as the monoidal product and the purely even super vector space as the unit object. The involutive braiding operator given by on homogeneous elements, turns into a symmetric monoidal category. This commutativity isomorphism encodes the "rule of signs" that is essential to super linear algebra. It effectively says that a minus sign is picked up whenever two odd elements are interchanged. One need not worry about signs in the categorical setting as long as the above operator is used wherever appropriate. is also a closed monoidal category with the internal Hom object, , given by the super vector space of all linear maps from to . The ordinary set is the even subspace therein: The fact that is closed means that the functor is left adjoint to the functor , given a natural bijection Superalgebra A superalgebra over can be described as a super vector space with a multiplication map that is a super vector space homomorphism. This is equivalent to demanding Associativity and the existence of an identity can be expressed with the usual commutative diagrams, so that a unital associative superalgebra over is a monoid in the category . Notes References Categories in category theory