On integral structure types

TL;DR

This paper introduces integral structure types as a categorical framework extending combinatorial species to include negative coefficients, enabling new algebraic and geometric insights such as describing Chern classes of hypersurfaces.

Contribution

It develops the theory of integral structure types, extending combinatorial species to handle negative coefficients and operators, and applies this to describe Chern classes combinatorially.

Findings

Defined integral structure types as a categorification of power series with negative coefficients.

Extended the notion of operators and commutators to integral structure types.

Provided a combinatorial description of Chern classes of projective hypersurfaces.

Abstract

We introduce integral structure types as a categorical analogue of virtual combinatorial species. Integral structure types then categorify power series with possibly negative coefficients in the same way that combinatorial species categorify power series with non-negative rational coefficients. The notion of an operator on combinatorial species naturally extends to integral structure types, and in light of their `negativity' we define the notion of the commutator of two operators on integral structure types. We then extend integral structure types to the setting of stuff types as introduced by Baez and Dolan, and then conclude by using integral structure types to give a combinatorial description for Chern classes of projective hypersurfaces.