M. Barr, Duality of Vector Spaces, Cahiers de Top. et G'eom. Diff. 17, (1976), pp. 3-14. 22  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of linear continuous maps, topologized with the topology of pointwise convergence. It is shown in [8] that the forgetful functor T VEC VEC is tensor preserving. Lefschetz proves that the embedding V V is always a bijection, but need not be an isomorphism. We then have: Theorem 1. 2 (Barr [5]) RT VEC, the full subcategory of reflexive objects in T VEC, is a complete, cocomplete autonomous category. The following definition and theorem can be found in [13] Definition 1.3 Let G be a group. A continuous G module is a linear action of G on a space V in T VEC, such that for all g 2 G, ....

M. Barr, Duality of Vector Spaces, Cahiers de Top. et G'eom. Diff. 17, (1976), pp. 3-14. 22

....the category has biproducts acting as tensor and par, then it is not hard to show that all ( nite) biproducts of the unit will also be in the core. This may then be a non trivial category. An example of this phenomenon is given by the category RTVec of re exive linear topological vector spaces [L63,Ba76,BS96], i.e. vector spaces equipped with a linear topology which are isomorphic to their double duals. In this category, all nite dimensional vector spaces are in the core but in nite dimensional vector spaces are not in the core. ii) As another example, consider the category of sup lattices, where ....

M. Barr \Duality of Vector Spaces", Cahiers de Topologie et Geometrie Differentielle, 17 (1976) 3-14.

....will be the base field. In an arbitrary symmetric monoidal closed category, objects for which is an isomorphism are called reflexive, or more precisely reflexive with respect to . 5. 1 Linear Topology The approach we use goes back to the work of Lefschetz [31] and has been studied by Barr [6] and the first author [10] The idea is to add to the linear structure an additional topological structure, and then define the dual space to be the linear continuous maps. This serves to decrease the size of the dual space and thus create a large class of reflexive objects, i.e. objects which are ....

....and then define the dual space to be the linear continuous maps. This serves to decrease the size of the dual space and thus create a large class of reflexive objects, i.e. objects which are canonically isomorphic to their second dual. The categorical structure so arising was studied by Barr in [6], where he shows that the resulting category is autonomous. In [10] the first author examines this category as a model of linear logic, and considers the representations of groups and Hopf algebras in such spaces. This theory leads to a large class of new models of commutative linear logic [19] ....

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M. Barr, Duality of Vector Spaces, Cahiers de Top. et G'eom. Diff. 17, (1976), pp. 3-14.

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