Actegories chapter

index.Rmd

@@ -11,11 +11,136 @@ github-repo: "jules-hedges/elements"
 favicon: "favicon.ico"
 ---
 
-\providecommand{\action}{\bullet}
 \newtheorem{theorem}{Theorem}
-\newtheorem{lemma}[theorem]{Lemma}
+<!--
+newtheorem{lemma}[theorem]{Lemma}
 \newtheorem{proposition}[theorem]{Proposition}
+-->
+
+\providecommand{\identity}{\mathrm{id}}
+\providecommand{\id}{\mathrm{id}}
+
+\providecommand{\Ob}{\operatorname{Ob}}
+\providecommand{\Hom}{\operatorname{Hom}}
+\providecommand{\End}{\operatorname{End}}
+\providecommand{\Aut}{\operatorname{Aut}}
+
+\providecommand{\Nat}{\operatorname{Nat}}
+
+\providecommand{\Lan}{\operatorname{Lan}}
+\providecommand{\Ran}{\operatorname{Ran}}
+
+\providecommand{\cat}[1]{\mathcal{#1}}
+\providecommand{\ncat}[1]{\mathbf{#1}}
+\providecommand{\twocat}[1]{\mathbb{#1}}
+
+\providecommand{\Cat}{\twocat{C}\ncat{at}}
+\providecommand{\CAT}{\twocat{C}\ncat{AT}}
+
+\providecommand{\Set}{\ncat{Set}}
+\providecommand{\FinSet}{\ncat{FinSet}}
+
+\providecommand{\Mon}{\ncat{Mon}}
+\providecommand{\Grp}{\ncat{Grp}}
+\providecommand{\Mod}{\ncat{Mod}}
+\providecommand{\Ab}{\ncat{Ab}}
+\reprovidecommand{\Vec}{\ncat{Vec}}
+
+\providecommand{\op}{\mathsf{op}}
+\providecommand{\co}{\mathsf{co}}
+\providecommand{\coop}{\mathsf{coop}}
+
+\providecommand{\iso}{\cong}
+\providecommand{\equi}{\simeq}
+
+\providecommand{\yo}{\operatorname{y}}
+
+\providecommand{\adj}{\dashv}
+
+\providecommand{\Para}{\ncat{Para}}
+\providecommand{\Copara}{\ncat{Copara}}
+\providecommand{\Optic}{\ncat{Optic}}
+\providecommand{\Lens}{\ncat{Lens}}
+
+\providecommand{\action}{\bullet}
+\providecommand{\actionn}{\circ}
+\providecommand{\ostar}{\circledast}
+
+\providecommand{\comp}{;}
+
+\providecommand{\opticto}{\rightleftarrows}
+
+\providecommand{\PsdMon}{\ncat{PsdMon}}
+\providecommand{\MonCat}{\twocat{M}\ncat{onCat}}
+\providecommand{\SymMonCat}{\twocat{S}\ncat{ymMonCat}}
+\providecommand{\BrMonCat}{\twocat{B}\ncat{rMonCat}}
+\providecommand{\Fib}{\twocat{F}\ncat{ib}}
+\providecommand{\OpFib}{\twocat{O}\ncat{pFib}}
+\providecommand{\deloop}{\twocat{B}}
+\providecommand{\Kl}{\ncat{Kl}}
+
+\providecommand{\dash}{\text{\textendash}}
+
+\providecommand{\Alg}[1]{{#1}\dash\twocat{A}\ncat{lg}}
+
+\providecommand{\lax}{\mathrm{lx}}
+\providecommand{\oplax}{\mathrm{ox}}
+\providecommand{\pseudo}{\mathrm{ps}}
+\providecommand{\strict}{\mathrm{s}}
+
+\providecommand{\VCat}[1]{{#1}\dash\Cat}
+
+\providecommand{\Actt}{\twocat{A}\ncat{ct}}
+\providecommand{\Act}[1]{{#1}\dash\Actt}
+\providecommand{\Biactt}{\twocat{B}\ncat{iact}}
+\providecommand{\Biact}[1]{{#1}\dash\Biactt}
+
+\providecommand{\rev}{\mathrm{rev}}
+
+\providecommand{\Bal}{\mathrm{Bal}}
+
+\providecommand{\equalto}{=\mathrel{\mkern-3mu}=}
+
+\providecommand{\Rho}{P}
+\providecommand{\Alpha}{A}
+
+\providecommand{\tensor}{\otimes}
+
+\providecommand{\Day}{\mathrm{Day}}
+
+\providecommand{\ot}{\leftarrow}
+\providecommand{\from}{\ot}
+
+\providecommand{\longto}{\longrightarrow}
+
+\providecommand{\mapsfrom}{\mathrel{\reflectbox{\ensuremath{\mapsto}}}}
+\providecommand{\longmapsfrom}{\mathrel{\reflectbox{\ensuremath{\longmapsto}}}}
+
+\providecommand{\into}{\hookrightarrow}
+\providecommand{\inot}{\hookleftarrow}
+\providecommand{\mono}{\rightarrowtail}
+
+\providecommand{\onto}{\twoheadrightarrow}
+\providecommand{\epi}{\twoheadrightarrow}
+
+\providecommand{\isoto}{\overset{\sim}\to}
+\providecommand{\isolongto}{\overset{\sim}\longto}
+
+\providecommand{\twoto}{\Rightarrow}
+\providecommand{\isotwoto}{\overset{\sim}\twoto}
+\providecommand{\longtwoto}{\Longrightarrow}
+\providecommand{\isolongtwoto}{\overset{\sim}\longtwoto}
+
+\providecommand{\threeto}{\Rrightarrow}
 
+\providecommand{\narrow}[2]{\overset{#1}{#2}}
+\providecommand{\nto}[1]{\narrow{#1}{\to}}
+\providecommand{\nfrom}[1]{\narrow{#1}{\from}}
+\providecommand{\nlongto}[1]{\xrightarrow{#1}}
+\providecommand{\ninto}[1]{\narrow{#1}{\into}}
+\providecommand{\nisoto}[1]{\narrow{#1}{\isolongto}}
+\providecommand{\nepi}[1]{\narrow{#1}{\epi}}
+\providecommand{\nmono}[1]{\narrow{#1}{\mono}}
 
 # Foreword {-}