On the category of structure species

TL;DR

This paper explores the mathematical relationships among different structures within objects and demonstrates that a category can be reconstructed from the category of structure species, extending ideas from structuralism and anabelian geometry.

Contribution

It provides a categorical reformulation of structure species and proves that a category can be reconstructed from its structure species category, with implications for structuralism and geometry.

Findings

Categories can be reconstructed from structure species categories.

The work extends the concept of structure in categorical terms.

Partial reconstruction is possible up to slight indeterminacy.

Abstract

The purpose of the present paper is to make a mathematical study of the differences and relations among possible structures inherent in an object, as well as of the whole structure constituted by them (i.e., the structure of structures), against the background of the structuralism by Claude L\'{e}vi-Strauss and others. Our discussion focuses on Blanchard's categorical reformulation of the notion of structure species introduced originally by Bourbaki. The main result of the present paper asserts that a category can be reconstructed, up to a certain slight indeterminacy, from the category of structure species on it. This result is partially motivated by various reconstruction theorems that have been shown in the context of anabelian geometry.