exterior algebra    The exterior algebra ΛV\Lambda V of a vector
space is the free graded-commutative algebra? over VV, where the
elements of VV are taken to be of degree 11. (That is, the
forgetful functor takes a graded-commutative algebra to its
vector space of degree-11 elements.)  This construction
generalizes to group representations, chain complexes, vector
bundles, coherent sheaves, and indeed objects in any symmetric
monoidal linear categories with enough colimits, where the
tensor product distributes over those colimits (as in a 2-rig).
Explicit definition  We begin with the construction for vector
spaces and then sketch how to generalize it.  For vector spaces
Suppose VV is a vector space over a field KK. Then the exterior
algebra ΛV\Lambda V is generated by the elements of VV using
these operations:    • addition and scalar multiplication   • an
associative binary operation \wedge called the exterior product
or wedge product,  subject to these identities:    • the
identities necessary for ΛV\Lam