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A vector space is finite-dimensional if it admits a finite basis. The category FinVect of finite-dimensional vector spaces is of course the full subcategory of Vect whose objects are finite-dimensional.
A vector space is finite-dimensional precisely if the hom-functor preserves filtered colimits.
Every vector space is the filtered colimit of the diagram of finite-dimensional subspaces and inclusions between them; applying this to , the condition that preserves filtered colimits implies that the canonical comparison map
is an isomorphism, so some element in the colimit represented by gets mapped to , i.e., for some inclusion . This implies is an isomorphism, so that is finite-dimensional.
In the converse direction, observe that has a right adjoint (and in particular preserves filtered colimits) if is finite-dimensional.
To see this, first notice that the dual vector space of functionals to the ground field is a dual object to in the monoidal category sense, so that there is a counit taking . The unit is uniquely determined from this counit and can be described using any basis of and dual basis as the map
taking . We thus have an adjunction , which is mated to an adjunction by familiar hom-tensor adjunctions; thus has a right adjoint.
This means that
Finite-dimensional vector spaces are exactly the compact objects of Vect in the sense of locally presentable categories, but also the compact = dualizable objects in the sense of monoidal category theory. In particular the category FinVect is a compact closed category.
Last revised on November 4, 2016 at 07:28:49. See the history of this page for a list of all contributions to it.