## Elementary remarks on units in monoidal categories

Citations: | 5 - 3 self |

### BibTeX

@TECHREPORT{Kock_elementaryremarks,

author = {Joachim Kock},

title = {Elementary remarks on units in monoidal categories},

institution = {},

year = {}

}

### OpenURL

### Abstract

We gather some not-very-well-known remarks on units in monoidal categories, motivated by (but independent of) higher-dimensional viewpoints. All arguments are elementary, some of them of a certain beauty. The first theme is uniqueness of units: we describe the semi-monoidal category of all possible unit structures on a given semi-monoidal category and show that it is contractible (if nonempty). The second theme is a redundancy in the classical definition of units, which is exhibited with clarity by comparison with an alternative definition of unit originally due to Saavedra: a Saavedra unit is a cancellable idempotent, in a certain sense. It is shown that the two notions are isomorphic in a strong functorial sense. One corollary of this comparison is that a (strong) semi-monoidal functor is compatible with the left constraint if and only if it is compatible with the right constraint, and in fact this compatibility can be measured on I alone. The unit compatibility condition for a (strong) monoidal functor is shown to be precisely the condition for the functor to lift to the categories of units. The notion of Saavedra unit leads naturally to the equivalent non-algebraic notion of fair monoidal category (treated elsewhere), where the contractible multitude of units is considered as a whole instead of choosing one unit. To finish, the lax version of the unit comparison is considered.

### Citations

319 | Homotopy associativity of H-spaces - Stasheff - 1963 |

140 | Introduction to bicategories - Bénabou - 1967 |

63 |
Higher Operads, Higher Categories
- LEINSTER
- 2004
(Show Context)
Citation Context ...her category theory, it is interesting to revisit even the most basic theory of monoidal categories, to test new viewpoints and experiment with new formulations (cf. also Chapter 3 in Leinster’s book =-=[9]-=-.) This note gathers some remarks on units in monoidal categories. Units are often swiped under the carpet, and in any case rarely get the same attention the multiplication laws of monoidal categories... |

57 | Natural associativity and commutativity - Lane - 1963 |

23 | Catégories Tannakiennes - RIVANO - 1972 |

22 |
On MacLane’s conditions for coherence of natural associativities, commutativities, etc
- Kelly
- 1964
(Show Context)
Citation Context ...ane [8] in 1963, including one axiom for associativity (the pentagon equation) and four axioms for the unit, expressed in terms of the left and right constraints. Shortly after, it was shown by Kelly =-=[6]-=- that one of these four axioms for units in fact implies the three others. His proof constitutes nowadays the first three lemmas in many treatments of monoidal categories, while other sources continue... |

12 | Catégories avec multiplication - BÉNABOU - 1963 |

12 | Weak identity arrows in higher categories
- Kock
(Show Context)
Citation Context ...e a by-product of a more general investigation of weak units and weak identity arrows in higher categories, and serves as a home for many easy arguments that did not fit into the more advanced papers =-=[7]-=-, [4], [5], but which nevertheless are too cute to keep secret. The notion of Saavedra unit dropped out of the theory of ‘fair categories’, cf. [7]. I am thankful to Georges Maltsiniotis for pointing ... |

10 | Homotopy types of strict 3-groupoids - Simpson - 1998 |

3 |
Coherence for weak units. Manuscript in preparation
- JOYAL, KOCK
(Show Context)
Citation Context ...y-product of a more general investigation of weak units and weak identity arrows in higher categories, and serves as a home for many easy arguments that did not fit into the more advanced papers [7], =-=[4]-=-, [5], but which nevertheless are too cute to keep secret. The notion of Saavedra unit dropped out of the theory of ‘fair categories’, cf. [7]. I am thankful to Georges Maltsiniotis for pointing out t... |

2 |
Algèbre élémentaire dans les catégories avec multiplication
- BÉNABOU
- 1964
(Show Context)
Citation Context ...onditions of 3.2. One important motivation for considering lax monoidal functors is that monoids are a special case: a monoid in C is essentially the same as a lax monoidal functor ∗ → C (cf. Bénabou =-=[2]-=-). 6.1 Lax monoidal functors in the Saavedra-unit setting. A Saavedra-unit compatibility for a lax multiplicative functor (C , I) → (D, J), X ↦→ X is a gentle semi-monoid homomorphism φ0 : J → I, such... |

2 |
Weak units and homotopy 3-types. Manuscript available from http://mat.uab.es/~ kock/cat/traintracks.html
- JOYAL, KOCK
(Show Context)
Citation Context ...duct of a more general investigation of weak units and weak identity arrows in higher categories, and serves as a home for many easy arguments that did not fit into the more advanced papers [7], [4], =-=[5]-=-, but which nevertheless are too cute to keep secret. The notion of Saavedra unit dropped out of the theory of ‘fair categories’, cf. [7]. I am thankful to Georges Maltsiniotis for pointing out that t... |

2 | Weak units and homotopy 3types. To appear - Joyal, Kock |