RCF1: Theories of PR Maps and Partial PR Maps

Covariant & Contravariant Homotopy Theories

Given a locally presentable category together with a suitable functorial cylinder object, we construct model structures which are sensitive to the `direction' of the cylinder. We show that the Covariant and Contravariant model structures on simplicial sets as well as the coCartesian and Cartesian model structures on marked simplicial sets are examples of our formalism. In this setting, notions of final and initial maps and smooth and proper maps arise very naturally and we will identify these maps in the examples.

On factorisation systems for Ord-enriched categories and categories of partial maps

In this work we discuss a new type of factorisation systems for \textbf{Ord}-enriched categories. We start by defining the new notion of lax weak orthogonality, which involves the existence of lax diagonal morphisms for lax squares. Using the usual theory of factorisation systems as a blueprint, we introduce a lax version of weak and functorial factorisation systems. We provide a characterisation of lax functorial weak factorisation systems, namely lax weak factorisation systems such that their factorisations are lax functorial. Then, we present a particular case of such lax functorial weak fa…

Latent Fibrations: Fibrations for Categories of Partial Maps

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 par…

Functorial Semantics for Partial Theories

We provide a Lawvere-style definition for partial theories, extending the classical notion of equational theory by allowing partially defined operations. As in the classical case, our definition is syntactic: we use an appropriate class of string diagrams as terms. This allows for equational reasoning about the class of models defined by a partial theory. We demonstrate the expressivity of such equational theories by considering a number of examples, including partial combinatory algebras and cartesian closed categories. Moreover, despite the increase in expressivity of the syntax we retain a …

Bilimits in categories of partial maps

The closure of chains of embedding-projection pairs (ep-pairs) under bilimits in some categories of predomains and domains is standard and well-known. For instance, Scott's $D_\infty$ construction is well-known to produce directed bilimits of ep-pairs in the category of directed-complete partial orders, and de Jong and Escardó have formalized this result in the constructive domain theory of a topos. The explicit construcition of bilimits for categories of predomains and partial maps is considerably murkier as far as constructivity is concerned; most expositions employ the constructive taboo th…

Quantale 值映射与偏映射

Quantale-valued maps and partial maps

本文研究交换含单位 quantale 上的 $\mathsf{Q}$-map 与偏 $\mathsf{Q}$-map，刻画其对称性及与普通集合映射的关系，并证明偏映射范畴在选择公理下具有 monadic 性。