Closed categories

TL;DR

This paper explores the generalization of adjoint functors to pairs of functors in directed graphs, aiming to extend categorical concepts beyond traditional categories and introduce new tools like relators for tensor products.

Contribution

It introduces a novel notion of joint functor pairs in directed graphs and discusses conditions for bijective graph transport mappings, expanding categorical theory.

Findings

Generalized adjoint functors to joint pairs in directed graphs

Identified conditions for bijective graph transport mappings

Proposed the concept of relators for tensor products

Abstract

We are checking the closed categories beginning with the category of sets and ending with the category of categories. The novelty is a generalizing the notion of adjoint functors to the joint pair of functors in the category of directed graphs. We have described for what condition we get a bijective name mapping for graphs transports. Graphs aren't instances of categories, however we believe that new notion of functors joint pair will become important also for the categorical studies. As an example of future applications we introduce the notion of relator to some tensor product.