2006/05/04 by Iovanov, Miodrag Cristian
#16W30 #18A30 #Category Theory (math.CT) #FOS: Mathematics #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.math/0605112
For a category C we investigate the problem of when the coproduct \bigoplus and the product functor ∏ from CI to C are isomorphic for a fixed set I, or, equivalently, when the two functors are Frobenius functors. We show that for an Ab-category C this happens if and only if the set I is finite. Moreover, this is true even in a much more general case, if there is a morphism in C that is invertible with respect to the addition of morphisms. If C does not have this property we give an example to see that the two functors can be isomorphic for infinite sets I. However we show that \bigoplus and ∏ are always isomorphic on a suitable subcategory of CI which is isomorphic to CI but is not a full subcategory. For the module category case we provide a different proof to display an interesting connection to the notion of Frobenius corings.