Negative quantities in objective linear algebra

Objective linear algebra works with slice categories instead of vector
spaces and with colimit-preserving functors instead of linear maps. It
serves as a method to turn algebro-combinatorial identities into bijective
proofs. I will explain an approach to negative quantities in this context,
and exemplify its features with an objective treatment of exterior powers
and determinants. Groupoids instead of sets are required to encode signs as
homotopies. This is joint work with Jesper M. Møller.