Results 1 
5 of
5
NOTE ON COMMUTATIVITY IN DOUBLE SEMIGROUPS AND TWOFOLD MONOIDAL CATEGORIES
"... A concrete computation — twelve slidings with sixteen tiles — reveals that certain commutativity phenomena occur in every double semigroup. This can be seen as a sort of EckmannHilton argument, but it does not use units. The result implies in particular that all cancellative double semigroups and a ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
A concrete computation — twelve slidings with sixteen tiles — reveals that certain commutativity phenomena occur in every double semigroup. This can be seen as a sort of EckmannHilton argument, but it does not use units. The result implies in particular that all cancellative double semigroups and all inverse double semigroups are commutative. Stepping up one dimension, the result is used to prove that all strictly associative twofold monoidal categories (with weak units) are degenerate symmetric. In particular, strictly associative oneobject, onearrow 3groupoids (with weak units) cannot realise all simplyconnected homotopy 3types. 1. Introduction and
unknown title
, 2006
"... Note on commutativity in double semigroups and twofold monoidal categories ..."
Abstract
 Add to MetaCart
Note on commutativity in double semigroups and twofold monoidal categories
SEMIUNITAL SEMIMONOIDAL CATEGORIES (APPLICATIONS TO SEMIRINGS AND SEMICORINGS)
"... Abstract. The category ASA of bisemimodules over a semialgebra A, with the so called Takahashi’s tensorlike product − ⊠A −, is semimonoidal but not monoidal. Although not a unit in ASA, the base semialgebra A has properties of a semiunit (in a sense which we clarify in this note). Motivated by this ..."
Abstract
 Add to MetaCart
Abstract. The category ASA of bisemimodules over a semialgebra A, with the so called Takahashi’s tensorlike product − ⊠A −, is semimonoidal but not monoidal. Although not a unit in ASA, the base semialgebra A has properties of a semiunit (in a sense which we clarify in this note). Motivated by this interesting example, we investigate semiunital semimonoidal categories (V, •, I) as a framework for studying notions like semimonoids (semicomonoids) as well as a notion of monads (comonads) which we call Jmonads (Jcomonads) with respect to the endofunctor J: = I• − ≃ −•I: V − → V. This motivated also introducing a more generalized notion of monads (comonads) in arbitrary categories with respect to arbitrary endofunctors. Applications to the semiunital semimonoidal variety (ASA, ⊠A, A) provide us with examples of semiunital Asemirings (semicounital Asemicorings) and semiunitary semimodules (semicounitary semicomodules) which extend the classical notions of unital rings (counital corings) and unitary modules (counitary comodules). 1.
units.tex 20090718 09:24 [1/37] Coherence for weak units
, 907
"... Abstract. We define weak units in a semimonoidal 2category C as cancellable pseudoidempotents: they are pairs (I, α) where I is an object such that tensoring with I from either side constitutes a biequivalence of C, and α: I ⊗ I → I is an equivalence in C. We show that this notion of weak unit ha ..."
Abstract
 Add to MetaCart
Abstract. We define weak units in a semimonoidal 2category C as cancellable pseudoidempotents: they are pairs (I, α) where I is an object such that tensoring with I from either side constitutes a biequivalence of C, and α: I ⊗ I → I is an equivalence in C. We show that this notion of weak unit has coherence built in: Theorem A: α has a canonical associator 2cell, which automatically satisfies the pentagon equation. Theorem B: every morphism of weak units is automatically compatible with those associators. Theorem C: the 2category of weak units is contractible if nonempty. Finally we show (Theorem E) that the notion of weak unit is equivalent to the notion obtained from the definition of tricategory: α alone induces the whole family of left and right maps (indexed by the objects), as well as the whole family of Kelly 2cells (one for each pair of objects), satisfying the relevant coherence axioms.
NOTES ON WEAK UNITS OF PICARD 1 AND 2STACKS
, 2011
"... The weak units of strict monoidal 1 and 2categories are defined respectively in [10] and [9]. In this paper, we recall them for Picard 1 and 2stacks. We show that they form a Picard 1 and 2stack, respectively. We deduce by [13, Theorem 6.4] that there exists a length 2 (resp. 3) complex of ab ..."
Abstract
 Add to MetaCart
The weak units of strict monoidal 1 and 2categories are defined respectively in [10] and [9]. In this paper, we recall them for Picard 1 and 2stacks. We show that they form a Picard 1 and 2stack, respectively. We deduce by [13, Theorem 6.4] that there exists a length 2 (resp. 3) complex of abelian sheaves that represent the Picard stack (resp. Picard 2stack) of the weak units. Lastly, we calculate such complexes.