nLab graded vector space



Differential-graded objects

Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Schanuel's lemma

Homology theories




Given a set GG, an GG-graded vector space is a map VV assigning to each element gGg \in G a vector space V gV_g. Given GG-graded vector spaces VV and WW, a morphism f:VWf\colon V \to W assigns to each element gGg \in G a linear operator f g:V gW gf_g\colon V_g \to W_g. That is, the category of GG-graded vector spaces is the functor category Vect GVect^G.

We can just as easily talk about a GG-graded module or a GG-graded object in any category. However, a graded algebra has additional requirements (using a monoid structure on GG, as below).

In other words, a GG-graded vector space is a functor V:GVectV\colon G \to Vect, where the set GG is treated as a discrete category, and Vect is the category of vector spaces. Similarly, a morphism of GG-graded vector spaces is a natural transformation between such functors. In short, the category of GG-graded vector spaces is the functor category Vect GVect^G.

People are usually interested in GG-graded vector spaces when the set GG is equipped with extra structure. If the set GG is a monoid, Vect GVect^G is a monoidal category, with the tensor product given by

(VW) g= hk=gV hW k. (V \otimes W)_g = \bigoplus_{h \cdot k = g} V_h \otimes W_k.

The unit is given by having the underlying field kk as V eV_e and the trivial vector space for other gradings. If GG is a commutative monoid, then Vect GVect^G is a symmetric monoidal category.

If GG is a group, every finite-dimensional GG-graded vector space has a left dual and a right dual. And if GG is an abelian group, these duals coincide.

By far the most widely-used examples are G=G = \mathbb{Z} and G=G = \mathbb{N}. Indeed, the term graded vector space is often used to mean a GG-graded vector space with one of these choices of GG. The case G=/2G = \mathbb{Z}/2 is also important: a /2\mathbb{Z}/2-graded vector space is also called a supervector space. However, in this case one often uses a different braiding on Vect GVect^G, one which uses the ring structure of \mathbb{N}; see Wikipedia.

Explicit definition

Let GG be a set with decidable equality. Then, given a commutative ring RR (usually a field FF), a GG-graded RR-module is an RR-module VV (usually a vector space VV) with a binary function ():V×GV\langle - \rangle_{(-)}: V \times G \to V called the grade projection operator such that

  • for all v:Vv:V, v= g:Gv gv = \sum_{g:G} \langle v \rangle_g

  • for all a:Ra:R, b:Rb:R, v:Vv:V, w:Vw:V, and g:Gg:G, av+bw g=av g+bw g\langle a v + b w \rangle_g = a \langle v \rangle_g + b \langle w \rangle_g

  • for all v:Vv:V and g:Gg:G, v g g=v g\langle \langle v \rangle_g \rangle_g = \langle v \rangle_g

  • for all v:Vv:V, g:Gg:G, and h:Gh:G, (gh)(v g h=0)(g \neq h) \Rightarrow (\langle \langle v \rangle_g \rangle_h = 0)

For an element g:Gg:G, the image of g\langle - \rangle_g under VV is denoted as V g\langle V \rangle_g.

V gim( g)\langle V \rangle_g \coloneqq \mathrm{im}(\langle - \rangle_g)


For the case that GG is a group, this means that the category of GG-graded vector spaces is a categorification of the group algebra of GG, where numbers are replaced by vector spaces. Recalling from the remark in category algebra that the group algebra of a group can be identified with the monoid of spans of the form

G pt pt Vect, \array{ && G \\ & \swarrow && \searrow \\ pt &&\Rightarrow&& pt \\ & \searrow && \swarrow \\ && Vect } \,,

where ptVectpt \to Vect goes to the ground field kk, the monoidal category of GG-graded vector spaces can be identified with the monoid of spans of the form

G pt pt 2Vect. \array{ && G \\ & \swarrow && \searrow \\ pt &&\Rightarrow&& pt \\ & \searrow && \swarrow \\ && 2Vect } \,.

