homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
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see also algebraic topology
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The tensor product of two vector spaces is a new vector space with the property that bilinear maps out of the Cartesian product of the two spaces are equivalently linear maps out of the tensor product.
The tensor product of vector spaces is just the special case of the tensor product of modules over some ring $R$ for the case that this ring happens to be a field.
The tensor product of vector spaces makes the category Vect of all vector spaces into a monoidal category, in fact a distributive monoidal category.
Given two vector spaces over some field $k$, $V_1, V_2 \in Vect_k$, their tensor product of vector spaces is the vector space denoted
whose elements are equivalence classes of tuples of elements $(v_1,v_2)$ with $v_i \in V_i$, for the equivalence relation given by
More abstractly this means that the tensor product of vector spaces is the vector space characterized by the fact that
it receives a bilinear map
(out of the Cartesian product of the underlying sets)
any other bilinear map of the form
factors through the above bilinear map via a unique linear map