Given a set , an -graded vector space is a map assigning to each element a vector space . Given -graded vector spaces and , a morphism assigns to each element a linear operator . That is, the category of -graded vector spaces is the functor category .
In other words, a -graded vector space is a functor , where the set is treated as a discrete category, and Vect is the category of vector spaces. Similarly, a morphism of -graded vector spaces is a natural transformation between such functors. In short, the category of -graded vector spaces is the functor category .
The unit is given by having the underlying field as and the trivial vector space for other gradings. If is a commutative monoid, then is a symmetric monoidal category.
By far the most widely-used examples are and . Indeed, the term graded vector space is often used to mean a -graded vector space with one of these choices of . The case is also important: a -graded vector space is also called a supervector space. However, in this case one often uses a different braiding on , one which uses the ring structure of ; see Wikipedia.
For the case that is a group, this means that the category of -graded vector spaces is a categorification of the group algebra of , 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
where goes to the ground field , the monoidal category of -graded vector spaces can be identified with the monoid of spans of the form
Here denotes some version of the category of 2-vector spaces with the property that the category is one of its objects and such that (in analogy to how ) and maps to . Possible choices for 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.
The case 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 -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.
A pre--graded vector space (pre-gvs) is a direct sum . The elements of are said to be homogeneous of degree . If , write .
Sometimes it may be convenient to write and . Another very useful piece of notation is . 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 or . (This effectively restricts from a -grading to one over
We consider the field to be a pre-gvs with , and if . We say is of finite type if for all .
A linear map between pre-gvs is of degree if for all . (Note this may also occur as .)
A morphism is a linear map of degree zero. Pregraded vector spaces and the morphisms between them define the category .
The set of all linear maps of degree from to will be denoted and we set
Of course, we now have two notations for the same object, .
If , the -suspension of is given by
We will need , the 1-suspension, and in particular. Of course, . It is also useful to note . Again, of course, is an isomorphism of degree having as its inverse.
(This is the basic example of the suspension functor discussed in triangulated category.)
The dual of a (pre-)gvs is defined by
If is of degree , then its transpose
is given by
for and . Thus if , then
} In particular, for an isomorphism
Let be a gvs, by convention in the duality
we will usually assume is non-negatively graded (so ), whilst the right hand side is non-positively graded.
If is of finite type then , of course. The suspension of the dual can be identified with and similarly . These identifications are via the rules:
This sign convention is needed to ensure that .
The tensor product of two pre-gvs, and , is , where
and is of degree .
In particular there is a natural injection , and this is an isomorphism if either or 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 , then -graded vector spaces may be equiped with a non-trivial braiding. For instance on the monoidal category of -graded vector spaces there is one non-trivial symmetric braiding, up to equivalence. Imposing this yields super vector spaces.