Here 2Vect2Vect denotes some version of the category of 2-vector spaces with the property that the category VectVect is one of its objects and such that End 2Vect(Vect)VectEnd_{2Vect}(Vect) \simeq Vect (in analogy to how End Vect(k)kEnd_{Vect}(k) \simeq k) and pt2Vectpt \to 2Vect maps to VectVect. Possible choices for 2Vect2Vect is the 2-category of Kapranov-Voevodsky 2-vector spaces or the bigger bicategory Bimod of algebras and bimodules.

More details on this perspective on graded vector spaces are in Nonabelian cocycles and their quantum symmetries.

In general, adding algebraic structure onto GG will add categorical structure onto Vect G\text{Vect}^G. A table keeping track of these structures is below:

Structure on GGStructure on Vect G\text{Vect}^G
MonoidMonoidal category
Finite groupFusion category
Finite abelian groupBraided fusion category

Special case of \mathbb{Z}-graded vector spaces

The case G=G = \mathbb{Z} serves as a base for many other applications of the same basic idea. It has some of its own ‘traditional’ terminology and structure that links it to differential objects, so that a ‘differential graded vector space’ is a chain complex of vector spaces. We will use ‘gvs’ as an abbreviation for this sort of graded vector space and ‘dgvs’ for the differential form. (Of course, the theory easily adapts to handle graded modules over a ring, and with some restriction, to graded groups.) Basing algebras on dgvs gives differential graded algebras (dg-algebra) and so on.


The entry here will be a sort of lexicon of some terms which are taken from a source on rational homotopy theory. This will be more or less ‘as-is’ from the source (except translating it from the original French that is!), i.e. without too much editing. This means that there may be conflicts with other entries, which will need resolving later. Some links to other entries have been given but more could be made. There WILL initially be some duplication but that will be eliminated later on.

The lexicon will be spread over a number of entries with links given in the table of contents ion the right hand side at the top.

Note With \mathbb{Z}-graded vector spaces (and sometimes with other examples as well), some authors work with a direct sum of the various vector spaces instead of using an indexed family.

(Pre-)graded vector spaces

A pre-\mathbb{Z}-graded vector space (pre-gvs) is a direct sum V= pV pV = \bigoplus_{p\in \mathbb{Z}} V_p. The elements of V pV_p are said to be homogeneous of degree pp. If xV px \in V_p, write |x|=p|x| = p.

Sometimes it may be convenient to write x¯=(1) |x|x\bar{x} = (-1)^{|x|}x and V += p>0V pV_+ = \bigoplus_{p\gt 0}V_p. Another very useful piece of notation is V p=V pV^p = V_{-p}. In this case we will refer to an ‘upper grading’ with, in contrast, the other notation being a ‘lower grading’. These are merely for convenience and have little or no mathematical significance.

For the purposes of this lexicon:

A graded vector space (gvs) is a positively or negatively graded pre-gvs that is either V= p0V pV = \sum_{p\geq 0}V_p or V= p0V pV = \sum_{p\leq 0}V^p. (This effectively restricts from a \mathbb{Z}-grading to one over \mathbb{N}

We consider the field kk to be a pre-gvs with (k) 0=k(k)_0 = k, and (k) p=0(k)_p =0 if p0p\neq 0. We say VV is of finite type if dim(V p)<dim(V_p) \lt \infty for all pp.

A linear map f:VWf :V\to W between pre-gvs is of degree pp if f(V q)W p+qf(V_q) \subseteq W_{p+q} for all qq. (Note this may also occur as f(V q)W qpf(V^q) \subseteq W^{q-p}.)

A morphism f:VWf : V\to W is a linear map of degree zero. Pregraded vector spaces and the morphisms between them define the category preGVS{pre GVS}.

The set of all linear maps of degree pp from VV to WW will be denoted Hom p(V,W)Hom_p(V,W) and we set

Hom(V,W)= pHom p(V,W).Hom(V,W) = \bigoplus_p Hom_p(V,W).

Of course, we now have two notations for the same object, preGVS(V,W)=Hom 0(V,W){pre GVS}(V,W) = Hom_0(V,W).


If rr\in \mathbb{Z}, the rr-suspension of VV is given by (s rV) n=V nr.(s^r V)_n = V_{n-r}.

