Anafunctor

An anafunctor is a notion introduced by Makkai (1996) that is a generalization of functors. In category theory, some statements require the axiom of choice, but the axiom of choice can sometimes be avoided when using an anafunctor. For example, the statement "every fully faithful and essentially surjective functor is an equivalence of categories" is equivalent to the axiom of choice, but we can usually follow the same statement without the axiom of choice by using anafunctor instead of functor.

Span formulation of anafunctors

Anafunctor (span)

Let X and A be categories. An anafunctor F with domain (source) X and codomain (target) A, and between categories X and A is a category $|F|$, in a notation $F:X\xrightarrow {a} A$, is given by the following conditions:

• $F_{0}$ is surjective on objects.

• Let pair $F_{0}:|F|\rightarrow X$ and $F_{1}:|F|\rightarrow A$ be functors, a span of ordinary functors ($X\leftarrow |F|\rightarrow A$), where $F_{0}$ is fully faithful.

See also

• Profunctor

Bibliography

• Hermida, Claudio; Makkai, Michael; Power, John (2000). "On weak higher dimensional categories I: Part 1". Journal of Pure and Applied Algebra. 154 (1–3): 221–246. doi:10.1016/S0022-4049(99)00179-6.
• Kelly, G. M. (1964). "Complete functors in homology I. Chain maps and endomorphisms". Mathematical Proceedings of the Cambridge Philosophical Society. 60 (4): 721–735. Bibcode:1964PCPS...60..721K. doi:10.1017/S0305004100038202.
• Makkai, M. (1996). "Avoiding the axiom of choice in general category theory". Journal of Pure and Applied Algebra. 108 (2): 109–173. doi:10.1016/0022-4049(95)00029-1.
• Makkai, M. (1998). "Towards a Categorical Foundation of Mathematics". Logic Colloquium '95: Proceedings of the Annual European Summer Meeting of the Association of Symbolic Logic, held in Haifa, Israel, August 9-18, 1995. Vol. 11. Association for Symbolic Logic. pp. 153–191. Zbl 0896.03051.
• Palmgren, Erik (2008). "Locally cartesian closed categories without chosen constructions". Theory and Applications of Categories. 20: 5–17.
• Roberts, David M. (2011). "Internal categories, anafunctors and localisations" (PDF). Theory and Application of Categories. arXiv:1101.2363.
• Schreiber, Urs; Waldorf, Konrad (2007). "Parallel Transport and Functors" (PDF). Journal of Homotopy and Related Structures. arXiv:0705.0452.

External links

• "anafunctor". ncatlab.org.
• Bartels, Toby (2004). "Higher gauge theory I: 2-Bundles". arXiv:math/0410328.