Fibered aspects of Yoneda's regular span

TL;DR

This paper reinterprets Yoneda's regular spans as fiberwise opfibrations within a fibrational framework, extending classical classification results to non-symmetric and non-additive contexts like crossed extensions.

Contribution

It introduces the concept of fiberwise opfibrations to analyze Yoneda's regular spans and extends classification theorems to broader fibrational settings and algebraic structures.

Findings

Interprets Yoneda's regular spans as fiberwise opfibrations.

Extends Yoneda's Classification Theorem to non-symmetric fibrations.

Applies the theory to crossed extensions in algebraic structures.

Abstract

In this paper we start by pointing out that Yoneda's notion of a regular span $S \colon \mathcal{X} \to \mathcal{A} \times \mathcal{B}$ can be interpreted as a special kind of morphism, that we call fiberwise opfibration, in the 2-category $\mathsf{Fib}(\mathcal{A})$. We study the relationship between these notions and those of internal opfibration and two-sided fibration. This fibrational point of view makes it possible to interpret Yoneda's Classification Theorem given in his 1960 paper as the result of a canonical factorization, and to extend it to a non-symmetric situation, where the fibration given by the product projection $Pr_0 \colon \mathcal{A} \times \mathcal{B} \to \mathcal{A}$ is replaced by any split fibration over $\mathcal{A}$. This new setting allows us to transfer Yoneda's theory of extensions to the non-additive analog given by crossed extensions for the cases of groups and other algebraic structures.