newcomm.tex

%newcomm.tex,  version 14/05/23

%all newcommands for notes and notesdev 

%% package names, categories etc., sans serif
\newcommand{\arXiv}    {{\sf arXiv}}
\newcommand{\Cat}      {{\sf Cat}}
\newcommand{\GAP}      {{\sf GAP}}
\newcommand{\GitHub}   {{\sf GitHub}}
%\newcommand{\Gpd}      {{\sf Groupoids}}
\newcommand{\Groupoid}{{\sf Groupoid}}
\newcommand{\Groupoids}{{\sf Groupoids}}
\newcommand{\SimpSet}  {{\sf SimpSet}}
\newcommand{\XMod}     {{\sf XMod}}
\newcommand{\PreXMod}  {{\sf PreXMod}}

%% names for categories
\newcommand{\catCat}    {{\bf Cat}}
\newcommand{\cattCat}   {{\bf 2-Cat}}
\newcommand{\catGp}     {{\bf Gp}}
\newcommand{\cattGp}    {{\bf 2-Gp}}
\newcommand{\catGpd}    {{\bf Gpd}}
\newcommand{\cattGpd}   {{\bf 2-Gpd}}
\newcommand{\catGpGpd}  {{\bf GpGpd}}
\newcommand{\catPreXMod}{{\bf PreXMod}}
\newcommand{\catSet}    {{\bf Set}}
\newcommand{\catXComp}  {{\bf XComp}}
\newcommand{\catXMod}   {{\bf XMod}}

%% math Roman
\newcommand{\ab}    {\mathrm{ab}\,}
\newcommand{\arr}   {\mathrm{arr}}
\newcommand{\Act}   {\mathrm{Act}}
\newcommand{\Arr}   {\mathrm{Arr}}
\newcommand{\Aut}   {\mathrm{Aut}}
\newcommand{\coker} {\mathrm{coker}\,}
\newcommand{\cosk}  {\mathrm{cosk}\,}
\newcommand{\Costar}{\mathrm{Costar}}
\newcommand{\Der}   {\mathrm{Der}}
\newcommand{\End}   {\mathrm{End}}
\newcommand{\even}  {\mathrm{even}\,}
\newcommand{\Hol}   {\mathrm{Hol}}
\newcommand{\grp}   {\mathrm{grp}}
\newcommand{\id}    {\mathrm{id}}
\newcommand{\ids}   {\mathrm{ids}}
\newcommand{\im}    {\mathrm{im}\,}
\newcommand{\inc}   {\mathrm{inc}}
\newcommand{\inn}   {\mathrm{inn}}
\newcommand{\Inn}   {\mathrm{Inn}}
\newcommand{\mon}   {\mathrm{mon}}
\newcommand{\Mor}   {\mathrm{Mor}}
\newcommand{\nat}   {\mathrm{nat}}
\newcommand{\Ner}   {\mathrm{Ner}}
\newcommand{\obj}   {\mathrm{obj}}
\newcommand{\Ob}    {\mathrm{Ob}}
\newcommand{\odd}   {\mathrm{odd}\,}
\newcommand{\Out}   {\mathrm{Out}}
\newcommand{\Sect}  {\mathrm{Sect}}
\newcommand{\Sing}  {\mathrm{Sing}} 
\newcommand{\sk}    {\mathrm{sk}\,}
\newcommand{\Star}  {\mathrm{Star}}
\newcommand{\Stab}  {\mathrm{Stab}}
\newcommand{\Symm}  {\mathrm{Symm}} 
\newcommand{\Top}   {\mathrm{Top}} 

%% brackets
\newcommand{\dlb} {\ [\hspace{-0.4em}[\ }
\newcommand{\drb} {\ ]\hspace{-0.4em}]\ }

