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\begin{document}

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\section*{adjoint string}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{adjoint_strings}{}\section*{{Adjoint strings}}\label{adjoint_strings}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In [[category theory]], an \textbf{adjoint string of length $n$}, or an \textbf{adjoint $n$-tuple}, is a sequence of $(n-1)$ [[adjunctions]] between $n$ [[functors]] (or more generally [[morphisms]] in a [[2-category]]):

\begin{displaymath}
f_1 \dashv f_2 \dashv \cdots \dashv f_n
\end{displaymath}
\hypertarget{special_cases}{}\subsection*{{Special cases}}\label{special_cases}

\begin{itemize}%
\item An adjoint $2$-tuple is just an ordinary [[adjunction]].
\item An adjoint $3$-tuple is an [[adjoint triple]].
\item An adjoint $4$-tuple is an [[adjoint quadruple]].

\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{enumerate}%
\item There is an adjoint $5$-tuple between $[Set^{op}, Set]$ and $Set$. Indeed, given a [[locally small category]] $B$, and the [[Yoneda embedding]], $y: B \to [B^{op}, Set]$, then $y$ being the rightmost functor of an adjoint $5$-tuple entails that $B$ is equivalent to [[Set]]; see \hyperlink{RWSets}{Rosebrugh-Wood}.


\item For any category $C$, there is a functor $ids: C\to Ar(C)$ from $C$ to its [[arrow category]] that assigns the identity morphism of each object. This functor always has both a left and a right adjoint which assign the codomain and domain of an arrow respectively; thus we have an adjoint triple $cod \dashv ids \dashv dom$. If $C$ has an initial object $0$, then $cod$ has a further left adjoint $I$ assigning to each object $x$ the morphism $0\to x$; and dually if $C$ has a terminal object $1$ then $dom$ has a further right adjoint $T$ assigning to $x$ the morphism $x\to 1$. Thus if $C$ has an initial and terminal object, we have an adjoint $5$-tuple.


\item Continuing from the last example, if $C$ is moreover a [[pointed category]] with [[pullbacks]] and [[pushouts]], then $I$ has a further left adjoint that constructs the [[cokernel]] of a morphism $x\to y$, i.e. the pushout of $y \leftarrow x \to 0$; and $T$ has a further right adjoint that constructs the [[kernel]] of a morphism $x \to y$, namely the pullback of $x\to y \leftarrow 0$. Thus we have an adjoint $7$-tuple. In fact, the existence of such an adjoint $7$-tuple characterizes pointed categories among categories with finite limits and colimits.


\item The previous two examples apply also to [[derivators]], and the extension of the analogous adjoint $5$-tuple to a $7$-tuple again characterizes the [[pointed derivators]]. Moreover, the [[stable derivators]] are characterized by the extension of this $7$-tuple to a doubly-infinite adjoint string with period 6 (\hyperlink{GrothShul17}{GrothShul17}).


\item Let $[n]$ denote the [[totally ordered]] $(n+1)$-element set, regarded as a category. For each positive integer $n$, we have $n+1$ order-preserving injections from $[n-1]$ to $[n]$, and $n$ order-preserving surjections from $[n]$ to $[n-1]$. Regarded as functors, these injections and surjections interleave to form an adjoint chain of length $2n + 1$. These categories, functors, and adjunctions form the [[simplex category]] [[simplex category\#As2Categories|regarded as a locally posetal 2-category]]; see below.


\item Let $C$ be a category with a [[terminal object]] but no [[initial object]]. Then there are functors

\begin{displaymath}
\itexarray{
   \delta_i \colon [n+1,C] \to [n,C] & 0\leq i \leq n; 
   \\
   \sigma_i\colon [n,C] \to [n+1,C] &  0\leq i \leq n
 }
\end{displaymath}
such that

\begin{displaymath}
\delta_0 \dashv \sigma_0 \dashv \cdots \dashv \delta_n \dashv \sigma_n
\end{displaymath}
is a maximal string of adjoint functors (all but $\sigma_n$ are obtained by applying $[-, C]$ to the simplex category example, and $\sigma_n$ exploits the presence of the terminal object of $C$).


\item Generalizing the simplex category example: if $P$ is a [[lax idempotent monad]] with unit $u: 1 \to P$ and multiplication $m: P P \to P$ (so that $m \dashv u P$), then there is an adjoint string

\begin{displaymath}
P^{n-1} m \dashv P^{n-1} u P \dashv P^{n-2}m P \dashv \ldots \dashv m P^{n-1} \dashv u P^n
\end{displaymath}
of length $2 n + 1$, back and forth between $P^{n+1}$ and $P^n$. The example of $[n]$ and $[n+1]$ above is based on the fact that the [[simplex category]] $\Delta$, regarded as a locally posetal [[bicategory]], is the [[walking structure|walking]] lax idempotent monoid.


\item Given an [[ambidextrous adjunction]], $F \dashv G$ and $G \dashv F$, we of course get an infinite adjoint string

\begin{displaymath}
\ldots \dashv F \dashv G \dashv F \dashv \ldots
\end{displaymath}
of period 2.



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

\begin{itemize}%
\item [[Bob Rosebrugh]] and R. J. Wood, \emph{Distributive Adjoint Strings}, Theory and Applications of Categories, Vol. 1, 1995, No. 6, pp 119-145, \href{http://www.tac.mta.ca/tac/volumes/1995/n6/1-06abs.html}{TAC}


\item [[Bob Rosebrugh]] and R. J. Wood, \emph{An adjoint characterization of the category of sets}, Proc. Amer. Math. Soc. \textbf{122} (1994), 409-413, doi:\href{https://doi.org/10.1090/S0002-9939-1994-1216823-2}{10.1090/S0002-9939-1994-1216823-2}



\end{itemize}
\begin{itemize}%
\item [[Moritz Groth]], [[Mike Shulman]], \emph{Generalized stability for abstract homotopy theories}, \href{https://arxiv.org/abs/1704.08084}{arXiv:1704.08084}.

\end{itemize}
[[!redirects adjoint string]] [[!redirects adjoint strings]] [[!redirects adjoint n-tuple]] [[!redirects adjoint n-tuples]] [[!redirects distributive adjoint string]] [[!redirects distributive adjoint strings]]

[[!redirects adjoint quintuple]] [[!redirects adjoint quintuples]]



\end{document}