Anafunctor

An anafunctor^{[note 1]} is a notion introduced by Makkai (1996) for ordinary categories that is a generalization of functors.^[1] In category theory, some statements require the axiom of choice, but the axiom of choice can sometimes be avoided when using an anafunctor.^[2] 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.^[1][3]

Definition

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 ${\displaystyle |F|}$, in a notation ${\displaystyle F:X\xrightarrow {a} A}$, is given by the following conditions:^{[1][4][5][6][7]}

• ${\displaystyle F_{0}}$ is surjective on objects.

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

Set-theoretic definition

(5)

An anafunctor ${\displaystyle F:X\xrightarrow {a} A}$ following condition:^[2][8][9]

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

See also

• Profunctor

Notes

1. The etymology of anafunctor is an analogy of the biological terms anaphase/prophase.^[1]