For the definition of 2-category, again see Wikipedia or nlab, or the classic Categories for the Working Mathematician, 2nd Edition.
Here we merely demonstrate some notation and recall a few related definitions.
Due to the importance of the subject, we present some of the diagrams in two different categories:
conventionally in a 2-category,
and in a double category where, for each 2-cell, the vertical arrows are identities (of course these two are isomorphic).
The most general adjunction is depicted as follows,
using fairly standard notation for the left and right adjoint 1-cells and the unit and counit 2-cells,
but unusual notation for the 0-cells, which are commonly called $\mathcal A, \mathcal B$ or some such.
First we give the definitions in a double category as above,
and consider 0-cells called $\Leftcat = \calL = \catB$, $\Rightcat = \calR = \catA$,
left and right adjoint 1-cells $\leftadj{L=F}$, $\rightadj{R=U}$, and
unit and counit 2-cells $\leftadj\eta$, $\rightadj\epsilon$.
For the triangular equation involving $1_\functL$,
start with the $\Leftcat$ at the left of the top row
and compose the top ($\eta$) 2-cell (the unit)
with the lower right counit ($\epsilon$) 2-cell.
For the triangular equation involving $1_\functR$,
start with the $\Rightcat$ at the left of the middle row
and compose the top ($\eta$) 2-cell (the unit again)
with the lower left counit ($\epsilon$) 2-cell. \[\begin{array}{ccccccc} && \Leftcat && \leftcat{\xrightarrow{1_\Leftcat}} && \Leftcat && \\ &&\leftcat\Vert && \leftadj{\Downarrow\rlap\eta} && \leftcat{\Vert} && \\ \Rightcat & \rightadj{\xrightarrow{\functR}} & \Leftcat & \leftadj{\xrightarrow[\functL]{}} & \Rightcat & \rightadj{\xrightarrow[\functR]{}} & \Leftcat & \leftadj{\xrightarrow{\functL}} & \Rightcat \\ \rightcat\Vert && \rightadj{\Downarrow\rlap\epsilon} && \rightcat\Vert && \rightadj{\Downarrow\rlap\epsilon} && \rightcat\Vert \\ \Rightcat && \rightcat{\xrightarrow[1_\Rightcat]{}} && \Rightcat && \rightcat{\xrightarrow[1_\Rightcat]{}} && \Rightcat \\ \end{array}\]
Now for a presentation in a 2-category.
For variety we use a slightly different notation for the 0-cells: $\Leftcat = \calL$, $\Rightcat = \calR$.
\[\begin{array}{} \source\calL && \source\longrightarrow && \source\calL && \source\longrightarrow && \source\calL \\ & \leftadj{ \llap \functL \searrow } & \leftadj{ \big\Downarrow \rlap\eta } & \rightadj{ \nearrow \mkern{-24mu} \functR } & \rightadj{ \big\Downarrow \rlap\epsilon } & \leftadj{ \searrow \mkern{-20mu} \functL } & \leftadj{ \big\Downarrow \rlap\eta } & \rightadj{ \nearrow \mkern{-24mu} \functR } & \rightadj{ \big\Downarrow \rlap\epsilon } & \leftadj{ \searrow \rlap \functL } \\ && \rightadj{ \target\calR } && \rightadj{ \target\longrightarrow } && \target\calR && \target\longrightarrow && \target\calR \\ \\ \mkern{-10mu} \rlap{\text{while the triangular equations are:}} \\ \\ & \leftadj \functL & \leftadj = &\leftcat{1_\calL} \leftadj \functL && \rightadj \functR \leftcat{1_\calL} & \rightadj = & \rightadj \functR \\ &&& \leftadj{ \llap\eta \Big\Downarrow \rlap\functL } && \llap{\rightadj\functR} \leftadj { \Big\Downarrow \rlap\eta } \\ \llap{\text{in 1-D} \mkern50mu} {} & \leftadj{ \llap{1_\functL} \Bigg\Downarrow } & \leftadj{ \xequiv[(\text{left }\bigtriangleup)]{\hom \functL {[\calL,\calR]} \functL} } & \leftadj \functL \rightadj \functR \leftadj \functL && \rightadj \functR \leftadj \functL \rightadj \functR & \rightadj{ \xequiv[(\text{right }\bigtriangleup)]{\hom \functR {[\calR,\calL]} \functR} } & \rightadj{ \Bigg\Downarrow \rlap{1_\functR} }\\ &&& \llap{\leftadj\functL} \rightadj{\Big\Downarrow \rlap\epsilon} && \rightadj{ \llap\epsilon \Big\Downarrow \rlap\functR } \\ & \leftadj \functL & \leftadj = &\leftadj \functL \rightcat{1_{\calR}} && \rightcat{1_{\calR}} \rightadj \functR & \rightadj = & \rightadj \functR \\ \\ \\ \llap{\text{in 0-D} \mkern50mu} {} & \leftadj{1_\functL} \rlap{ {} \mathrel{\leftadj\equiv} (\leftadj\eta \ncomp0 \leftadj \functL) \ncomp1 (\leftadj \functL \ncomp0 \rightadj\epsilon) } &&&&&& \llap{ ( \rightadj \functR \ncomp0 \leftadj\eta ) \ncomp1 ( \rightadj\epsilon \ncomp0 \rightadj \functR ) \mathrel{\rightadj\equiv} {} } \rightadj{1_\functR} \\ \end{array}\]
