Anafunctor

An anafunctor is a notion introduced by for ordinary categories 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.

Definition

Span formulation of anafunctors

Let and be categories. An anafunctor with domain ( source) and codomain (target) , and between categories and is a category $, F,$, in a notation $F:X \xrightarrow A$, 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.

Set-theoretic definition

An anafunctor $F: X \xrightarrow A$ following condition:

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

See also

*Profunctor

Notes

References

Bibliography

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Further reading

* - 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.

External links

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*{{cite web, title=anafunctor , url=https://ncatlab.org/nlab/show/anafunctor , website=ncatlab.org, ref={{harvid, anafunctor in nlab
Axiom of choice
category:Functors