### Abstract

Let C be an arbitrary category. We study strongly involutory functors on C, defined as involutory contravariant endofunctors of C acting as identity on objects. Motivating examples can be constructed if we think at the transpose of a matrix, the adjoint of a linear continuous operator between two Hilbert spaces, and the inverse of a morphism in a groupoid. We show how a strongly involutory functor on a skeleton of C extends to C, and we apply this to find all such functors for a groupoid. We describe and classify up to a natural equivalence all strongly involutory functors on the category of finite dimensional vector spaces over a field. Strongly involutory functors with a special property related to generalized inverses of morphisms are studied.

Original language | English (US) |
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Pages (from-to) | 1677-1689 |

Number of pages | 13 |

Journal | Communications in Algebra |

Volume | 37 |

Issue number | 5 |

DOIs | |

State | Published - May 2009 |

### Keywords

- Generalized inverse
- Involutory functor
- Vector space

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Communications in Algebra*,

*37*(5), 1677-1689. https://doi.org/10.1080/00927870802210027