Exponential Objects in Categories of Generalized Uniform Hypergraphs

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.As a consequence of our constructions we address the problem that the category of k-uniform hypergraphs (as defined in [4]) lacks connected colimits, exponentials and does not continuously embed into the category of hypergraphs. We prove there is a continuous embedding of the category of k-uniform hypergraphs in a category of (X, M)-graphs (Proposition 5.5) which preserves any relevant categorical structures (e.g., colimits, exponentials, injectives, projectives). Therefore, working in a category of (reflexive) (X, M)-graphs provides a better categorical environment for constructions on uniform hypergraphs. (X, M )-GraphsWe begin with a definition.Definition 2.1.1. Let M be a monoid and X a right M-set. The theory for (X, M)-graphs, G (X,M ) , is the category with two objects V and A and homsets given byComposition is defined as m • x = x.m (the right-action via M), m • m ′ = m ′ m (monoid operation of M). 2. Let M be a monoid such that the set Fix(M) := { m ′ ∈ M | ∀m ∈ M, m ′ m = m ′ } is non-empty. Let X := { x m ′ | m ′ ∈ Fix(M) } be the right M-set with rightaction x m .m ′ := x mm ′ for each m ∈ M and x m ′ ∈ X. The theory for reflexive (X, M)-graphs, rG (X,M ) is the same as for G (X,M ) but with rG (X,M ) (A, V ) := {ℓ}, and composition ℓ • m = ℓ, ℓ • x m ′ = id V , and x • ℓ = x for each m ∈ M and m ′ ∈ Fix(M).