Fibered aspects of Yoneda's regular span

Abstract

In this paper we start by pointing out that Yoneda's notion of a regular span S : X → A × B can be interpreted as a special kind of morphism, that we call fiberwise opfibration, in the 2-category Fib(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 P r0 : A × B → A is replaced by any split fibration over 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.

Discrete and Conservative Factorizations in Fib(B)

“…We cannot repeat the above argument because internal opfibrations in Fib(B) are not opfibrations in Cat. Let us recall from [2] the following definition. From Theorem 2.8 in [2] it follows that every internal opfibration in Fib(B) is a fibrewise opfibration, while the latter is exactly a morphism in Fib(B) which is an internal opfibration in Cat/B (see Propositions 2.5 and 2.7 in [2]).…”

“…Let us recall from [2] the following definition. From Theorem 2.8 in [2] it follows that every internal opfibration in Fib(B) is a fibrewise opfibration, while the latter is exactly a morphism in Fib(B) which is an internal opfibration in Cat/B (see Propositions 2.5 and 2.7 in [2]). By Corollary 2.9 in [2], the two notions coincide in the discrete case.…”

“…The first aim of the present work is to show that the above mentioned result, obtained in [2] via an ad hoc construction, is actually an application of an orthogonal factorization system in Fib(B), whose right class is given by internal discrete opfibrations. This is explained in Theorem 4.10, which generalizes to Fib(B) the well known comprehensive factorization system of Cat [15].…”

“…The crucial point of the work [2] was to recover Yoneda's Classification Theorem in [17, §3.2] as a result of the factorization of a regular span S : X → B × A through an internal discrete opfibration, in the 2-category Fib(B) of fibrations over B. Yoneda's Theorem was the base to give a new interpretation of cohomology groups in the additive case. Theorem 3.2…”

“…in [2], which extends the above factorization to any fibrewise opfibration (see Definition 4.7) with codomain a split fibration, allows us to enlarge this point of view to non-additive cases, such as cohomology of groups and of associative algebras.…”