Super-Algebra and Super-Geometry
Paths and cylinders
A super vector space is an object in the non-trivial symmetric monoidal category structure on the monoidal category of -graded vector spaces: as an object it is just a -graded vector space, but the braiding of the underlying tensor product of vector spaces is taken to be the non-trivial linear map which on elements of homogeneous degree is given by
We make this precise as definition 8 below.
Super vector spaces form the basis of superalgebra (over ground rings which are fields) in direct analogy of how ordinary vector spaces form the basis of ordinary algebra. For more on this see below, and for yet more see at geometry of physics -- superalgebra.
For a field, we write for the category whose
When the ground field is understood or when its precise nature is irrelevant, we will often notationally suppress it and speak of just the category Vect of vector spaces.
This is the category inside which linear algebra takes place.
Of course the category Vect has some special properties. Not only are its objects “linear spaces”, but the whole category inherits linear structure of sorts. This is traditionally captured by the following terminology for additive and abelian categories. Notice that there are several different but equivalent ways to state the following properties (discussed behind the relevant links).
We also make the following definition of -linear category, but notice that conventions differ as to which extra properties beyond Vect-enrichment to require on a linear category:
The category Vect of vector spaces (def. 1) is a -linear category according to def. 3.
Here the abstract direct sum is the usual direct sum of vector spaces, whence the name of the general concept.
For two -vector spaces, the vector space structure on the hom-set of linear maps is given by “pointwise” multiplication and addition of functions:
for all and .
Recall the basic construction of the tensor product of vector spaces:
Given two vector spaces over some field , , their tensor product of vector spaces is the vector space denoted
whose elements are equivalence classes of tuples of elements with , 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
The existence of the tensor product of vector spaces, def. 4, equips the category Vect of vector spaces with extra structure, which is a “categorification” of the familiar structure of a semi-group. One also says “monoid” for semi-group and therefore categories equipped with a tensor product operation are also called monoidal categories:
A monoidal category is a category equipped with
out of the product category of with itself, called the tensor product,
called the unit object or tensor unit,
a natural isomorphism
called the associator,
a natural isomorphism
called the left unitor, and a natural isomorphism
called the right unitor,
such that the following two kinds of diagrams commute, for all objects involved:
the pentagon identity:
As expected, we have the following basic example:
For a field, the category Vect of -vector spaces becomes a monoidal category (def. 5) as follows
the abstract tensor product is the tensor product of vector spaces from def. 4;
the tensor unit is the field itself, regarded as a 1-dimensional vector space over itself;
the associator is the map that on representing tuples acts as
the left unitor is the map that on representing tuples is given by
and the right unitor is similarly given by
That this satisifes the pentagon identity (def. 5) and the left and right unit identities is immediate on representing tuples.
But the point of the abstract definition of monoidal categories is that there are also more exotic examples. The followig one is just a minimal enrichment of example 2, and yet it will be important.
Let be a group (or in fact just a monoid/semi-group). A -graded vector space is a direct sum of vector spaces labeled by the elements in :
of -graded vector spaces is a linear map that respects this direct sum structure, hence equivalently a direct sum of linear maps
for all , such that
This defines a category, denoted . Equip this category with a tensor product which on the underlying vector spaces is just the tensor product of vector spaces from def. 4, equipped with the -grading which is obtained by multiplying degree labels in :
The tensor unit for the tensor product is the ground field , regarded as being in the degree of the neutral element
The associator and unitors are just those of the monoidal structure on plain vector spaces, from example 2.
One advantage of abstracting the concept of a monoidal category is that it allows to prove general statements uniformly for all kinds of tensor products, familar ones and more exotic ones. The following lemma 1 and remark 1 are two important such statements.
Let be a monoidal category, def. 5. Then the left and right unitors and satisfy the following conditions:
for all objects the following diagrams commutes:
For proof see at monoidal category this lemma and this lemma.
