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

and of pointed sets

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:
    • 2002: Restriction categories I: categories of partial maps (doi)
    • 2003: Restriction categories II: partial map classification (doi)
    • 2007: Restriction categories III: colimits, partial limits, and extensivity (doi, arxiv)
      • Sec 4.2: Comparison with p-categories and other formalisms
  • Cockett & Manes, 2009: Boolean and classical restriction categories (doi)
  • Cockett, Cruttwell, Gallagher, 2011: Differential restriction categories (pdf)
  • Cockett, Xiuzhan Guo and Pieter Hofstra, 2012:
    • Range categories I: General theory (pdf)
    • Range categories II: Towards regularity (pdf)
  • 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