nLab
ananatural transformation

Idea

Just as natural transformations go between functors, ananatural transformations go between anafunctors.

Given two functors interpreted as anafunctors, the natural transformations and ananatural transformations between them are the same, so the term ‘ananatural’ is overkill; one only needs it to emphasise the ana-context and otherwise can just say ‘natural’. That is, a ‘natural transformation’ between anafunctors unambigously means an ananatural transformation.

Definitions

Given two categories $C$ and $D$ and two anafunctors $F,G:C\to D$, let us interpret $F,G$ as spans $C\leftarrow \overline{F}\to D$ and $C\leftarrow \overline{G}\to D$ of strict functors (where each backwards-pointing arrow is strictly surjective and faithful; see the definition of anafunctor). Form the strict $2$-pullback $P≔\overline{F}{\times }_C\overline{G}$ and consider the strict functors $P\to \overline{F}\to D$ and $P\to \overline{G}\to D$. Then an ananatural transformation from $F$ to $G$ is simply a natural transformation between these two strict functors.

More explicitly, if $F,G$ are given by sets $\mid F\mid ,\mid G\mid$ of specifications and additional maps (see the other definition of anafunctor), then an ananatural transformation from $F$ to $G$ consists of a coherent family of morphisms of $D$ indexed by the elements of $\mid F\mid$ and $\mid G\mid$ with common values in $C$. That is:

• for each object $x$ of $C$, each $F$-specification $s$ over $x$, and each $G$-specification $t$ over $x$, we have a morphism

  $${\eta }_{s,t}(x):F_s(x)\to G_t(x)$$\eta_{s,t}(x)\colon F_s(x) \to G_t(x)

  in $D$;

• for each morphism $f:x\to y$ in $C$, each pair of $F$-specifications $s,t$ over $x,y$, and each pair of $G$-specifications $u,v$ over $x,y$, the diagram

  $$\begin{array}{ccc}{F}_s(x)& \stackrel{{\eta }_{s,u}(x)}{\to }& G_u(x)\\ F_{s,t}(f)\downarrow & & \downarrow F_{u,v}(f)\\ F_t(y)& \underset{{\eta }_{t,v}(y)}{\to }& G_v(y)\end{array}$$\array { F_s(x) & \overset{\eta_{s,u}(x)}\rightarrow & G_u(x) \\ F_{s,t}(f) \downarrow & & \downarrow F_{u,v}(f) \\ F_t(y) & \underset{\eta_{t,v}(y)}\rightarrow & G_v(y) }

  is a commutative square.

Of course, an ananatural isomorphism is an invertible ananatural transformation.

Composition

Just as natural transformations can be composed vertically to form the morphisms of a functor category, so ananatural transformations can be composed vertically to form an anafunctor category.

Just as natural transformations can also be whiskered by functors and composed horizontally to make a strict 2-category $\mathrm{Str}\mathrm{Cat}$ of (strict) categories, (strict) functors and natural transformations, so ananatural transformations can also be whiskered by anafunctors and composed horizontally to make a bicategory ${\mathrm{Cat}}_{\mathrm{ana}}$ of (strict) categories, anafunctors and (ana)natural transformations. Assuming the axiom of choice, ${\mathrm{Cat}}_{\mathrm{ana}}$ is equivalent to $\mathrm{Str}\mathrm{Cat}$; without choice (and internally), ${\mathrm{Cat}}_{\mathrm{ana}}$ has better properties than $\mathrm{Str}\mathrm{Cat}$ and we will usually identify the former with Cat.