From 3a030e87a694fa9ac91af99c4c450ac61d75014d Mon Sep 17 00:00:00 2001 From: Derived Cat Date: Sat, 27 Jan 2024 23:19:15 -0500 Subject: [PATCH] update Universal Morphism Equivalent Characterizations of Universal Morphism --- category_theory.tex | 181 +++++++++++++++++++++++++++++++++++++++----- 1 file changed, 161 insertions(+), 20 deletions(-) diff --git a/category_theory.tex b/category_theory.tex index a18671b..b3c6d4c 100644 --- a/category_theory.tex +++ b/category_theory.tex @@ -105,6 +105,20 @@ \section{Category} The functor ${}^{\mathrm{op}}$ is an involution, i.e. ${}^{\mathrm{op}}\circ {}^{\mathrm{op}}=\mathrm{id}_{\mathsf{CAT}}$. Hence ${}^{\mathrm{op}}$ is an automorphism of $\mathsf{CAT}$. } +\prop{}{ + $$ +(C \times D)^{\mathrm{op}} \cong C^{\mathrm{op}} \times D^{\mathrm{op}} \text { (see product category) } +$$ + +Opposite preserves functors: +$(\text { Funct }(C, D))^{\mathrm{op}} \cong \operatorname{Funct}\left(C^{\mathrm{op}}, D^{\mathrm{op}}\right)^{[2][3]}$ (see functor category, opposite functor) + +Opposite preserves slices: +$$ +(F \downarrow G)^{\mathrm{op}} \cong\left(G^{\mathrm{op}} \downarrow F^{\mathrm{op}}\right) \text { (see comma category) } +$$ +} + \dfn{$\mathrm{Hom}$ Functors}{ Let $\mathsf{C}$ be a locally small category. We can define a functor $\mathrm{Hom}_{\mathsf{C}}\left(-{,}-\right)$ as follows \[ @@ -220,7 +234,7 @@ \section{Category} \] } \dfn{Diagonal Functor}{ - Suppose $\mathsf{J}$, $\mathsf{C}$ are categories. The \textbf{diagonal functor} $\Delta:\mathsf{C}\to [\mathsf{J},\mathsf{C}]$ is defined by + Suppose $\mathsf{J}$, $\mathsf{C}$ are categories. The \textbf{diagonal functor} $\diagarrow:\mathsf{C}\to [\mathsf{J},\mathsf{C}]$ is defined by \[ \begin{tikzcd}[ampersand replacement=\&] \mathsf{C}\&[-25pt]\&[+10pt]\&[-30pt][\mathsf{J},\mathsf{C}]\&[-30pt]\&[-30pt] \&\&\&\\ [-15pt] @@ -1798,7 +1812,7 @@ \section{Representable Functor} \] -\prop[universal_element_characterization]{Equivalent Characterizations of Representable Functor}{ +\prop[representable_functor_by_universal_element]{Equivalent Characterizations of Representable Functor}{ Suppose $F:\mathsf{C}\to \mathsf{Set}$ is a functor. Then the following statements are equivalent: \begin{enumerate}[(i)] \item $F$ is representable by universal element $(A,u)$ @@ -1808,7 +1822,7 @@ \section{Representable Functor} \item $(A,\phi^u)$ is initial in the category $\left( \text{\hyperref[th:yoneda_embedding_functor]{$Y_{\mathsf{C}^{\mathrm{op}}}$}}\downarrow F\right)$, which corresponds to $ \begin{tikzcd}[ampersand replacement=\&] \mathsf{C}^{\mathrm{op}} \arrow[r, " Y_{\mathsf{C}^{\mathrm{op}}}"] \& \left[\mathsf{C},\mathsf{Set}\right] \& \mathsf{1} \arrow[l, "\Delta F"'] \end{tikzcd}$. - \item $\left(A,\Delta u:\{*\}\to F(A)\right)$ is a universal morphism from $\{*\}$ to $F$. + \item $\left(A,\Delta u:\{*\}\to F(A)\right)$ is a \hyperref[th:universal_morphism]{universal morphism} from $\{*\}$ to $F$. \item For any $(X,x)\in \mathrm{Ob}(\int_{\mathsf{C}}F)$, there is a unique morphism $(A,u)\to (X,x)$ in $\int_{\mathsf{C}}F$ (which is a morphism $f:A\to X$ in $\mathsf{C}$ such that $F(f)(u)=x$). \end{enumerate} Suppose $F:\mathsf{C}^{\mathrm{op}}\to \mathsf{Set}$ is a functor. Then the following statements are equivalent: @@ -1820,7 +1834,7 @@ \section{Representable Functor} \item $(A,\phi^u)$ is terminal in the category $\left( \text{\hyperref[th:yoneda_embedding_functor]{$Y_{\mathsf{C}}$}}\downarrow F\right)$, which corresponds to $ \begin{tikzcd}[ampersand replacement=\&] \mathsf{C} \arrow[r, " Y_{\mathsf{C}}"] \& \left[\mathsf{C}^{\mathrm{op}},\mathsf{Set}\right] \& \mathsf{1} \arrow[l, "\Delta F"'] \end{tikzcd}$. - \item $\left(A,\Delta u:\{*\}\to F(A)\right)$ is a universal morphism from $\{*\}$ to $F$. + \item $\left(A,\Delta u:\{*\}\to F(A)\right)$ is a \hyperref[th:universal_morphism]{universal morphism} from $\{*\}$ to $F$. \item For any $(X,x)\in \mathrm{Ob}\left(\int_{\mathsf{C}^{\mathrm{op}} }F\right)$, there is a unique morphism $(A,u)\to (X,x)$ in $\int_{\mathsf{C}^{\mathrm{op}} }F$ (which is a morphism $f: X\to A$ in $\mathsf{C}$ such that $F(f)(x)=u$). \end{enumerate} } @@ -1893,11 +1907,41 @@ \section{Representable Functor} p:\int_\mathsf{C}\Delta \{*\}&\longrightarrow \mathsf{C}\\ (C,*) &\longmapsto C \end{align*} - As established in \Cref{th:universal_element_characterization}, + As established in \Cref{th:representable_functor_by_universal_element}, $(A,*)\in\mathrm{Ob}(\mathsf{C})$ is initial in $\int_\mathsf{C}\Delta \{*\}$ if and only if $\Delta\{*\}$ is a representable functor with a universal element $(A,*)$, which proves the first statement. The second statement can be obtained by applying the first statement to $\mathsf{C}^{\mathrm{op}}$.\\ In addition, an alternative ad-hoc proof is conceivable. If $\Delta\{*\}$ is naturally isomorphic to $\mathrm{Hom}_\mathsf{C}\left(A,-\right)$ through $\theta:\mathrm{Hom}_\mathsf{C}\left(A,-\right)\xRightarrow{\sim} \Delta \{*\}$, we have no choice but to define $\theta_X(\mathrm{id}_X)=*$. Note that $\Delta \{*\}(A)=\{*\}$. Yoneda lemma also implies that $\theta$ must correspond to $*\in \Delta \{*\}(A)$ and accordingly $\theta$ is the unique natural isomorphism from $\mathrm{Hom}_\mathsf{C}\left(A,-\right)$ to $\Delta\{*\}$. } +\prop[universal_morphism_by_representability]{Equivalent Characterizations of Universal Morphism}{ + Let $F:\mathsf{C}\to \mathsf{D}$ be a functor and $A\in \mathrm{Ob}(\mathsf{D})$. Then the following statements are equivalent: + \begin{enumerate}[(i)] + \item $(A,u)$ is initial in the category $\left(X \downarrow F\right)$. + \item $\mathrm{Hom}_{\mathsf{D}}\left(X,F(-)\right)$ is representable by universal element $(A,u)$. + \item $(A,u)$ is initial in the category $\int_{\mathsf{C}} \mathrm{Hom}_{\mathsf{D}}\left(X,F(-)\right)=\left( \{*\}\downarrow \mathrm{Hom}_{\mathsf{D}}\left(X,F(-)\right)\right)$. + \end{enumerate} + Dually, the following statements are equivalent: + \begin{enumerate}[(i)] + \item $(A,u)$ is terminal in the category $\left(F \downarrow X\right)$. + \item $\mathrm{Hom}_{\mathsf{D}}\left(F(-),X\right)$ is representable by universal element $(A,u)$. + \item $(A,u)$ is initial in the category $\int_{\mathsf{C}^{\mathrm{op}}} \mathrm{Hom}_{\mathsf{D}}\left(F(-),X\right)=\left( \{*\}\downarrow \mathrm{Hom}_{\mathsf{D}}\left(F(-),X\right)\right)$. + \end{enumerate} +} +\pf{ + It suffices to show (i)$\iff$ (iii). + \[ + \begin{tikzcd}[ampersand replacement=\&] + X \arrow[r, "u"] \arrow[rd, "g"'] \&[+20pt] F(A) \arrow[d, "F\left(h\right)", dashed]\&[+50pt] \{*\} \arrow[r, "\Delta u"] \arrow[rd, "\Delta g"'] \&[+15pt] \mathrm{Hom}_{\mathsf{D}}\left(X,F(A)\right)\arrow[d, "F(h)_*", dashed] \\[+15pt]\& F(B)\&\&\mathrm{Hom}_{\mathsf{D}}\left(X,F(B)\right) + \end{tikzcd} + \] + \begin{align*} + (A,u)\text{ is initial in }\left(X \downarrow F\right)&\iff \forall(B,g)\in \mathrm{Ob}\left(X \downarrow F\right),\;\exists! h\in \mathrm{Hom}_{\mathsf{C}}(A,B),\;F(h)\circ u=g\\ + &\iff \forall B\in \mathrm{Ob}\left(\mathsf{C}\right),\;\forall g\in \mathrm{Hom}_{\mathsf{C}}(X,F(B)),\;\exists! h\in \mathrm{Hom}_{\mathsf{C}}(A,B),\;F(h)\circ u=g\\ + &\iff \forall (B,\Delta g)\in \mathrm{Ob}\left(\int_{\mathsf{C}} \mathrm{Hom}_{\mathsf{D}}\left(X,F(-)\right)\right),\;\exists! h\in \mathrm{Hom}_{\mathsf{C}}(A,B),\;F(h)_*\circ\Delta u=\Delta g\\ + &\iff (A,u)\text{ is initial in }\int_{\mathsf{C}} \mathrm{Hom}_{\mathsf{D}}\left(X,F(-)\right) + \end{align*} + The dual version is similar. + +} \section{Limit and Colimit} \dfn{Cone}{ @@ -2037,7 +2081,7 @@ \section{Limit and Colimit} \] } \pf{ - According to \Cref{th:universal_element_characterization}, $\mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(\Delta\left(-\right), F\right)$ is representable by the universal element $(A,u)$ if and only if $(A,u)$ is initial in $\int_{\mathsf{C}^{\mathrm{op}}}\mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(\Delta\left(-\right), F\right)$, which is equivalent to saying that for any $C\in\mathrm{Ob}\left(\mathsf{C}\right)$ and $(h_i)\in \mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(\Delta C, F\right)$, there is a unique morphism $f:C\to A$ in $\mathsf{C}$ such that $h_i=f^*(u_i)=u_i\circ f$ for each $i\in \mathrm{Ob}(\mathsf{J})$. This is exactly the universal property of limit. + According to \Cref{th:representable_functor_by_universal_element}, $\mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(\Delta\left(-\right), F\right)$ is representable