2-dinatural transformation

2-dinatural transformations


A 2-dinatural transformation is a categorification of a dinatural transformation. It is more complicated since there are three kinds of opposite 2-category, and also because it could be lax, colax, or pseudo as well as strict.


Let CC and DD be 2-categories and let

F,G:C coop×C op×C co×CDF,G: C^{coop}\times C^{op}\times C^{co} \times C\to D

be 2-functors. A lax 2-dinatural transformation α:FG\alpha:F\to G consists of

  1. For each xCx\in C, a 1-morphism component

    α x:F(x,x,x,x)G(x,x,x,x) \alpha_x : F(x,x,x,x) \to G(x,x,x,x)
  2. For each 1-morphism f:xyf:x\to y in CC, a 2-morphism component from the composite

    F(y,y,x,x)F(f,f,1,1)F(x,x,x,x)α xG(x,x,x,x)G(1,1,f,f)G(x,x,y,y) F(y,y,x,x) \xrightarrow{F(f,f,1,1)} F(x,x,x,x) \xrightarrow{\alpha_x} G(x,x,x,x) \xrightarrow{G(1,1,f,f)} G(x,x,y,y)

    to the composite

    F(y,y,x,x)F(1,1,f,f)F(y,y,y,y)α yG(y,y,y,y)G(f,f,1,1)G(x,x,y,y). F(y,y,x,x) \xrightarrow{F(1,1,f,f)} F(y,y,y,y) \xrightarrow{\alpha_y} G(y,y,y,y) \xrightarrow{G(f,f,1,1)} G(x,x,y,y).

    (This is a generalization of the hexagon identity for an ordinary dinatural transformation.)

  3. For any 2-morphism μ:fg:xy\mu:f\to g:x\to y in CC, the two composites

    G(1,1,g,f)α xF(g,f,1,1)G(1,1,μ,1)*1*F(μ,1,1,1)G(1,1,f,f)α xF(f,f,1,1)α fG(f,f,1,1)α yF(1,1,f,f)G(1,μ,1,1)*1*F(1,1,1,μ)G(f,g,1,1)α yF(1,1,f,g) G(1,1,g,f)\circ \alpha_x \circ F(g,f,1,1) \xrightarrow{G(1,1,\mu,1) \ast 1 \ast F(\mu,1,1,1)} G(1,1,f,f) \circ \alpha_x \circ F(f,f,1,1) \xrightarrow{\alpha_f} G(f,f,1,1) \circ \alpha_y \circ F(1,1,f,f) \xrightarrow{G(1,\mu,1,1) \ast 1 \ast F(1,1,1,\mu)} G(f,g,1,1) \circ \alpha_y \circ F(1,1,f,g)


    G(1,1,g,f)α xF(g,f,1,1)G(1,1,1,μ)*1*F(1,μ,1,1)G(1,1,g,g)α xF(g,g,1,1)α gG(g,g,1,1)α yF(1,1,g,g)G(μ,1,1,1)*1*F(1,1,μ,1)G(f,g,1,1)α yF(1,1,f,g) G(1,1,g,f)\circ \alpha_x \circ F(g,f,1,1) \xrightarrow{G(1,1,1,\mu) \ast 1 \ast F(1,\mu,1,1)} G(1,1,g,g) \circ \alpha_x \circ F(g,g,1,1) \xrightarrow{\alpha_g} G(g,g,1,1) \circ \alpha_y \circ F(1,1,g,g) \xrightarrow{G(\mu,1,1,1) \ast 1 \ast F(1,1,\mu,1)} G(f,g,1,1) \circ \alpha_y \circ F(1,1,f,g)

    are equal.

  4. Each 2-morphism component α 1 x\alpha_{1_x} is the identity 1 α x1_{\alpha_x}.

  5. For any composable 1-morphisms xfygzx\xrightarrow{f} y \xrightarrow{g} z, the composite of α f\alpha_f and α g\alpha_g (together with two functoriality commutative squares for FF and GG) is equal to α gf\alpha_{g f}.


If one of FF and GG is constant and the other depends only on C op×CC^{op}\times C, we obtain a notion of 2-extranatural transformation.