%% blackboard font
\newcommand{\bbA} {\mathbb{A}}
\newcommand{\bbAut} {\mathbb{A}{\mathrm ut}}
\newcommand{\bbB} {\mathbb{B}}
\newcommand{\bbC} {\mathbb{C}}
\newcommand{\bbD} {\mathbb{D}}
\newcommand{\bbE} {\mathbb{E}}
\newcommand{\bbF} {\mathbb{F}}
\newcommand{\bbG} {\mathbb{G}}
\newcommand{\bbH} {\mathbb{H}}
\newcommand{\bbI} {\mathbb{I}}
\newcommand{\bbJ} {\mathbb{J}}
\newcommand{\bbN} {\mathbb{N}}
\newcommand{\bbO} {\mathbb{O}}
\newcommand{\bbQ} {\mathbb{Q}}
\newcommand{\bbR} {\mathbb{R}}
\newcommand{\bbS} {\mathbb{S}}
\newcommand{\bbT} {\mathbb{T}}
\newcommand{\bbZ} {\mathbb{Z}}

%% bold symbol for vectors
\newcommand{\bsyb}  {\boldsymbol{b}}
\newcommand{\bsyc}  {\boldsymbol{c}}
\newcommand{\bsye}  {\boldsymbol{e}}
\newcommand{\bsyg}  {\boldsymbol{g}}
\newcommand{\bsyx}  {\boldsymbol{x}}
\newcommand{\bsyy}  {\boldsymbol{y}}
\newcommand{\bsyz}  {\boldsymbol{z}}

%% calligraphic letters
\newcommand{\calA}{\mathcal{A}}
\newcommand{\calB}{\mathcal{B}}
\newcommand{\calC}{\mathcal{C}}
\newcommand{\calD}{\mathcal{D}}
\newcommand{\calE}{\mathcal{E}}
\newcommand{\calF}{\mathcal{F}}
\newcommand{\calG}{\mathcal{G}}
\newcommand{\calH}{\mathcal{H}}
\newcommand{\calI}{\mathcal{I}}
\newcommand{\calJ}{\mathcal{J}}
\newcommand{\calK}{\mathcal{K}}
\newcommand{\calL}{\mathcal{L}}
\newcommand{\calM}{\mathcal{M}}
\newcommand{\calN}{\mathcal{N}}
\newcommand{\calP}{\mathcal{P}}
\newcommand{\calQ}{\mathcal{Q}}
\newcommand{\calR}{\mathcal{R}}
\newcommand{\calS}{\mathcal{S}}
\newcommand{\calT}{\mathcal{T}}
\newcommand{\calU}{\mathcal{U}}
\newcommand{\calW}{\mathcal{W}}
\newcommand{\calX}{\mathcal{X}}
\newcommand{\calY}{\mathcal{Y}}
\newcommand{\calZ}{\mathcal{Z}}

