nLab
ananatural transformation

Ananatural transformations

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 CC and DD and two anafunctors F,G:CDF, G\colon C \to D, let us interpret F,GF,G as spans CF¯DC \leftarrow \overline{F} \rightarrow D and CG¯DC \leftarrow \overline{G} \rightarrow D of strict functors (where each backwards-pointing arrow is strictly surjective and faithful; see the definition of anafunctor). Form the strict 22-pullback PF¯× CG¯P \coloneqq \overline{F} \times_C \overline{G} and consider the strict functors PF¯DP \to \overline{F} \to D and PG¯DP \to \overline{G} \to D. Then an ananatural transformation from FF to GG is simply a natural transformation between these two strict functors.

More explicitly, if F,GF,G are given by sets |F|,|G|{|F|}, {|G|} of specifications and additional maps (see the other definition of anafunctor), then an ananatural transformation from FF to GG consists of a coherent family of morphisms of DD indexed by the elements of |F||F| and |G||G| with common values in CC. That is:

  • for each object xx of CC, each FF-specification ss over xx, and each GG-specification tt over xx, we have a morphism

    η s,t(x):F s(x)G t(x) \eta_{s,t}(x)\colon F_s(x) \to G_t(x)

    in DD;

  • for each morphism f:xyf\colon x \to y in CC, each pair of FF-specifications s,ts,t over x,yx,y, and each pair of GG-specifications u,vu,v over x,yx,y, the diagram

    F s(x) η s,u(x) G u(x) F s,t(f) F u,v(f) F t(y) η t,v(y) G v(y) \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 StrCatStr 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 Cat anaCat_{ana} of (strict) categories, anafunctors and (ana)natural transformations. Assuming the axiom of choice, Cat anaCat_{ana} is equivalent to StrCatStr Cat; without choice (and internally), Cat anaCat_{ana} has better properties than StrCatStr Cat and we will usually identify the former with Cat.

Revised on November 1, 2011 00:36:11 by Toby Bartels (64.89.53.43)