Anafunctor

File:Anafunctor (span).svg

Let {{mvar|X}} and {{mvar|A}} be categories. An anafunctor {{mvar|F}} with domain (source) {{mvar|X}} and codomain (target) {{mvar|A}}, and between categories {{mvar|X}} and {{mvar|A}} is a category $|F|$, in a notation $F:X\; \backslash xrightarrow\{a\}\; A$, is given by the following conditions:{{harv|Roberts|2011}}{{harv|Makkai|1996|loc = §1.1. and 1.1*. Anafunctors }}{{harv|Palmgren|2008|loc=§2. Anafunctors}}{{harv|Schreiber|Waldorf|2007|loc=§7.4. Anafunctors}}{{harv|anafunctor in nlab|loc=§2. Definitions}}

• $F\_0$ is surjective on objects.

• Let pair $F\_0:|F|\; \backslash rightarrow\; X$ and $F\_1:|F|\; \backslash rightarrow\; A$ be functors, a span of ordinary functors ($X\; \backslash leftarrow\; |F|\; \backslash rightarrow\; A$), where $F\_0$ is fully faithful.

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An anafunctor $F:\; X\; \backslash xrightarrow\{a\}\; A$ following condition:{{R|Makkai1998}}{{harv|Makkai|1996|loc=§1.1. Anafunctor}}{{harv|anafunctor in nlab|loc=§2. Anafunctors (Explicit set-theoretic definition)}}

1. A set $|F|$ of specifications of $F$, with maps $\backslash sigma\; :\; |F|\; \backslash to\; \backslash mathrm\{Ob\}\; (X)$ (source), $\backslash tau\; :\; |F|\; \backslash to\; \backslash mathrm\{Ob\}\; (A)$ (target). $|F|$ is the set of specifications, $s\; \backslash in\; |F|$ specifies the value $\backslash tau\; (s)$ at the argument $\backslash sigma\; (s)$. For $X\; \backslash in\; \backslash mathrm\{Ob\}\; (X)$, we write $|F|\; \backslash ;\; X$ for the class $\backslash \{s\; \backslash in\; |F|\; :\; \backslash sigma\; (s)\; =\; X\backslash \}$ and $F\_\{s\}\; (X)$ for $\backslash tau\; (s)$ the notation $F\_\{s\}\; (X)$ presumes that $s\; \backslash in\; |F|\; \backslash ;\; X$.
2. For each $X,\; \backslash ;\; Y\; \backslash in\; \backslash mathrm\{Ob\}\; (X)$, $x\; \backslash in\; |F|\; \backslash ;\; X$, $y\; \backslash in\; |F|\; \backslash ;\; Y$ and $f\; :\; X\; \backslash to\; Y$ in the class of all arrows $\backslash mathrm\{Arr\; (X)\}$ an arrows $F\_\{x,y\}\; (f)\; :\; F\_\{x\}\; (X)\; \backslash to\; F\_\{y\}\; (Y)$ in $A$.
3. For every $X\; \backslash in\; \backslash mathrm\{Ob\}\; (X)$, such that $|F|\; \backslash ;\; X$ is inhabited (non-empty).
4. $F$ hold identity. For all $X\; \backslash in\; \backslash mathrm\{Ob\}\; (X)$ and $x\; \backslash in\; |F|\; \backslash ;\; X$, we have $F\_\{x,x\}\; (\backslash mathrm\{id\}\_x)\; =\; \backslash mathrm\{id\}\_\{F\_\{x\}X\}$
5. $F$ hold composition. Whenever $X,\; Y,\; Z\; \backslash in\; \backslash mathrm\{Ob\}\; (X)$, $x\; \backslash in\; |F|\; \backslash ;\; X$, $y\; \backslash in\; |F|\; \backslash ;\; Y$, $z\; \backslash in\; |F|\; \backslash ;\; Z,$ and $F\_\{x,z\}\; (gf)\; =\; F\_\{y,z\}\; (g)\; \backslash circ\; F\_\{x,y\}\; (f)$

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• Profunctor

{{Reflist|group=note}}

{{reflist}}

• {{cite journal |doi=10.1016/0022-4049(95)00029-1 |title=Avoiding the axiom of choice in general category theory |date=1996 |last1=Makkai |first1=M. |author-link1= Michael Makkai|journal=Journal of Pure and Applied Algebra |volume=108 |issue=2 |pages=109–173 }}
• {{cite book|last1=Makkai |first1=M. |chapter-url=https://projecteuclid.org/ebooks/lecture-notes-in-logic/Logic-Colloquium-95--Proceedings-of-the-Annual-European-Summer/chapter/Towards-a-Categorical-Foundation-of-Mathematics/lnl/1235415906?tab=ArticleLinkCited | title=Logic Colloquium '95: Proceedings of the Annual European Summer Meeting of the Association of Symbolic Logic, held in Haifa, Israel, August 9-18, 1995 | chapter=Towards a Categorical Foundation of Mathematics | date=1998 | volume=11 | pages=153–191 | publisher=Association for Symbolic Logic|zbl = 0896.03051 }}
• {{cite journal |url=https://eudml.org/doc/129995 |title=Locally cartesian closed categories without chosen constructions |journal=Theory and Applications of Categories |date=2008 |volume=20 |pages=5–17 |last1=Palmgren |first1=Erik }}
• {{cite journal |url=http://www.tac.mta.ca/tac/volumes/26/29/26-29.pdf |title=Internal categories, anafunctors and localisations |last1=Roberts |first1=David M. |date=2011 |arxiv=1101.2363|journal=Theory and Application of Categories }}
• {{cite journal|arxiv=0705.0452 |last1=Schreiber |first1=Urs |last2=Waldorf |first2=Konrad |title=Parallel Transport and Functors |date=2007|url=http://tcms.org.ge/Journals/JHRS/xvolumes/2009/n1a10/v4n1a10.pdf|journal=Journal of Homotopy and Related Structures }}

• {{cite journal |doi=10.1017/S0305004100038202 |title=Complete functors in homology I. Chain maps and endomorphisms |date=1964 |last1=Kelly |first1=G. M. |journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=60 |issue=4 |pages=721–735 |bibcode=1964PCPS...60..721K }} - Kelly had already noticed a notion that was essentially the same as anafunctor in this paper, but did not seem to develop the notion further.

• {{cite arXiv |eprint=math/0410328 |last1=Bartels |first1=Toby |title=Higher gauge theory I: 2-Bundles |date=2004 }}
• {{cite web|title=anafunctor |url=https://ncatlab.org/nlab/show/anafunctor |website=ncatlab.org|ref={{harvid|anafunctor in nlab}}}}

Category:Axiom of choice

category:Functors