A GENERALISATION OF HIGHER-RANK GRAPHS

We introduce ‘generalised higher-rank k-graphs’ as a class of categories equipped with a notion of size. They extend not only higher-rank k-graphs, but also the Levi categories introduced by the first author as a categorical setting for graphs of groups. We prove that examples of generalised higher-rank k-graphs can be constructed using Zappa–Szép products of groupoids and higher-rank graphs.

CATEGORICAL HARMONY AND PATH INDUCTION

This paper responds to recent work in the philosophy of Homotopy Type Theory by James Ladyman and Stuart Presnell. They consider one of the rules for identity, path induction, and justify it along ‘pre-mathematical’ lines. I give an alternate justification based on the philosophical framework of inferentialism. Accordingly, I construct a notion of harmony that allows the inferentialist to say when a connective or concept is meaning-bearing and this conception unifies most of the prominent conceptions of harmony through category theory. This categorical harmony is stated in terms of adjoints and says that any concept definable by iterated adjoints from general categorical operations is harmonious. Moreover, it has been shown that identity in a categorical setting is determined by an adjoint in the relevant way. Furthermore, path induction as a rule comes from this definition. Thus we arrive at an account of how path induction, as a rule of inference governing identity, can be justified on mathematically motivated grounds.

Completeness types for uniformity theory on textures

In this paper the author considers the various types of completeness for di-uniform texture spaces and especially for complemented ones. Following that, the relationships between completeness of uniform spaces and these types of completeness for complemented di-uniform texture spaces are investigated in a categorical setting, just as interrelations between quasi-uniform spaces and di-uniform texture spaces are pointed out insofar as completeness is concerned. Additionally, useful requirements among the various types of completeness of a di-uniformity and real dicompactness of the uniform ditopological space generated by that di-uniformity are presented as a diagram.