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\begin{document}

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\section*{internal diagram}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{internal_category_theory}{}\paragraph*{{Internal category theory}}\label{internal_category_theory}

[[!include internal infinity-categories contents]]

\hypertarget{internal_diagrams}{}\section*{{Internal diagrams}}\label{internal_diagrams}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_category_theory}{In category theory}\dotfill \pageref*{in_category_theory} \linebreak
\noindent\hyperlink{in_dependent_type_theory}{In dependent type theory}\dotfill \pageref*{in_dependent_type_theory} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{diagrams_in_an_indexed_category}{Diagrams in an indexed category}\dotfill \pageref*{diagrams_in_an_indexed_category} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Given a [[finitely complete category]] $E$, one can consider the [[bicategory]] $Cat(E)$ of [[internal categories]] in $E$, and thus [[internal functors]], which are the morphisms in $Cat(E)$. If $E = Set$ then one can consider not only functors among small categories but also functors of the type $F: C\to Set$ from a small category $C$ to a large category of sets. In that case one can describe $F$ as consisting of a $C_0$-indexed family of objects and an action of $C_1$ on the diagram.

Compare the ideas discussed on this page with those at [[internal profunctor]] and [[discrete fibration]]. All three notions intersect --- an internal diagram on $C$ is the same thing as an internal profunctor $C &#8696; 1$, which is the same thing as a discrete opfibration in $Cat(E)$. The three generalize the basic idea in different ways.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{in_category_theory}{}\subsubsection*{{In category theory}}\label{in_category_theory}

Given an [[internal category]] $C\in Cat(E)$, with the usual structure maps $s,t,i,c$, an \textbf{internal [[diagram]]} $F$ on $C$ (or, of type $C$) is given by

\begin{itemize}%
\item a [[morphism]] $d : F_0\to C_0$ in $E$ together with
\item a morphism $e : F_1= F_0\times_{C_0} C_1 \to F_0$

\end{itemize}
where $F_1$ is the [[pullback]]

\begin{displaymath}
\itexarray{
  F_0 \times_{C_0} C_1 & \to & F_0 \\
  \mathllap{s^* d} \downarrow & & \downarrow \mathrlap{d} \\
  C_1 & \underset{s}{\to} & C_0
}
\end{displaymath}
These data must satisfy the following conditions:

\begin{itemize}%
\item $e$ respects the source and target maps of $C$, in that $d \circ e = t \circ s^* d$. Equivalently, $e$ is a morphism from $s^* d$ to $t^* d$ in $E/C_1$.


\item $e$ is an \emph{action} in the sense that $e \circ (1 \times i) = 1$ and $e(e \times 1) = e(1 \times c)$.



\end{itemize}
It is clear how to define [[homomorphisms]] of internal diagrams: a morphism $F \to G$ is given by an $E/C_0$-morphism $F_0 \to G_0$ that commutes with the actions $e$. Internal diagrams on $C$ in $E$ form a category denoted by $E^C$.

An internal diagram on $C^{op}$ is sometimes called an \textbf{[[internal presheaf]]} on $C$.

\hypertarget{in_dependent_type_theory}{}\subsubsection*{{In dependent type theory}}\label{in_dependent_type_theory}

Using the [[dependent type theory|language of]] [[dependent type|dependent types]], the map $d: F_0 \to C_0$ can be seen as the [[categorical semantics|interpretation]] of a dependent type $(X:C_0) \,\vdash\, (F(X):Type)$. The action of $C_1$ on $F_0$ can equivalently be given by the interpretation of a term in [[context]]:

\begin{displaymath}
(X:C_0), (Y:C_0), (f:C_1(X,Y)), (a:F(X)) \;\vdash\; (p(X,Y,f,a) : F(Y)).
\end{displaymath}
Here we consider $C_1$ to depend on $C_0 \times C_0$ by the canonical morphism $C_1 \to C_0 \times C_0$ induced by $s$ and $t$, as in the [[type-theoretic definition of category]].