%%% dotted and barred Greek letters
\newcommand{\da}  {\dot{\alpha}}
\newcommand{\dda} {\ddot{\alpha}}
\newcommand{\bara}{\bar{\alpha}}
\newcommand{\db}  {\dot{\beta}}
\newcommand{\ddb} {\ddot{\beta}}
\newcommand{\barb}{\bar{\beta}}
\newcommand{\dd}  {\dot{\delta}}
\newcommand{\ddd} {\ddot{\delta}}
\newcommand{\bard}{\bar{\delta}}
\newcommand{\dg}  {\dot{\gamma}}
\newcommand{\ddg} {\ddot{\gamma}}
\newcommand{\barg}{\bar{\gamma}}
\newcommand{\ddgo} {\ddot{\gamma}_1}
\newcommand{\ddgt} {\ddot{\gamma}_2}
\newcommand{\dgo} {\dot{\gamma}_1}
\newcommand{\dgt} {\dot{\gamma}_2}
\newcommand{\di}  {\dot{\iota}}
\newcommand{\ddi} {\ddot{\iota}}
\newcommand{\bari}{\bar{\iota}}
\newcommand{\dk}  {\dot{\kappa}}
\newcommand{\ddk} {\ddot{\kappa}}
\newcommand{\bark}{\bar{\kappa}}
\newcommand{\ddl} {\ddot{\lambda}}
\newcommand{\dl} {\dot{\lambda}}
\newcommand{\ddlo} {\ddot{\lambda}_1}
\newcommand{\ddlt} {\ddot{\lambda}_2}
\newcommand{\dlo} {\dot{\lambda}_1}
\newcommand{\dlt} {\dot{\lambda}_2}
\newcommand{\lot} {\lambda_{[2]}}
\newcommand{\barl}{\bar{\lambda}}
\newcommand{\dm}  {\dot{\mu}}
\newcommand{\ddm} {\ddot{\mu}}
\newcommand{\barm}{\bar{\mu}}
\newcommand{\dmo} {\dot{\mu}_1}
\newcommand{\ddmo}{\ddot{\mu}_1}
\newcommand{\dmt} {\dot{\mu}_2}
\newcommand{\ddmt}{\ddot{\mu}_2}
\newcommand{\dn}  {\dot{\nu}}
\newcommand{\ddn} {\ddot{\nu}}
\newcommand{\barn}{\bar{\nu}}
\newcommand{\ddph}{\ddot{\phi}}
\newcommand{\ddpho}{\ddot{\phi}_1}
\newcommand{\ddpht}{\ddot{\phi}_2}
\newcommand{\ddphod}{\ddot{\phi}_1{}^{\prime}}
\newcommand{\ddphtd}{\ddot{\phi}_2{}^{\prime}}
\newcommand{\dph} {\dot{\phi}}
\newcommand{\dpho} {\dot{\phi}_1}
\newcommand{\dpht} {\dot{\phi}_2}
\newcommand{\dphod}{\dot{\phi}_1{}^{\prime}}
\newcommand{\dphtd}{\dot{\phi}_2{}^{\prime}}
\newcommand{\phot} {\phi_{[2]}}
\newcommand{\ddps}{\ddot{\psi}}
\newcommand{\dps} {\dot{\psi}}
\newcommand{\ddch}{\ddot{\chi}}
\newcommand{\dch} {\dot{\chi}}
\newcommand{\barch}{\bar{\chi}}
\newcommand{\dds} {\ddot{\sigma}}
\newcommand{\ds} {\dot{\sigma}}
\newcommand{\ddr} {\ddot{\rho}}
\newcommand{\dr} {\dot{\rho}}
\newcommand{\ddro} {\ddot{\rho}_1}
\newcommand{\ddrt} {\ddot{\rho}_2}
\newcommand{\dro} {\dot{\rho}_1}
\newcommand{\drt} {\dot{\rho}_2}
\newcommand{\ddth} {\ddot{\theta}}
\newcommand{\dth} {\dot{\theta}}
\newcommand{\ddtho} {\ddot{\theta}_1}
\newcommand{\ddtht} {\ddot{\theta}_2}
\newcommand{\dtho} {\dot{\theta}_1}
\newcommand{\dtht} {\dot{\theta}_2}
\newcommand{\thot} {\theta_{[2]}}
\newcommand{\bare} {\bar{\eta}}
\newcommand{\dde} {\ddot{\eta}}
\newcommand{\de} {\dot{\eta}}
\newcommand{\ddx} {\ddot{\xi}}
\newcommand{\dx} {\dot{\xi}}
\newcommand{\ddxo} {\ddot{\xi}_1}
\newcommand{\ddxt} {\ddot{\xi}_2}
\newcommand{\dxo} {\dot{\xi}_1}
\newcommand{\dxt} {\dot{\xi}_2}
\newcommand{\xot} {\xi_{[2]}}
\newcommand{\ddz} {\ddot{\zeta}}
\newcommand{\dz} {\dot{\zeta}}
\newcommand{\ddzo} {\ddot{\zeta}_1}
\newcommand{\ddzt} {\ddot{\zeta}_2}
\newcommand{\dzo} {\dot{\zeta}_1}
\newcommand{\dzt} {\dot{\zeta}_2}
\newcommand{\zot} {\zeta_{[2]}}
\newcommand{\ddD}{\ddot{\Delta}}
\newcommand{\dD}{\dot{\Delta}}

