Crossed Modules of Monoids I: Relative Categories

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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 spanscrossed modules of monoids relative to this class of spanssimplicial 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.

Original languageEnglish
Pages (from-to)641-662
Number of pages22
JournalApplied Categorical Structures
Issue number6
Publication statusPublished - Dec 1 2019



  • Crossed module
  • Internal category
  • Relative pullback

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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