Exponential Objects in Categories of Generalized Uniform Hypergraphs

TL;DR

This paper explores the construction of exponential objects within categories of generalized uniform hypergraphs and explains why traditional categories of graphs and hypergraphs lack such structures, using nerve-realization adjunctions.

Contribution

It introduces a method to construct exponential objects in generalized hypergraph categories and clarifies the absence of these objects in conventional graph categories.

Findings

Exponential objects are constructible in generalized hypergraph categories.

Conventional graph categories lack exponential objects due to structural limitations.

Nerve-realization adjunctions explain the non-existence of exponentials in standard categories.

Abstract

We construct exponential objects in categories of generalized uniform hypergraphs and use embeddings induced by nerve-realization adjunctions to show why conventional categories of graphs and hypergraphs do not have exponential objects.