%%% dotted partials
\newcommand{\dbdy}   {\dot{\partial}}
\newcommand{\dbdyo}  {\dot{\partial}_1}
\newcommand{\dbdyt}  {\dot{\partial}_2}
\newcommand{\ddbdy}  {\ddot{\partial}}
\newcommand{\ddbdyo} {\ddot{\partial}_1}
\newcommand{\ddbdyt} {\ddot{\partial}_2}
\newcommand{\bdyot}  {\partial_{[2]}}
\newcommand{\barbdy} {\bar{\partial}}
\newcommand{\barbdyo}{\bar{\partial}_1}
\newcommand{\barbdyt}{\bar{\partial}_2}

%%% dotted Roman letters
\newcommand{\dee}  {\dot{e}}
\newcommand{\deeo} {\dot{e}_1}
\newcommand{\deet} {\dot{e}_2}
\newcommand{\deeod}{\dot{e}_1{}^{\prime}}
\newcommand{\deetd}{\dot{e}_2{}^{\prime}}
\newcommand{\ddee} {\ddot{e}}
\newcommand{\ddeeo}{\ddot{e}_1}
\newcommand{\ddeet}{\ddot{e}_2}
\newcommand{\ddeeod}{\ddot{e}_1{}^{\prime}}
\newcommand{\ddeetd}{\ddot{e}_2{}^{\prime}}
\newcommand{\eeot} {e_{[2]}}
\newcommand{\bareeo}{\bar{e}_1}
\newcommand{\bareet}{\bar{e}_2}
\newcommand{\dhh}  {\dot{h}}
\newcommand{\dhho} {\dot{h}_1}
\newcommand{\dhht} {\dot{h}_2}
\newcommand{\dhhod}{\dot{h}_1{}^{\prime}}
\newcommand{\dhhtd}{\dot{h}_2{}^{\prime}}
\newcommand{\ddhh} {\ddot{h}}
\newcommand{\ddhho}{\ddot{h}_1}
\newcommand{\ddhht}{\ddot{h}_2}
\newcommand{\ddhhod}{\ddot{h}_1{}^{\prime}}
\newcommand{\ddhhtd}{\ddot{h}_2{}^{\prime}}
\newcommand{\hhot} {h_{[2]}}
\newcommand{\barhho}{\bar{h}_1}
\newcommand{\barhht}{\bar{h}_2}
\newcommand{\dii}  {\dot{i}}
\newcommand{\diio} {\dot{i}_1}
\newcommand{\diit} {\dot{i}_2}
\newcommand{\ddii} {\ddot{i}}
\newcommand{\ddiio}{\ddot{i}_1}
\newcommand{\ddiit}{\ddot{i}_2}
\newcommand{\iiot} {i_{[2]}}
\newcommand{\dtt}  {\dot{t}}
\newcommand{\dtto} {\dot{t}_1}
\newcommand{\dttt} {\dot{t}_2}
\newcommand{\dttod}{\dot{t}_1{}^{\prime}}
\newcommand{\dtttd}{\dot{t}_2{}^{\prime}}
\newcommand{\ddtt} {\ddot{t}}
\newcommand{\ddtto}{\ddot{t}_1}
\newcommand{\ddttt}{\ddot{t}_2}
\newcommand{\ddttod}{\ddot{t}_1{}^{\prime}}
\newcommand{\ddtttd}{\ddot{t}_2{}^{\prime}}
\newcommand{\ttot} {t_{[2]}}
\newcommand{\bartto}{\bar{t}_1}
\newcommand{\barttt}{\bar{t}_2}
\newcommand{\duuo} {\dot{u}_1}
\newcommand{\duut} {\dot{u}_2}
\newcommand{\dduu} {\ddot{u}}
\newcommand{\dduuo}{\ddot{u}_1}
\newcommand{\dduut}{\ddot{u}_2}
\newcommand{\uuot} {u_{[2]}}
\newcommand{\baruuo}{\bar{u}_1}
\newcommand{\baruut}{\bar{u}_2}

