 for groupoids.…”

“…We also note from [25] that any internal category in C is an internal groupoid since, given a ∈ C, a −1 = ǫd 1 (a) − a + ǫd 0 (a) satisfies a −1 • a = ǫd 1 (a) and a • a −1 = ǫd 0 (a). So we use the term internal groupoid rather than internal category and write G for an internal groupoid.…”

“…group-groupoids, is equivalent to the category of crossed modules of groups. Then in [25,Section 3], Porter proved that a similar result holds for certain algebraic categories C, introduced by Orzech [23], the definition of which was adapted by him and called category of groups with operations. Applying Porter's result, the study of internal category theory in C was continued in the works of Datuashvili [10] and [12].…”

“…Moreover, she developed the cohomology theory of internal categories, equivalently, crossed modules, in categories of groups with operations [9,11]. In a similar way, the results of [8] and [25] enabled us to prove some properties of covering groupoids for internal groupoids.…”