If the ambient category $E$ is a [[locally cartesian closed category]], so that its internal type theory has [[dependent product types]], then this can be interpreted instead as a closed term

\begin{displaymath}
p : \Pi_{X,Y:C_0} \Pi_{f:C_1(X,Y)} (F(X) \to F(Y)).
\end{displaymath}
The axioms then take a particularly familiar form, also to be interpreted in the [[internal language]] of $E$:

\begin{itemize}%
\item $(X:C_0), (a:F(X)) \;\vdash\; p(X,X,id_X,a) = a$
\item $(X,Y,Z:C_0), (f:C_1(X,Y)), (g:C_1(Y,Z)), (a:F(X)) \;\vdash\; p(X,Z,g \circ f,a) = p(Y,Z,g,p(X,Y,f,a))$

\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

From an internal diagram $(F,C,\lambda,e)$ one can equip $F =(F_0,F_1)$ with a structure of an internal category over $C$. In other words, there is a forgetful functor $E^C\to Cat(E)/C$ (where $Cat(E)/C$ is the corresponding [[slice category]]). This functor is fully faithful and its essential image consists precisely of all objects in $Cat(E)/C$ which are [[discrete opfibrations]]. Similarly, the objects of $E^{C^{op}}$ are the [[discrete fibrations]] in $Cat(E)/C$.

There is a composite forgetful functor $U \colon E^C \to Cat(E)/C \to E/C_0$. This functor $U$ is monadic --- its left adjoint takes $p \colon X \to C_0$ to $t \circ s^* p \colon X \times_{C_0} C_1 \to C_1 \to C_0$.

\hypertarget{diagrams_in_an_indexed_category}{}\subsection*{{Diagrams in an indexed category}}\label{diagrams_in_an_indexed_category}

An internal diagram as above may take values in any [[Grothendieck fibration]] over $E$. Given a fibration in the guise of an [[indexed category]] $F : E^{op} \to Cat$, a \textbf{$C$-diagram in $F$} is given by

\begin{itemize}%
\item an object $P \in F(C_0)$, together with
\item a morphism $\phi : s^* P \to t^* P$ in $F(C_1)$

\end{itemize}
satisfying `cocycle equations'

\begin{itemize}%
\item $i^*\phi = 1_P$
\item $c^*\phi = p_1^* \phi \circ p_2^* \phi$

\end{itemize}
modulo coherent isos, where the $p_i$ are the projections out of $C_2$.

By the [[Yoneda lemma for bicategories]], the object $P$ determines (up to canonical isomorphism) a pseudonatural $\alpha^0 : E(-,C_0) \to F_0$ in $[E^{op},Cat]$, where $E$ is considered as a locally discrete bicategory, and $F_0(X) = ob F X$ considered as a discrete category, such that $\alpha^0(f) \cong f^* P$. Similarly, $\phi$ determines $\alpha^1 : E(-,C_1) \to F_1 = arr \circ F$, and $\alpha^1(g) \cong g^* \phi$. It is not hard to check that the conditions above correspond to requiring that the $\alpha^i_X$ form a functor $E(X,C) \to F X$ for each $X$, and pseudonaturality then makes the $C$-diagram $(P,\phi)$ equivalent to an [[indexed functor]] $E(-,C) \to F$. The category of $C$-diagrams in $F$ is then simply the hom-category $[E^{op},Cat](E(-,C),F)$.

\hypertarget{examples}{}\subsubsection*{{Examples}}\label{examples}

\begin{itemize}%
\item An internal diagram on $C$ in the sense above is a $C$-diagram in the [[codomain fibration]] of $E$, that is the pseudofunctor $X \mapsto E/X$.


\item If $E$ is equipped with a [[coverage]] and $C$ is the [[Cech nerve]] associated to a cover $p : U \to X$ in $E$, then the category of $C$-diagrams in $F$ is the [[descent]] category $Des_p(F)$.



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[diagram]], [[commuting diagram]], [[free diagram]]


\item [[internal presheaf]], [[internal sheaf]]


\item [[internal (co-)limit]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[P. T. Johnstone]], \emph{Topos theory}, 1977, chapter 2


\item [[Elephant]], B2.3



\end{itemize}
[[!redirects internal diagrams]]



\end{document}