# Tagged Questions

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### Universal property of tensor products / Categorial expression of bilinearity

Let $V$ and $W$ be linear spaces. According to Wikipedia, the universal definition of the tensor product $V \otimes W$ satisfies the following property: There is a bilinear map (i.e., linear in each ...
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### In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual?

I've read in several places that one motivation for category theory was to be able to give precise meaning to statements like, "finite dimensional vector spaces are canonically isomorphic to their ...
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### Choice of the right isomorphisms

The question makes sense in every abelian category, but for the moment let's work in the category of vector spaces over a field. PS: I previously posted a similar question which didn't make a lot of ...
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Let $X$ be a Hilbert space with associated canonical isomorphism $I:X\rightarrow X^\ast$ (by the Riesz representation theorem). If $A:X\rightarrow X$ is a linear operator on $X$, then its dual ...
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### Category equivalence of sets and vector spaces

It seems to me that for a field $K$, the functor $\mathbf{Set}\to\mathbf{Vec}_K$, sending a set to the free $K$-vector space on it, is a category equivalence. It is full and faithful because a linear ...
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### Why are (representations of ) quivers such a big deal?

Quivers are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, ...
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