nLab Vect



Category theory

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




Given a field kk, the category of kk-vector spaces Vect kVect_k is the category whose

  1. objects are vector spaces,

  2. morphisms are linear maps.

If the field kk is understood, one often just writes VectVect.

Via direct sum and tensor product of vector spaces k\otimes_k, this becomes a symmetric monoidal category in two compatible ways, making it a distributive monoidal category, in particular a rig category.

The study of VectVect is called linear algebra.



For any field kk, the category Vect kVect_k is complete, cocomplete and closed monoidal with respect to the tensor product of vector spaces.

Splitting lemma

Assuming the axiom of choice (and essentially by the basis theorem):


In VectVect every short exact sequence splits.

(See there.)


On FinDimVect this is a categorification of the rank-nullity theorem.

Finite-dimensional vector spaces

The full subcategory of Vect consisting of finite-dimensional vector spaces may be denoted FinDimVect.

This is a compact closed category (see here).

FinDimVectFinDimVect is where most of ordinary linear algebra lives, although much of it makes sense in all of VectVect. See also at quantum information theory in terms of dagger-compact categories.

On the other hand, anything involving transposes or inner products really takes place in FinFin Hilb.


More generally, for RR any ring (not necessarily a field) then the analog of VectVect is the category RRMod of RR-modules and module homomorphisms between them.

Vector bundles

For XX a suitable space of sorts, there is the category Vect(X) of vector bundles over XX. Specifically for XX a topological space, there is the category of topological vector bundles over XX. For X=*X = \ast the point space, then this is equivalently the category of plain vector spaces:

Vect(*)Vect. Vect(\ast) \simeq Vect \,.

Topological vector spaces

There are various categories of topological vector spaces, for instance bornological topological vector spaces.

category: category

Last revised on May 1, 2024 at 04:03:35. See the history of this page for a list of all contributions to it.