According to the wikipedia, the exterior algebra of a $\Bbbk$-vector space $V$ is initial with respect to being unital and there existing a $\Bbbk$-linear map $j\colon V\to A$ such that $j(v)^2=0$ for all $v\in V$.
This is a reasonable algebra to consider if one is interested in measuring $k$-dimensional volumes, which are specified by $k$ linearly independent vectors, and which are degenerate if the vectors are not linearly independent (equivalently, for $char \Bbbk\neq 2$, measuring $k$-dimensional signed volumes). Then multiplication consists simply of throwing in extra vectors.
My question, inspired by the recent question on the meaning of addition in the exterior algebra, is about the meaning of co-multiplication. I happen to know virtually nothing about co-algebras outside of their formalism, so in part I am looking for answers that will help me build some intuition, geometric and otherwise, about what's happening.
- What is a categorical argument that the exterior algebra satisfying the above universal property has co-multiplication (and is thus a bi-algbera)?
- Is there a (heurisitc) geometric interpretation of the exterior algebra's co-multiplication similar to the $k$-dimensional volume tracking I sketched above?
- (Bonus question): Is there any geometric significance to the anti-pode that makes the exterior (bi-)algebra into a Hopf algebra?