On the Notion of Abstract Mathematical Structure in (Higher) Category Theory
DOI:
https://doi.org/10.35588/cc.v7d8359Keywords:
theory of n-categories, structure, isomorphism, equivalence, abstraction, principle of identity, monadAbstract
In this article we analyze two of the senses that the term structure takes on in (higher) category theory. To this end, we understand a category as a relational ontological context associated with a kind of abstract mathematical structure (e.g. the category of groups). In the expression abstract mathematical structure, the term structure has a twofold meaning. On the one hand, it refers to the kinds of structure that can be defined over a set-theoretic matter (e.g. the group structure) and to the models of each kind of structure (each particular group). On the other hand, the term structure refers to the abstract concept produced by the relation of isomorphism (and, more generally, by the relation of equivalence in the general case of a higher category). Isomorphic objects of a category can be understood as distinct presentations of the same abstract structure, each of which is a model of the corresponding kind of structure. In this article, we attempt to develop both senses of the term structure – as well as the relations among the three central concepts: kind of structure, model and presentation – within a unified conceptual framework.
Since a category is itself a referent of mathematical discourse, we will show that the concepts of kind of structure, model and presentation apply reflexively to the notion of category itself. Indeed, a category can be understood as a presentation of a model of a kind of structure whose associated relational ontological context is the 2-category of categories Cat. We will see that this kind of structure can also be understood as a (two-sorted) set-theoretic matter enriched with a category structure. The models of this kind of structure are defined by the identity criterion for categories given by the notion of equivalence of categories.
Finally, we will argue that this “structuralist” characterization of the notion of category itself (and of higher category) acts retroactively on the way we understand the notion of abstract mathematical structure in general. Each model of a structure will not be associated with an equivalence class, as is the case in the framework of Fregean abstraction, but rather (in the general case) with a connected component of the ∞-groupoid underlying the corresponding ∞-category.
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