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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Given a field , the category of -vector spaces is the category whose
objects are vector spaces,
morphisms are linear maps.
If the field is understood, one often just writes .
Via direct sum and tensor product of vector spaces , this becomes a symmetric monoidal category in two compatible ways, making it a distributive monoidal category, in particular a rig category.
The study of is called linear algebra.
The full subcategory of Vect consisting of finite-dimensional vector spaces is denoted FinVect.
This is a compact closed category (see here).
is where most of ordinary linear algebra lives, although much of it makes sense in all of . See also at finite quantum mechanics in terms of dagger-compact categories.
On the other hand, anything involving transposes or inner products really takes place in Hilb.
More generally, for any ring (not necessarily a field) then the analog of is the category Mod of -modules and module homomorphisms between them.
For a suitable space of sorts, there is the category Vect(X) of vector bundles over . Specifically for a topological space, there is the category of topological vector bundles over . For the point space, then this is equivalently the category of plain vector spaces:
Last revised on March 24, 2021 at 04:19:29. See the history of this page for a list of all contributions to it.