% dotted capital Roman letters
\newcommand{\dCCo} {\dot{C}_1}
\newcommand{\dCCt} {\dot{C}_2}
\newcommand{\ddCCo}{\ddot{C}_1}
\newcommand{\ddCCt}{\ddot{C}_2}
\newcommand{\CCot} {C_{[2]}}
\newcommand{\CCbt} {C_{\{2\}}}
\newcommand{\CCbo} {C_{\{1\}}}
\newcommand{\CCe} {C_{\emptyset}}
\newcommand{\dGGo} {\dot{G}_1}
\newcommand{\dGGt} {\dot{G}_2}
\newcommand{\ddGGo}{\ddot{G}_1}
\newcommand{\ddGGt}{\ddot{G}_2}
\newcommand{\barGGo}{\bar{\calG}_1}
\newcommand{\barGGt}{\bar{\calG}_2}
\newcommand{\GGot} {G_{[2]}}
\newcommand{\GGbt} {G_{\{2\}}}
\newcommand{\GGbo} {G_{\{1\}}}
\newcommand{\GGe} {G_{\emptyset}}
\newcommand{\dSSo} {\dot{S}_1}
\newcommand{\dSSt} {\dot{S}_2}
\newcommand{\ddSSo}{\ddot{S}_1}
\newcommand{\ddSSt}{\ddot{S}_2}
\newcommand{\SSot} {S_{[2]}}
\newcommand{\SSe} {S_{\emptyset}}

%% dotted calligraphic commands
\newcommand{\dcalC}   {\dot{\calC}}
\newcommand{\dcalCo}  {\dot{\calC}_1}
\newcommand{\dcalCt}  {\dot{\calC}_2}
\newcommand{\ddcalC}  {\ddot{\calC}}
\newcommand{\ddcalCo} {\ddot{\calC}_1}
\newcommand{\ddcalCt} {\ddot{\calC}_2}
\newcommand{\calCot}  {\calC_{[2]}}
\newcommand{\dcalD}   {\dot{\calD}}
\newcommand{\dcalDo}  {\dot{\calD}_1}
\newcommand{\dcalDt}  {\dot{\calD}_2}
\newcommand{\ddcalD}  {\ddot{\calD}}
\newcommand{\ddcalDo} {\ddot{\calD}_1}
\newcommand{\ddcalDt} {\ddot{\calD}_2}
\newcommand{\calDot}  {\calD_{[2]}}
\newcommand{\dcalE}   {\dot{\calE}}
\newcommand{\dcalEo}  {\dot{\calE}_1}
\newcommand{\dcalEt}  {\dot{\calE}_2}
\newcommand{\ddcalE}  {\ddot{\calE}}
\newcommand{\ddcalEo} {\ddot{\calE}_1}
\newcommand{\ddcalEt} {\ddot{\calE}_2}
\newcommand{\calEot}  {\calE_{[2]}}
\newcommand{\dcalG}   {\dot{\calG}}
\newcommand{\dcalGo}  {\dot{\calG}_1}
\newcommand{\dcalGt}  {\dot{\calG}_2}
\newcommand{\ddcalG}  {\ddot{\calG}}
\newcommand{\ddcalGd} {\ddot{\calG}{}^{\prime}}
\newcommand{\ddcalGo} {\ddot{\calG}_1}
\newcommand{\ddcalGt} {\ddot{\calG}_2}
\newcommand{\ddcalGod}{\ddot{\calG}_1{}^{\prime}}
\newcommand{\ddcalGtd}{\ddot{\calG}_2{}^{\prime}}
\newcommand{\calGot}  {\calG_{[2]}}
\newcommand{\dcalM}   {\dot{\calM}}
\newcommand{\dcalMo}  {\dot{\calM}_1}
\newcommand{\dcalMt}  {\dot{\calM}_2}
\newcommand{\ddcalM}  {\ddot{\calM}}
\newcommand{\ddcalMo} {\ddot{\calM}_1}
\newcommand{\ddcalMt} {\ddot{\calM}_2}
\newcommand{\calMot}  {\calM_{[2]}}
\newcommand{\dcalN}   {\dot{\calN}}
\newcommand{\ddcalN}  {\ddot{\calN}}
\newcommand{\dcalR}   {\dot{\calR}}
\newcommand{\dcalRo}  {\dot{\calR}_1}
\newcommand{\dcalRt}  {\dot{\calR}_2}
\newcommand{\ddcalR}  {\ddot{\calR}}
\newcommand{\ddcalRo} {\ddot{\calR}_1}
\newcommand{\ddcalRt} {\ddot{\calR}_2}
\newcommand{\calRot}  {\calR_{[2]}}
\newcommand{\dcalS}   {\dot{\calS}}
\newcommand{\dcalSo}  {\dot{\calS}_1}
\newcommand{\dcalSt}  {\dot{\calS}_2}
\newcommand{\ddcalS}  {\ddot{\calS}}
\newcommand{\ddcalSo} {\ddot{\calS}_1}
\newcommand{\ddcalSt} {\ddot{\calS}_2}
\newcommand{\calSot}  {\calS_{[2]}}