Work in progress:
\[\begin{array}{} \catI && \leftcat{ \xrightarrow[]{\textstyle \mkern{12mu} \objb \mkern{12mu}} } && \leftcat\catB && \\ & \rightcat { \llap{\objap \searrow \buildrel \alpha \over \Leftarrow \leftcat\objb\leftadj\functF \mkern{-20mu} } \searrow } & \leftadj{\Downarrow \rlap { \hom {\leftcat\objb} {\leftadj\eta} {} } } & \rightadj{ \nearrow \rlap{\mkern-20mu\functU} } & \Uparrow \rlap{\hat{ \hom {\leftcat\objb} {\leftadj\eta} {} }} & \leftcat{ \searrow \rlap{ \hom \objb \catB {?'}} } \\ && \rightcat\catA && \rightcat{ \xrightarrow[\textstyle \hom {\leftcat\objb\leftadj\functF} {\rightcat\catA} {-'}]{} } && \Set\\ \end{array}\]
| $\bbox[3ex,border:4px groove black]{\begin{array}{} \catI & \xrightarrow{\textstyle 1} & \Set \\ \llap{\leftcat\objb\leftadj\functF} \rightcat{ \Bigg\downarrow } & \llap{\leftadj{ \lower14pt\hbox{$\llap{{ \hom {\leftcat\objb} {\leftadj\eta} {} }\mkern-5mu} \Downarrow$} } \mkern-12mu} \leftcat{\searrow \rlap{\mkern-20mu \objb \raise10pt\hbox{$\mkern-10mu \Downarrow \mkern-5mu 1_\objb$}} } & \leftcat{ \Bigg\uparrow \rlap{\mkern-20mu \hom \objb \catB {?'}} } \\ \rightcat\catA & \rightadj{ \xrightarrow[\textstyle \mkern10mu \functU \mkern10mu]{} } & \leftcat\catB \\ \end{array}}$ | $\xlongequal[\begin{array}{} b \\ \text{left} \\ \text{lift} \end{array}]{\text {YS2}}$ | $\bbox[3ex,border:4px groove black]{\begin{array}{} \catI & \xrightarrow{\textstyle 1} & \Set \\ \llap{\leftcat\objb\leftadj\functF} \rightcat{ \Bigg\downarrow } & \leftadj{ \Bigg\Downarrow \rlap{\mkern-26mu \name{ \hom {\leftcat\objb} {\leftadj\eta} {} }} } & \leftcat{ \Bigg\uparrow \rlap{\mkern-20mu \hom \objb \catB {?'}} } \\ \rightcat\catA & \rightadj{ \xrightarrow[\textstyle \mkern10mu \functU \mkern10mu]{} } & \leftcat\catB \\ \end{array}}$ | $\xlongequal[\begin{array}{} \hom {\leftcat\objb\leftadj\functF} {\rightcat\catA} {-'} \\ \text{left} \\ \text{extension}\end{array}]{\text {YS1}}$ | $\bbox[3ex,border:4px groove black]{\begin{array}{} \catI & \xrightarrow{\textstyle 1} & \Set \\ \llap{\leftcat\objb\leftadj\functF} \rightcat{ \Bigg\downarrow } & \llap{\leftadj{ \raise10pt\hbox{$\llap{{\rightcat 1}_{\leftcat\objb\leftadj\functF} \mkern-8mu} \Downarrow$} } \mkern-16mu} \leftcat{\nearrow \rlap{\mkern-36mu \lower3pt\hbox{$\hom {\leftcat\objb\leftadj\functF} {\rightcat\catA} {-'}$} \lower14pt\hbox{$\mkern-26mu \Downarrow \mkern-5mu \hom \objb {\leftadj{\hat\eta}} {}$}} } & \leftcat{ \Bigg\uparrow \rlap{\mkern-20mu \hom \objb \catB {?'}} } \\ \rightcat\catA & \rightadj{ \xrightarrow[\textstyle \mkern10mu \functU \mkern10mu]{} } & \leftcat\catB \\ \end{array}}$ |
| $1 \leftcat {{} \xrightarrow[]{1_\objb} \hom \objb \catB \objb} \leftadj{ {} \xrightarrow[]{\leftcat{\hom \objb \catB {\hom \objb {\leftadj\eta} {}}}} {} } \leftcat { \hom \objb \catB {\objb\leftadj\functF\rightadj\functU} }$ | $ 1 \leftadj { {} \xrightarrow[]{\name{\hom {\leftadj\objb} \eta {}}} {} } \leftcat { \hom \objb \catB {\objb\leftadj\functF\rightadj\functU} }$ | $1 \rightcat {{} \xrightarrow[]{1_{\leftcat\objb\leftadj\functF}} \hom {\leftcat\objb\leftadj\functF} \catA {\leftcat\objb\leftadj\functF}} \leftadj{ {} \xrightarrow[]{ \hom {\leftcat\objb} {\leftadj{\hat\eta}} {\leftcat\objb\leftadj\functF} } \leftcat { \hom \objb \catB {\objb\leftadj\functF\rightadj\functU} }}$ |
Here are some standard definitions using the concept of adjunction:
- isomorphism (in a 2-category)
- The unit and counit are both identities: $\leftadj\eta \mathrel{\leftadj=} \leftcat{1_{1_\Leftcat}}$ and $\rightadj\epsilon \mathrel{\rightadj=} \rightcat{1_{1_\Rightcat}}$.
- reflection
- The unit is an identity: $\leftadj\eta \mathrel{\leftadj=} \leftcat{1_{1_\Leftcat}}$.
- coreflection
- The counit is an identity: $\rightadj\epsilon \mathrel{\rightadj=} \rightcat{1_{1_\Rightcat}}$.
- adjoint equivalence
- The unit $\eta$ and the counit $\epsilon$ are both invertible 2-cells, i.e., are isomorphisms.
No comments:
Post a Comment