On Internal Categories and Crossed Objects in the Category of Monoids

IF 0.5 4区数学 Q3 MATHEMATICS

Applied Categorical Structures Pub Date : 2025-08-14 DOI:10.1007/s10485-025-09822-2

Ilia Pirashvili

引用次数: 0

Abstract

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.

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论一元群范畴中的内范畴和交叉对象

在之前关于二次代数的工作中（Pirashvili In glass Math J 61: 151-167, 2018），我在monoids范畴中构造了一个内部范畴，在3.2.1节中回顾。在此基础上，本文引入了交叉半双模的概念。这个新的构造推广了交叉半模块的概念，交叉半模块的概念是由R. Street和a . Patchkoria分别引入的，参见Joyal （Macquarie Math Reports 860081, 1986）和Patchkoria （Georg Math J 5: 575-581, 1986）。我们还证明了在满足cc条件的变换范畴上，交叉半双模和严格一元范畴结构之间存在一一对应关系，见第4节和第5节。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Applied Categorical Structures 数学-数学

CiteScore

1.30

自引率

16.70%

发文量

审稿时长

>12 weeks

期刊介绍： Applied Categorical Structures focuses on applications of results, techniques and ideas from category theory to mathematics, physics and computer science. These include the study of topological and algebraic categories, representation theory, algebraic geometry, homological and homotopical algebra, derived and triangulated categories, categorification of (geometric) invariants, categorical investigations in mathematical physics, higher category theory and applications, categorical investigations in functional analysis, in continuous order theory and in theoretical computer science. In addition, the journal also follows the development of emerging fields in which the application of categorical methods proves to be relevant. Applied Categorical Structures publishes both carefully refereed research papers and survey papers. It promotes communication and increases the dissemination of new results and ideas among mathematicians and computer scientists who use categorical methods in their research.