### Abstract

Weak Cayley table functions between groups are generalized conjugacy-preserving homomorphisms, under which products of images are conjugate to images of products. There is a weak Cayley table bijection between two groups iff they have the same 2-characters. In this paper, weak Cayley table functions are augmented to include the specific conjugating elements, leading to the concept of a weak (Cayley table) morphism. If the conjugating elements are chosen subject to a crossed-product condition, then the weak morphisms between groups form a category. The forgetful functor to this category from the category of group homomorphisms is shown to possess a left adjoint. Two weak morphisms are said to be homotopic if they project to the same weak Cayley table function. As a first step in the analysis of the category of weak morphisms, the group of units of the monoid of weak morphisms homotopic to the identity automorphism of a group is described.

Original language | English (US) |
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Pages (from-to) | 57-67 |

Number of pages | 11 |

Journal | Selecta Mathematica, New Series |

Volume | 13 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2007 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Physics and Astronomy(all)

### Cite this

*Selecta Mathematica, New Series*,

*13*(1), 57-67. https://doi.org/10.1007/s00029-007-0032-x

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*Selecta Mathematica, New Series*, vol. 13, no. 1, pp. 57-67. https://doi.org/10.1007/s00029-007-0032-x

**On the category of weak Cayley table morphisms between groups.** / Johnson, K. W.; Smith, J. D.H.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the category of weak Cayley table morphisms between groups

AU - Johnson, K. W.

AU - Smith, J. D.H.

PY - 2007/3/1

Y1 - 2007/3/1

N2 - Weak Cayley table functions between groups are generalized conjugacy-preserving homomorphisms, under which products of images are conjugate to images of products. There is a weak Cayley table bijection between two groups iff they have the same 2-characters. In this paper, weak Cayley table functions are augmented to include the specific conjugating elements, leading to the concept of a weak (Cayley table) morphism. If the conjugating elements are chosen subject to a crossed-product condition, then the weak morphisms between groups form a category. The forgetful functor to this category from the category of group homomorphisms is shown to possess a left adjoint. Two weak morphisms are said to be homotopic if they project to the same weak Cayley table function. As a first step in the analysis of the category of weak morphisms, the group of units of the monoid of weak morphisms homotopic to the identity automorphism of a group is described.

AB - Weak Cayley table functions between groups are generalized conjugacy-preserving homomorphisms, under which products of images are conjugate to images of products. There is a weak Cayley table bijection between two groups iff they have the same 2-characters. In this paper, weak Cayley table functions are augmented to include the specific conjugating elements, leading to the concept of a weak (Cayley table) morphism. If the conjugating elements are chosen subject to a crossed-product condition, then the weak morphisms between groups form a category. The forgetful functor to this category from the category of group homomorphisms is shown to possess a left adjoint. Two weak morphisms are said to be homotopic if they project to the same weak Cayley table function. As a first step in the analysis of the category of weak morphisms, the group of units of the monoid of weak morphisms homotopic to the identity automorphism of a group is described.

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UR - http://www.scopus.com/inward/citedby.url?scp=34249097748&partnerID=8YFLogxK

U2 - 10.1007/s00029-007-0032-x

DO - 10.1007/s00029-007-0032-x

M3 - Article

AN - SCOPUS:34249097748

VL - 13

SP - 57

EP - 67

JO - Selecta Mathematica, New Series

JF - Selecta Mathematica, New Series

SN - 1022-1824

IS - 1

ER -