1998/08/15 by Lucian M. Ionescu, Ionescu, Lucian M.
Mathematics · #18D10 (Primary) 20J05 #18G50 (Secondary) #Category Theory (math.CT) #FOS: Mathematics #Group Theory (math.GR) #math.CT #math.GR #msc:18D10 #msc:18G50 #msc:20J05
paper · pdf · doi:10.48550/arxiv.math/9808068
AMS-LaTex, 31 pages
arxiv created 1998/08/15 · arxiv updated 2009/11/30
To characterize categorical constraints - associativity, commutativity and monoidality - in the context of quasimonoidal categories, from a cohomological point of view, we define the notion of a parity (quasi)complex. Applied to groups gives non-abelian cohomology. The categorification - functor from groups to monoidal categories - provides the correspondence between the respective parity (quasi)complexes and allows to interpret 1-cochains as functors, 2-cocycles - monoidal structures, 3-cocycles - associators. The cohomology spaces H3, H2, H1, H0 correspond as usual to quasi-extensions, extensions, split extensions and invariants, as in the abelian case. A larger class of commutativity constraints for monoidal categories is identified. It is naturally associated with coboundary Hopf algebras.