caa a22 4500 467882517 CHVBK 20180406152724.0 cr unu---uuuuu 170328e20060601xx s 000 0 eng 10.1007/s10485-006-9015-x doi (NATIONALLICENCE)springer-10.1007/s10485-006-9015-x Zeibig G. Department of Mathematical Sciences, Kent State University, 44240, Kent, OH, USA aut Coherence for Product Monoids and their Actions [Elektronische Daten] [G. Zeibig] Let $(A,\mu^A, \eta^A)$ and $(B, \mu^B, \eta^B)$ be two monoids (algebras) in a monoidal category $${\left( {{\user1{\mathcal{V}}}{\text{,}}\,{\user1{\square }},\,e} \right)}$$ . Further let $\iota: B\Box A \to A\Box B$ be a distributive law in the sense of [J. Beck, Lect. Notes Math., 80:119-140, 1969]; $\iota$ naturally yields a monoid $(A\Box B, \eta, \mu)$ . Consider a word $W'$ in the symbols $A$ , $B$ , and $e$ . The first coherence theorem proved in this paper asserts that all morphisms $W' \to A\Box B$ coincide in $${\user1{\mathcal{V}}}$$ , provided they arise as composites of morphisms which are $\Box$ -products of $${\user1{\mathcal{V}}}$$ 's ‘canonical' structure morphisms, and of $1_A$ , $1_B$ , $1_e$ , $\eta^A$ , $\mu^A$ , $\eta^B$ , $\mu^B$ , and $\iota$ . Assume now that an object $X$ is endowed with both an ${\left( {A,\mu ^{A} ,\eta ^{A} } \right)}$ -object structure $(X,\nu^A)$ , and an ${\left( {B,\mu ^{B} ,\eta ^{B} } \right)}$ -object structure $(X,\nu^B)$ . Further assume that these two structures are compatible, in the sense that they naturally yield an ${\left( {A\Box B,\mu ,\eta } \right)}$ -object $(A\Box B, \nu)$ . Let $W''$ be a word in $A$ , $B$ , $e$ , and $X$ , which contains a single instance of $X$ , in the rightmost position. The second coherence theorem states that all morphisms $W'' \to X$ coincide in $${\user1{\mathcal{V}}}$$ , provided they arise as composites of morphisms which are $\Box$ -products of $${\user1{\mathcal{V}}}$$ 's ‘canonical' structure morphisms, and of $1_A$ , $1_B$ , $1_e$ , $\eta^A$ , $\mu^A$ , $\eta^B$ , $\mu^B$ , $\iota$ , $\nu^A$ , and $$\nu ^{B} $$ . Springer Science+Business Media B.V., 2006 Monoidal category nationallicence monoid nationallicence action nationallicence $A$ -object nationallicence distributive law nationallicence coherence theorem nationallicence crossed product nationallicence semi-direct product nationallicence Applied Categorical Structures Kluwer Academic Publishers 14/3(2006-06-01), 215-227 0927-2852 14:3<215 2006 14 10485 https://doi.org/10.1007/s10485-006-9015-x text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10485-006-9015-x text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Zeibig G. Department of Mathematical Sciences, Kent State University, 44240, Kent, OH, USA aut NATIONALLICENCE 773 0- Applied Categorical Structures Kluwer Academic Publishers 14/3(2006-06-01), 215-227 0927-2852 14:3<215 2006 14 10485 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer