  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.