The above discussion makes it clear that a monoidal category is like a monoid/semi-group, but “categorified”. Accordingly we may consider additional properties of monoids/semi-groups and correspondingly lift them to monoidal categories. A key such property is commutativity. But while for a monoid commutativity is just an extra property, for a monoidal category it involves choices of commutativity-isomorphisms and hence is extra structure. We will see below that this is the very source of superalgebra.
The categorification of “commutativity” comes in two stages: braiding and symmetric braiding.
A braided monoidal category, is a monoidal category (def. 5) equipped with a natural isomorphism
(for all objects ) called the braiding, such that the following two kinds of diagrams commute for all objects involved (“hexagon identities”):
where denotes the components of the associator of .
Consider the simplest non-trivial special case of -graded vector spaces from example 3, the case where is the cyclic group of order two.
A -graded vector space is a direct sum of two vector spaces
where we think of as the summand that is graded by the neutral element in , and of as being the summand that is graded by the single non-trivial element.
A homomorphism of -graded vector spaces
is a linear map of the underlying vector spaces that respects the grading, hence equivalently a pair of linear maps
between then summands in even degree and in odd degree, respectively:
The tensor product of -graded vector space is the tensor product of vector spaces of the underlying vector spaces, but with the grading obtained from multiplying the original gradings in . Hence
As in example 3, this definition makes a monoidal category def. 5.
There are, up to braided monoidal equivalence of categories, precisely two choices for a symmetric braiding (def. 7)
on the monoidal category of -graded vector spaces from def. 4:
the trivial braiding which is the natural linear map given on tuples representing an element in (according to def. 4) by
the super-braiding which is the natural linear function given on tuples of homogeneous degree (i.e. , for ) by
For a monoidal category, write
for the full subcategory on those which are invertible objects under the tensor product, i.e. such that there is an object with and . Since the tensor unit is clearly in (with ) and since with also (with ) the monoidal category structure on restricts to .
Accordingly any braiding on restricts to a braiding on . Hence it is sufficient to show that there is an essentially unique non-trivial symmetric braiding on , and that this is the restriction of a braiding on .
Now is necessarily a groupoid (the “Picard groupoid” of ) and in fact is what is called a 2-group. As such we may regard it equivalently as a homotopy 1-type with group structure, and as such it it is equivalent to its delooping
regarded as a pointed homotopy type. (See at looping and delooping).
The Grothendieck group of is
the fundamental group of the delooping space.
Now a symmetric braiding on is precisely the structure that makes it a symmetric 2-group which is equivalently the structure of a second delooping (for the braiding) and then a third delooping (for the symmetry), regarded as a pointed homotopy type.
This way we have rephrased the question equivalently as a question about the possible k-invariants of spaces of this form.
Now in the case at hand, has precisely two isomorphism classes of objects, namely the ground field itself, regarded as being in even degree and regarded as being in odd degree. We write and for these, respectively. By the rules of the tensor product of graded vector spaces we have
In other words
Now under the above homotopical identification the non-trivial braiding is identified with the elements
Due to the symmetry condition (def. 7) we have
which implies that
Therefore for classifying just the symmetric braidings, it is sufficient to restrict the hom-spaces in from being either or empty, to hom-sets being or empty. Write for the resulting 2-group.
In conclusion then the equivalence classes of possible k-invariants of , hence the possible symmetric braiding on are in the degree-4 ordinary cohomology of the Eilenberg-MacLane space with coefficients in . One finds (…)
The symmetric monoidal category (def. 7)
Structures internal to super-vector spaces
By internalizing algebra and geometry in the category of super vector spaces, one obtains the corresponding superalgebra and supergeometry.
By the above definition, any structure in works just like the corresponding structure in Vect, but with a sign inserted whenever two odd-graded symbols are interchanged. For more on this see also at signs in supergeometry.
Deligne’s theorem on tensor categories
Deligne's theorem on tensor categories says that all suitable tensor categories of subexponential growth have a fiber functor to and are equivalent to categories of representations of affine algebraic supergroups.