If FF and GG each depend on only one factor in C coop×C op×C co×CC^{coop}\times C^{op}\times C^{co} \times C, we obtain a notion of strict 2-natural transformation from a 2-functor of any arbitrary variance to a 2-functor of any other arbitrary variance. For example, if F:C coDF:C^{co}\to D and G:C opDG:C^{op}\to D, then a lax 2-dinatural transformation α:FG\alpha:F\to G consists of an ob(C)\ob(\mathbf{C})-indexed family of 11-morphisms FXα XGXFX\stackrel{\alpha_X}{\to} GX in D\mathbf{D}, and for each two objects X,YX,Y of C\mathbf{C}, an ob[X,Y]\ob[X,Y]-indexed family of 22-morphisms α f\alpha_f, so that for every 22-morphism fγgf\stackrel{\gamma}{\Rightarrow} g, we have the commutative diagram of 22-morphisms in D\mathbf{D}:

FX Ff FY α X α f α Y FX GX Gf GY Ff FX id Gγ.(α YFf) FY α X α Y FY id (Ggα Y).Fγ GY FX Fg FY Gg α X α g α Y GX GX Gg GY \array{ &&FX&\stackrel{Ff}{\rightarrow}&FY\\ &&\alpha_X\downarrow&\stackrel{\alpha_f}{\Rightarrow}&\downarrow\alpha_Y&&FX\\ &&GX&\stackrel{Gf}{\leftarrow}&GY&&\downarrow Ff\\ FX&\stackrel{id}{\neArrow}&&&& \stackrel{G\gamma.(\alpha_Y\circ Ff)}{\seArrow}&FY\\ \alpha_X\downarrow&&&&&&\downarrow\alpha_Y\\ FY&\stackrel{id}{\seArrow}&&&&\stackrel{(Gg\circ\alpha_Y).F\gamma}{\neArrow}&GY\\ &&FX&\stackrel{Fg}{\rightarrow}&FY&&\downarrow Gg\\ &&\alpha_X\downarrow&\stackrel{\alpha_g}{\Rightarrow}&\downarrow\alpha_Y&&GX\\ &&GX&\stackrel{Gg}{\leftarrow}&GY }

where . is whiskering/horizontal composition. Furthermore, given composable 11-morphisms XfYhZX\stackrel{f}{\rightarrow}Y\stackrel{h}{\rightarrow} Z, the 22-morphisms α Xα fGfα YFf\alpha_X\stackrel{\alpha_f}{\Rightarrow}Gf\circ\alpha_Y\circ Ff and α Yα hGhα ZFh\alpha_Y\stackrel{\alpha_h}{\Rightarrow}Gh\circ\alpha_Z\circ Fh are related via the formula α hf=(Gf.α h.Ff)α f\alpha_{h\circ f}=(Gf.\alpha_h.Ff)\circ\alpha_f, which says that the pasting diagram of 22-morphisms:

FX Ff FY Fh FZ α X α f α Y α h α Z GX Gf GY Gh GZ \array{ FX&\stackrel{Ff}{\rightarrow}&FY&\stackrel{Fh}{\rightarrow}&FZ\\ \alpha_X\downarrow&\stackrel{\alpha_f}{\Rightarrow}&\downarrow\alpha_Y&\stackrel{\alpha_h}{\Rightarrow}&\downarrow\alpha_Z\\ GX&\stackrel{Gf}{\leftarrow}&GY&\stackrel{Gh}{\leftarrow}&GZ }

reduces to

FX FhFf FZ α X α hf α Z GX GhGf GZ \array{ FX&\stackrel{Fh\circ Ff}{\rightarrow}&FZ\\ \alpha_X\downarrow&\stackrel{\alpha_{h\circ f}}{\Rightarrow}&\downarrow\alpha_Z\\ GX&\stackrel{Gh\circ Gf}{\leftarrow}&GZ }

This sort of transformation appears in the category of V-enriched categories, which is a 22-category which comes with a unit enriched category \mathcal{I} and either a lax natural transformation [,] op[[,],V 0][\mathcal{I},-]^{op}\Rightarrow[[\mathcal{I},-],V_0] (in the case of 𝒱\mathcal{V} a monoidal structure on VV), or a lax natural transformation [,] op[,V e][\mathcal{I},-]^{op}\Rightarrow[-,V^e] (in the case of 𝒱\mathcal{V} a closed structure on [,V e]V 0[\mathcal{I},V^e]\cong V_0).

Created on June 17, 2016 at 15:18:00. See the history of this page for a list of all contributions to it.