We will need ss, the 1-suspension, and s 1s^{-1} in particular. Of course, (s 1V) n=V n+1(s^{-1}V)_n = V_{n+1}. It is also useful to note (s rV) p=V p+r(s^r V)^p = V^{p+r}. Again, of course, s r:Vs rVs^r: V\to s^r V is an isomorphism of degree r-r having s rs^{-r} as its inverse.

(This is the basic example of the suspension functor discussed in triangulated category.)


The dual of a (pre-)gvs VV is #V\#V defined by

(#V) p :=Hom p(V,k) Vect(V p,k) #(V p) =#(V p). \begin{aligned} (\#V)_p &: = Hom_p(V,k)\\ &\cong{Vect}(V_{-p},k)\\ &\cong\#(V_{-p})\\ & =\#(V^p). \end{aligned}

If f:VWf : V\to W is of degree |f||f|, then its transpose

tf:#W#V^t f : \#W \to \#V

is given by

( tf)(ψ)(x)=(1) |f||ψ|ψf(x),(^t f)(\psi)(x) = (-1)^{|f||\psi|}\psi f(x),

for ψ#W\psi \in \#W and xVx\in V. Thus if VfWgXV\stackrel{f}{\to}W\stackrel{g}{\to}X, then

t(gf)=(1) |f||g|( tf tg).^t (g\circ f) = (-1)^{|f||g|}(^t f\circ {}^t g).} In particular, for ff an isomorphism

( tf) 1=(1) |f|t(f 1).( ^t f)^{-1} = (-1)^{|f|} ^t (f^{-1}).


Let VV be a gvs, by convention in the duality

;:V#V,\langle \quad ; \quad \rangle: V \leftrightarrow \#V,

we will usually assume VV is non-negatively graded (so V= p0V pV = \bigoplus_{p\geq 0}V_p), whilst the right hand side is non-positively graded.

If VV is of finite type then ##VV\#\#V \cong V, of course. The suspension of the dual s(#V)s(\#V) can be identified with #(s 1V)\#(s^{-1}V) and similarly s 1(#V)=#s(V)s^{-1}(\#V) = \#s(V). These identifications are via the rules:

s 1z;su =(1) |z|z;u, sz;s 1u =(1) |z|+1z;u.\begin{aligned} \langle s^{-1}z;su\rangle &= (-1)^{|z|} \langle z;u \rangle,\\ \langle s z;s^{-1}u \rangle &= (-1)^{|z|+1}\langle z;u \rangle.\end{aligned}

This sign convention is needed to ensure that ss 1=ids s^{-1} = id.

Tensor products

The tensor product of two pre-gvs, VV and WW, is VWV\otimes W, where

(VW) n= p+q=nV pW q.(V\otimes W)_n = \bigoplus_{p + q = n}V_p\otimes W_q.

On morphisms

(fg)(vw)=(1) |g||f|(f(v)g(w))(f\otimes g) (v\otimes w) = (-1)^{|g||f|}(f(v) \otimes g(w))

and is of degree |f|+|g||f| + |g|.

In particular there is a natural injection (#V)(#W)#(VW)(\#V)\otimes (\#W) \to \#(V\otimes W), and this is an isomorphism if either VV or WW is of finite type.

This tensor product makes the category of graded vector spaces into a monoidal category. This becomes a symmetric monoidal category with braiding the evident braiding inherited from the underlying vector spaces.

Notice that given suitable cocycles on GG, then GG-graded vector spaces may be equiped with a non-trivial braiding. For instance on the monoidal category of /2\mathbb{Z}/2-graded vector spaces there is one non-trivial symmetric braiding, up to equivalence. Imposing this yields super vector spaces.


  • G. Aragón, J.L. Aragón, M.A. Rodríguez (1997), Clifford Algebras and Geometric Algebra, Advances in Applied Clifford Algebras Vol. 7 No. 2, pg 91–102, doi:10.1007/BF03041220, S2CID:120860757

Last revised on May 7, 2023 at 00:51:10. See the history of this page for a list of all contributions to it.