%crossed pairing commands
\newcommand{\ugxh}{G \times H}
\newcommand{\ugth}{G \otimes H}
\newcommand{\gth} {g \otimes h}
\newcommand{\ugtg}{G \otimes G}
\newcommand{\gtg} {g \otimes g}
\newcommand{\ctd} {c \otimes d}
\newcommand{\ugoh}{G \odot H}
\newcommand{\goh} {g \odot h}
\newcommand{\cod} {c \odot d}
\newcommand{\goho}{g_1 \otimes h_1}
\newcommand{\goht}{g_2 \otimes h_2}
\newcommand{\gbh} {g \boxtimes h}
\newcommand{\cbd} {c \boxtimes d}
\newcommand{\gbho}{g_1 \boxtimes h_1}
\newcommand{\gbht}{g_2 \boxtimes h_2}


\newcommand{\asto}    {\;\ast_1\;}
\newcommand{\astt}    {\;\ast_2\;}
\newcommand{\ddasto}  {~\ddot{\ast}_1~}
\newcommand{\ddastt}  {~\ddot{\ast}_2~}
\newcommand{\dasto}   {~\dot{\ast}_1~}
\newcommand{\dastt}   {~\dot{\ast}_2~}

\newcommand{\ot}      {\otimes}
\newcommand{\bt}      {\boxtimes}
\newcommand{\btt}     {\ \tilde{\boxtimes}\ }
\newcommand{\normeq}  {\trianglelefteq}
\newcommand{\sharpz}  {\,\sharp_0\,}
\newcommand{\sharpo}  {\,\sharp_1\,}

%% Gareth's macro for \hash
\def\hashatom{\hbox{{$||$}\hspace*{-0.687em}\raisebox{0.037em}{\rotatebox[origin=c]{90}{$||$}}}}
\def\hashatomscript{\hspace*{0.75mm}\scalebox{0.5}{\hashatom}}
\def\hashatomscriptscript{\hspace*{0.5mm}\scalebox{0.4}{\hashatom}}
\newcommand*{\hashop}{\mathrel{\mathchoice{\hashatom\displaystyle}
{\hashatom\textstyle}
{\hashatomscript\scriptstyle}
{\hashatomscriptscript\scriptscriptstyle}}}
\newcommand{\hash}{\mathop{\hashop}\nolimits}