Partiality in category theory
Everything starts with \(\mathbf{Par}\), the category of sets and partial functions , or the isomorphic category \(\mathbf{Set}_*\) of pointed sets and basepoint-preserving functions. As a subcategory of \(\mathbf{Rel}\), the category \(\mathbf{Par}\) is actually a locally posetal 2-category, and this structures transfer to \(\mathbf{Set}_*\). The category of pointed sets is also complete and cocomplete, because the category of sets is and because of a general fact about categories of pointed objects.
Many authors have proposed generalizations and abstractions of partial maps
- Bicategories of partial maps (Carboni, 1987)
- Same idea as Carboni and Walters’ bicategory of relations
- p-categories (Robinson & Rosolini, 1988)
- Partial cartesian categories (Curien & Obtułowicz, 1989)
- Restriction categories , by Cockett and Lack
and of pointed sets
- Categories of pointed objects
- More generally, pointed categories (categories with a zero object)
- Categories with nulls, by Grandis
- Generalizes pointed categories by having a null ideal
Literature
Partial morphisms
- Carboni, 1987: Bicategories of partial maps (pdf)
- Robinson & Rosolini, 1988: Categories of partial maps (doi, pdf)
- Sec. 3 compares p-categories with other approaches, including partial cartesian categories and bicategories of partial maps
- Curien & Obtułowicz, 1989: Partiality, cartesian closedness, and toposes (doi)
- Jay, 1991: Partial functions, ordered categories, limits and cartesian closure
(doi)
- Short version of: Jay, 1990, tech report: Extending properties to categories of partial maps (pdf)
- Unlike approaches above, does not assume existence of cartesian products
- A nice approach for those who, like me, prefer to axiomatize the morphism order through locally preordered 2-categories, rather than constructing categories of partial maps via spans in existing categories
Restriction categories
Robin Cockett and coauthors have written a long series of papers on restriction categories.
- Cockett & Lack:
- Cockett & Manes, 2009: Boolean and classical restriction categories (doi)
- Cockett, Cruttwell, Gallagher, 2011: Differential restriction categories (pdf)
- Cockett, Xiuzhan Guo and Pieter Hofstra, 2012:
- Cockett & Garner, 2014: Restriction categories as enriched categories (doi, arxiv)
- Capobianco, 2017: Notes on restriction categories (pdf)
- Notes based on lectures by Cockett
Pointed structures
Categories with nulls
Under the name semiexact categories, Marco Grandis has studied categories with an ideal of null morphisms.
- Grandis, 1992: On the categorical foundations of homological and homotopical algebra (pdf)
- Grandis, 2013: Homological algebra in strongly non-abelian settings, Sec. 1.3: The main definitions