Some basic steps in objective linear algebra

The 'objective method', advocated by Lawvere, seeks to calculate directly
with combinatorial objects, rather than with their numbers. In algebraic
combinatorics, this is largely a question of doing linear algebra over the
'ground field' of finite sets (or groupoids or infinity-groupoids). The
role of vector spaces is played by slice categories, and the role of linear
maps is played by linear functors (which here means colimit-preserving
functors), which in turn can be represented by spans, so that 'matrix
multiplication' is given by pullbacks. The ordinary vector-space level is
recovered from the objective level by taking (homotopy) cardinality. After
explaining the basic theory, the talk will focus on finiteness conditions
and the objective version of the duality between vector spaces and
pro-finite-dimensional vector spaces, including a neat combinatorial
interpretation of continuity. To finish I will briefly outline how the
basic set-up supports more elaborate algebraic structures, such as
incidence bialgebras, Möbius inversion, and antipodes.

This is based on joint work with Imma Gálvez and Andy Tonks.