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<p>In a previous work on quadratic algebras (Pirashvili in Glas Math J 61: 151–167, 2018), I constructed an internal category in the category of monoids, recalled in Sect.&#xa0;<InternalRef RefID="Sec7">3.2.1</InternalRef>. Based on this, we introduce the notion of a crossed semi-bimodule in this paper. This new construction generalises the notion of a crossed semi-module, introduced independently by R. Street and A. Patchkoria, see Joyal (Macquarie Math Reports 860081, 1986) and Patchkoria (Georg Math J 5: 575–581, 1986) respectively. We also show that there is a one to one correspondence between crossed semi-bimodules and strict monoidal category structures on transformation categories satisfying the cc-condition, see Sects.&#xa0;<InternalRef RefID="Sec10">4</InternalRef> and <InternalRef RefID="Sec12">5</InternalRef>.</p>

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On Internal Categories and Crossed Objects in the Category of Monoids

  • 期刊论文

  • 发表日期：

    2025年8月14日

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  • Ilia Pirashvili

摘要

In a previous work on quadratic algebras (Pirashvili in Glas Math J 61: 151–167, 2018), I constructed an internal category in the category of monoids, recalled in Sect. 3.2.1. Based on this, we introduce the notion of a crossed semi-bimodule in this paper. This new construction generalises the notion of a crossed semi-module, introduced independently by R. Street and A. Patchkoria, see Joyal (Macquarie Math Reports 860081, 1986) and Patchkoria (Georg Math J 5: 575–581, 1986) respectively. We also show that there is a one to one correspondence between crossed semi-bimodules and strict monoidal category structures on transformation categories satisfying the cc-condition, see Sects. 4 and 5.