Latent Fibrations: Fibrations for Categories of Partial Maps

TL;DR

This paper introduces latent fibrations, a new adaptation of fibrations for categories of partial maps, developing their basic theory and exploring examples relevant to computational and reverse differential programming contexts.

Contribution

It develops the foundational theory of latent fibrations and identifies key properties, including hyperconnected latent fibrations, relevant to computational and categorical applications.

Findings

Latent fibrations generalize standard fibrations to partial map categories.

Hyperconnected latent fibrations enable fibrational dual constructions.

Examples include partial versions of standard fibrations and applications in computational settings.

Abstract

Latent fibrations are an adaptation, appropriate for categories of partial maps (as presented by restriction categories), of the usual notion of fibration. The paper initiates the development of the basic theory of latent fibrations and explores some key examples. Latent fibrations cover a wide variety of examples, some of which are partial versions of standard fibrations, and some of which are particular to partial map categories (particularly those that arise in computational settings). Latent fibrations with various special properties are identified: hyperconnected latent fibrations, in particular, are shown to support the construction of a fibrational dual; this is important to reverse differential programming and, more generally, in the theory of lenses.