When is the Product isomorphic to the Coproduct TeX Export

@article{citeulike:5419213, abstract = {For a category C we investigate the problem of when the coproduct \$\bigoplus\$ and the product functor \$\prod\$ from C^I 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 \$\prod\$ are always isomorphic on a suitable subcategory of C^I which is isomorphic to C^I 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.}, archivePrefix = {arXiv}, author = {Iovanov, Miodrag C.}, citeulike-article-id = {5419213}, citeulike-linkout-0 = {http://arxiv.org/abs/math/0605112}, citeulike-linkout-1 = {http://arxiv.org/pdf/math/0605112}, day = {4}, eprint = {math/0605112}, keywords = {category-theory, coproduct, math, product}, month = {May}, posted-at = {2009-08-11 23:07:42}, priority = {2}, title = {When is the Product isomorphic to the Coproduct}, url = {http://arxiv.org/abs/math/0605112}, year = {2006} }

TY - JOUR ID - citeulike:5419213 L3 - citeulike-article-id:5419213 N2 - For a category C we investigate the problem of when the coproduct $\bigoplus$ and the product functor $\prod$ from C^I 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 $\prod$ are always isomorphic on a suitable subcategory of C^I which is isomorphic to C^I 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. TI - When is the Product isomorphic to the Coproduct KW - category-theory KW - coproduct KW - math KW - product AU - Iovanov, Miodrag PY - 2006/May/04/ UR - http://arxiv.org/abs/math/0605112 ER -