by the universal element $(A,u)$ if and only if $(A,u)$ is initial in $\int_{\mathsf{C}^{\mathrm{op}}}\mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(\Delta\left(-\right), F\right)$, which is equivalent to saying that for any $C\in\mathrm{Ob}\left(\mathsf{C}\right)$ and $(h_i)\in \mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(\Delta C, F\right)$, there is a unique morphism $f:C\to A$ in $\mathsf{C}$ such that $h_i=f^*(u_i)=u_i\circ f$ for each $i\in \mathrm{Ob}(\mathsf{J})$. This is exactly the universal property of limit. } \dfn{Complete Category}{ @@ -2809,10 +2853,10 @@ \section{Adjoint Functor} } -\prop{Equivalent Definition of Adjoint Functor Using Representable Functor}{ +\prop[adjunction_by_representability]{Equivalent Definition of Adjoint Functor Using Representable Functor}{ \begin{enumerate}[(i)] - \item A functor $L:\mathsf{C}\to\mathsf{D}$ has right adjoint if and only for each $Y\in \mathrm{Ob}(\mathsf{D})$, $\mathrm{Hom}_{\mathsf{D}}\left(L(-),Y\right)$ is representable. - \item A functor $R:\mathsf{D}\to\mathsf{C}$ has left adjoint if and only for each $X\in \mathrm{Ob}(\mathsf{C})$, $\mathrm{Hom}_{\mathsf{C}}\left(X,R(-)\right)$ is representable. + \item A functor $L:\mathsf{C}\to\mathsf{D}$ has right adjoint if and only for each $Y\in \mathrm{Ob}(\mathsf{D})$, $\mathrm{Hom}_{\mathsf{D}}\left(L(-),Y\right)$ is representable. If so, the universal element of $\mathrm{Hom}_{\mathsf{D}}\left(L(-),Y\right)$ is $\left(R(Y), \varepsilon_Y\right)$. + \item A functor $R:\mathsf{D}\to\mathsf{C}$ has left adjoint if and only for each $X\in \mathrm{Ob}(\mathsf{C})$, $\mathrm{Hom}_{\mathsf{C}}\left(X,R(-)\right)$ is representable. If so, the universal element of $\mathrm{Hom}_{\mathsf{C}}\left(X,R(-)\right)$ is $\left(L(X), \eta_X\right)$. \end{enumerate} } \pf{ @@ -2821,13 +2865,24 @@ \section{Adjoint Functor} $$ \Phi_{\text{-},Y}:\mathrm{Hom}_{\mathsf{D}}\left(L(-),Y\right)\xRightarrow{\sim} \mathrm{Hom}_{\mathsf{D}}\left(-,R(Y)\right). $$ - Hence $\mathrm{Hom}_{\mathsf{D}}\left(L(-),Y\right)$ is representable. + where $\Phi_{\text{-},Y}$ is the horizontal composition + \[ + \begin{tikzcd}[ampersand replacement=\&] + \mathsf{C}^{\mathrm{op}}\times\mathsf{1}\arrow[r,"{\mathrm{id}\times\Delta Y}"]\&[+15pt]\mathsf{C}^{\mathrm{op}}\times\mathsf{D} \arrow[r, "\scalebox{1.2}{$\mathrm{Hom}_{\mathsf{D}}\left(L(-){,}-\right)$}"{name=A, above}, bend left] \arrow[r, "\scalebox{1.2}{$\mathrm{Hom}_{\mathsf{C}}\left(-{,}R(-)\right)$}"'{name=B, below}, bend right] \&[+60pt] \mathsf{Set} + \arrow[Rightarrow, shorten <=5.5pt, shorten >=5.5pt, from=A.south-|B, to=B, "\Phi", "\sim\hspace{1.5pt}"'] + \end{tikzcd} +\] + Since + \[ + \left(\Phi_{-,Y}^{-1}\right)_{R(Y)}\left(\mathrm{id}_{R(Y)}\right)=\Phi_{R(Y),Y}^{-1}\left(\mathrm{id}_{R(Y)}\right)=\varepsilon_Y, + \] + we see $\mathrm{Hom}_{\mathsf{D}}\left(L(-),Y\right)$ is representable by $\left(R(Y), \varepsilon_Y\right)$. Conversely, suppose for each $Y\in \mathrm{Ob}(\mathsf{D})$, $\mathrm{Hom}_{\mathsf{D}}\left(L(-),Y\right)$ is representable. Then for each $Y\in \mathrm{Ob}(\mathsf{D})$, there exist a natural isomorphism $$ \phi(Y):\mathrm{Hom}_{\mathsf{D}}\left(-,R_Y\right)\xRightarrow{\sim} \mathrm{Hom}_{\mathsf{D}}\left(L(-),Y\right) $$ - for some $R_Y\in \mathrm{Ob}(\mathsf{D})$. \Cref{th:universal_element_characterization} implies that $(R_Y,\phi(Y))$ is terminal in the category $\left(Y_{\mathsf{C}}\downarrow \mathrm{Hom}_{\mathsf{D}}\left(L(-),Y\right)\right)$, which corresponds to + for some $R_Y\in \mathrm{Ob}(\mathsf{D})$. \Cref{th:representable_functor_by_universal_element} implies that $(R_Y,\phi(Y))$ is terminal in the category $\left(Y_{\mathsf{C}}\downarrow \mathrm{Hom}_{\mathsf{D}}\left(L(-),Y\right)\right)$, which corresponds to $$ \begin{tikzcd}[ampersand replacement=\&] \mathsf{C} \arrow[r, " Y_{\mathsf{C}}"] \& \left[\mathsf{C}^{\mathrm{op}},\mathsf{Set}\right] \&[+45pt] \mathsf{1} \arrow[l, "{\Delta \mathrm{Hom}_{\mathsf{D}}\left(L(-),Y\right)}"'] @@ -2859,9 +2914,7 @@ \section{Adjoint Functor} Therefore, $(L,R,\Phi)$ is an adjoint pair of functors. } - - -\prop{Equivalent Definition of Adjoint Functor Using Universal Morphism}{ +\cor{Equivalent Definition of Adjoint Functor Using Universal Morphism}{ Given pair of functors $\begin{tikzcd}[ampersand replacement=\&] \mathsf{C} \arrow[r, "L", bend left] \& \mathsf{D} \arrow[l, "R", bend left] \end{tikzcd}$, then the following are equivalent @@ -2872,16 +2925,21 @@ \section{Adjoint Functor} \end{enumerate} } \pf{ - If $L \dashv R$, then for every $X\in\mathrm{Ob}(\mathsf{C})$, we can let $X\xrightarrow{\eta_X} R(L(X))$ be the unit of the adjunction. Then for any $X\xrightarrow{g} R(Y)$, we have the following commutative diagram + This is a direct consequence of \Cref{th:adjunction_by_representability}. According to \Cref{th:universal_morphism_by_representability}, for each $X\in\mathrm{Ob}(\mathsf{C})$, $\mathrm{Hom}_{\mathsf{C}}\left(X,R(-)\right)$ is representable by $\left(L(X),\eta_X\right)$ is equivalent to that $\left(L(X), X\xrightarrow{\eta_X} R(L(X))\right)$ is initial in $\left(X\downarrow R\right)$. + \[ \begin{tikzcd}[ampersand replacement=\&] - X \arrow[rd, "f"'] \arrow[r, "\eta_X"] \& R(L(X))\arrow[d, "{R\left(\Phi_{X,Y}^{-1}(g)\right)}"] \\ - \& R(Y) - \end{tikzcd} + X \arrow[r, "\eta_X"] \arrow[rd, "g"'] \&[+20pt] R(L(X)) \arrow[d, "R\left(g^{\triangleleft}\right)", dashed] \&[+50pt] \{*\} \arrow[r, "\Delta \eta_X"] \arrow[rd, "\Delta g"'] \&[+15pt] \mathrm{Hom}_{\mathsf{C}}\left(X,R(L(X))\right) \arrow[d, "R\left(g^{\triangleleft}\right)_*", dashed] \\[+20pt] + \& R(Y) \&\& \mathrm{Hom}_{\mathsf{C}}\left(X,R(Y)\right) + \end{tikzcd} \] - because + + Similarly, for each $Y\in\mathrm{Ob}(\mathsf{D})$, $\mathrm{Hom}_{\mathsf{D}}\left(L(-),Y\right)$ is representable by $\left(R(Y), \varepsilon_Y\right)$ is equivalent to that $\left(R(Y), \varepsilon_Y\right)$ is terminal in $\left(L\downarrow Y\right)$. \[ - R\left(\Phi_{X,Y}^{-1}(g)\right)\circ \eta_X=R\left(\eta_Y\circ L(g)\right)\circ R\left(\eta_X\right)=R\left(\eta_Y\circ L(g)\circ \eta_X\right) + \begin{tikzcd}[ampersand replacement=\&] + L(X) \arrow[d, "L\left(f^{\triangleright}\right)"', dashed] \arrow[rd, "f"] \&[+20pt] \&[+50pt] {\mathrm{Hom}_{\mathsf{D}}\left(L(X),Y\right)} \&[+15pt] \\[+20pt] + L(R(Y)) \arrow[r, "\eta_Y"'] \& Y \& {\mathrm{Hom}_{\mathsf{D}}\left(L(R(Y)),Y\right)} \arrow[u, "L\left(f^{\triangleright}\right)^*", dashed] \& \{*\} \arrow[lu, "\Delta f"'] \arrow[l, "\Delta\eta_Y"] + \end{tikzcd} \] } @@ -2943,7 +3001,30 @@ \section{Monoidal Category} \end{enumerate} } +\dfn{Braided Monoidal Category}{ + A \textbf{braided monoidal category} is a monoidal category $\mathsf{V}$ equipped with an isomorphism natural in $X,Y\in \mathrm{Ob}(\mathsf{V})$ + \[ + B_{X,Y} : X \otimes Y \to Y \otimes X + \] + called the \textbf{braiding} such that the following two conditions hold: + \begin{enumerate}[(i)] + \item The hexagon axiom: the following diagram commutes + \[ + \begin{tikzcd}[ampersand replacement=\&, column sep=small] + \&[-50pt] (X\otimes Y)\otimes Z\arrow[rr, "{B_{X\otimes Y,Z}}"]\&[+45pt] \&[-25pt] Z\otimes (X\otimes Y) \arrow[rd, "{a_{Z,X,Y}^{-1}}"] \&[-50pt] \\[+20pt] + X\otimes (Y\otimes Z)\arrow[ru, "{a_{X,Y,Z}^{-1}}"] \arrow[rd, "{\mathrm{id}_X\otimes B_{Y,Z}}"'] \& \& \& \& (Z\otimes X)\otimes Y \\[+35pt] + \& X\otimes (Z\otimes Y) \arrow[rr, "{a_{X,Z,Y}^{-1}}"'] \& \& (X\otimes Z)\otimes Y\arrow[ru, "{B_{X,Z}\otimes\mathrm{id}_Y}"'] \& + \end{tikzcd} + \] + \end{enumerate} +} +\dfn{Symmetric Monoidal Category}{ + A \textbf{symmetric monoidal category} is a braided monoidal category $\mathsf{V}$ satisfying the following condition: + \[ + B_{Y,X}\circ B_{X,Y}=\mathrm{id}_{X\otimes Y}. + \] +} \section{Enriched Category} \dfn{Enriched Category}{ Let $\mathsf{V}$ be a monoidal category. An \textbf{$\mathsf{V}$-enriched category} $\mathsf{C}$ consists of @@ -3026,6 +3107,66 @@ \section{2-Category} \item 2-morphism set: $\mathrm{Hom}_{\mathsf{Cat}}(F,G):=\mathrm{Hom}_{[\mathsf{C},\mathsf{D}]}\left(F,G\right)$ for any $\mathsf{C},\mathsf{D}\in \mathrm{Ob}(\mathsf{Cat})$ and $F,G\in [\mathsf{C},\mathsf{D}]$ \end{itemize} } +\section{Internalization} +Traditional Bourbaki-style mathematical structures are formulated within set theory, or put differently, within the ambient category $\mathsf{Set}$. The concept of \textbf{Internalization} entails reformulating these mathematical structures in a broader ambient category $\mathsf{C}$, which typically need some extra structures to express the correponding mathematics. The extra structure required on an ambient category $\mathsf{C}$ is referred to as a \textbf{doctrine} for internalization. + +\subsection{Monoid Object} +Monoids can be internalized in the doctrine of monoidal categories. + +\ex{Monoid Objects in Monoidal Categories}{ + Monoid objects internal to Cartesian monoidal categories + \begin{itemize} + \item $\mathsf{Set}$: traditional monoid. + \item $\mathsf{Cat}$: (small) strict monoidal category. + \item $\mathsf{Top}$: topological monoid. + \item $\mathsf{Mon}$: commutative monoid. + \end{itemize} + Monoid objects internal to general monoidal categories + \begin{itemize} + \item $(R\text{-}\mathsf{Mod},\otimes_{R}, R)$ for $R\in \mathrm{Ob}(\mathsf{CRing})$: associative $R$-algebra. + \item $(\mathsf{Ab},\otimes_{\mathbb{Z}}, \mathbb{Z})$: ring. + \item $\left(\mathsf{Ch}\left(R\text{-}\mathsf{Mod}\right),\otimes_{R}, (R)_{n\in\mathbb{Z}}\right)$ for $R\in \mathrm{Ob}(\mathsf{CRing})$: differential graded $R$-algebra. + \item $\left([\mathsf{C},\mathsf{C}], \circ, \mathrm{id}_{\mathsf{C}}\right)$ for $\mathsf{C}\in \mathrm{Ob}(\mathsf{Cat})$: monad on $\mathsf{C}$. + \end{itemize} +} +Commutative monoids can be internalized in the doctrine of symmetric monoidal categories. + + +\subsection{Internal Category} + +\dfn{Internal Category}{ + Let $\mathsf{C}$ be a category with pullbacks. A \textbf{category internal to $\mathsf{C}$} consists of + \begin{enumerate}[(i)] + \item Object of 0-morphisms: an object $C_0\in \mathrm{Ob}(\mathsf{C})$. + \item Object of 1-morphisms: an object $C_1\in \mathrm{Ob}(\mathsf{C})$. + \item Source and target morphisms: two morphisms $s,t:C_1\to C_0$. + \item Identity assignment: a morphism $e:C_0\to C_1$. + \item Composition: a morphism $c:C_1\times_{C_0}C_1\to C_1$. + \end{enumerate} + such that the following diagrams commute + \begin{enumerate}[(i)] + \item Source and target of identity morphisms + \[ + \begin{tikzcd}[ampersand replacement=\&] + \& C_0\arrow[rd, "\mathrm{id}_{C_0}"] \arrow[ld, "\mathrm{id}_{C_0}"']\arrow[d, "e"] \& \\ + C_0 \& C_1 \arrow[l, "s"] \arrow[r, "t"'] \& C_0 + \end{tikzcd} + \] + \item Source and target of composition + \[ + \begin{tikzcd}[ampersand replacement=\&] + C_1\arrow[d, "s"'] \&C_1\times_{C_0}C_1 \arrow[d, "c"]\arrow[l, "\pi_1"']\arrow[r, "\pi_2"] \& C_1 \arrow[d, "t"] \\ + C_0\&C_1 \arrow[l, "s"]\arrow[r, "t"'] \& C_0 + \end{tikzcd} + \] + \item Associativity of composition + \item Left and right unit laws + \end{enumerate} +} + +\dfn{Double Category}{ + A \textbf{double category} $\mathsf{D}$ is a category internal to $\mathsf{Cat}$. +} \section{Abelian Category} Some literature refers to $\mathsf{Ab}$-categories as preadditive categories. We will not use this term in this note.