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\begin{document}

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\section*{Frobenius reciprocity}

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

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

[[!include category theory - contents]]

\hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{InRepresentationTheory}{In representation theory}\dotfill \pageref*{InRepresentationTheory} \linebreak
\noindent\hyperlink{InCategoryTheory}{In category theory}\dotfill \pageref*{InCategoryTheory} \linebreak
\noindent\hyperlink{InWirthmuellerContexts}{In Wirthm\"u{}ller contexts of six-operations yoga}\dotfill \pageref*{InWirthmuellerContexts} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_frobenius_laws_in_frobenius_algebras}{Relation to Frobenius laws (in Frobenius algebras)}\dotfill \pageref*{relation_to_frobenius_laws_in_frobenius_algebras} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


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

The term \emph{Frobenius reciprocity} has a meaning

\begin{itemize}%
\item \hyperlink{InRepresentationTheory}{In representation theory}


\item \hyperlink{InCategoryTheory}{In category theory}.



\end{itemize}
(For different statements of a similar name see the disambiguation at \emph{[[Frobenius theorem]]}.)

\hypertarget{InRepresentationTheory}{}\subsubsection*{{In representation theory}}\label{InRepresentationTheory}

In [[representation theory]], \textbf{Frobenius reciprocity} (sometimes \emph{Frobenious}) is the statement that the [[induction functor]] for [[group representation|representations of groups]] (or in some other [[algebraic categories]]) is [[left adjoint]] to the [[restriction]] functor. Sometimes it is used for a [[decategorification|decategorified]] version of this statement as well, on [[characters]].

Specifically for $H \hookrightarrow G$ an [[subgroup]] inclusion, there is an [[adjunction]]

\begin{displaymath}
(Ind \dashv Red) \;\colon\;
  Rep_G
    \stackrel{\overset{Ind}{\leftarrow}}{\underset{Red}{\longrightarrow}} 
  Rep_H
\end{displaymath}
between the [[categories]] of $G$-[[representations]] and $H$-[[representations]], where for $\rho$ an $H$-representation, $Ind(\rho) \in Rep(G)$ is the [[induced representation]].

Sometimes also the \emph{[[projection formula]]}

\begin{displaymath}
Ind(Red(W) \otimes V) \simeq W \otimes Ind(V)
\end{displaymath}
is referred to as \emph{Frobenius reciprocity} in representation theory (e.g. \href{http://planetmath.org/frobeniusreciprocity}{here on PlanetMath}). See below the general discussion \hyperlink{InWirthmuellerContexts}{in Wirthm\"u{}ller contexts}.

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

In [[category theory]], Frobenius reciprocity is a condition on a pair of [[adjoint functors]] $f_! \dashv f^*$. If both categories are [[cartesian closed]], then the adjunction is said to satisfy \textbf{Frobenius reciprocity} if the right adjoint $f^* \colon Y \to X$ is a [[cartesian closed functor]]; that is, if the canonical map $f^*(B^A) \to f^*(B)^{f^*(A)}$ is an [[isomorphism]] for all objects $B,A$ of $Y$.

Each of the functors $-^A$, $-^{f^*A}$ and $f^*$ has a [[left adjoint]], so by the calculus of [[mates]], this condition is equivalent to asking that the canonical ``[[projection formula|projection]]'' morphism

\begin{displaymath}
f_!(C \times f^*A) \to (f_! C) \times A
\end{displaymath}
is an isomorphism for each $A$ in $Y$ and $C$ in $X$.

This clearly makes sense also if the categories are [[cartesian monoidal category|cartesian]] but not necessarily [[closed category|closed]], and is the usual formulation found in the literature. It is equivalent to saying that the adjunction is a [[Hopf adjunction]] relative to the cartesian monoidal structures.

This terminology is most commonly used in the following situations:

\begin{itemize}%
\item When $f^*$ and $f_!$ are the [[inverse image|inverse]] and [[direct image]] functors along a map $f$ in a [[hyperdoctrine]]. Here $S$ is a category and $P \colon S^{op} \to Cat$ is an $S$-[[indexed category]] such that each category $P X$ is [[cartesian closed]] and each functor $f^* = P f$ has a [[left adjoint]] $\exists_f$ ([[existential quantifier]], also written $f_!$). Then $P$ is said to satisfy Frobenius reciprocity, or the \textbf{Frobenius condition}, if each of the [[adjunctions]] $\exists_f\dashv f^*$ does. If the categories $P X$ are cartesian but not closed then it still makes sense to ask for Frobenius reciprocity in the second form above, and in that case its logical interpretation is that $\exists x . (\phi \wedge \psi)$ is equivalent to $(\exists x.\phi) \wedge \psi$ if $x$ is not free in $\psi$.


\item When $f^*$ is the [[inverse image]] part of a [[geometric morphism]] between [[(n,1)-topoi]] and $f_!$ is a [[left adjoint]] of it, if the [[adjunction]] $f_!\dashv f^*$ satisfies Frobenius reciprocity, then the geometric morphism is called [[locally n-connected (n+1,1)-topos|locally (n-1)-connected]]. In particular, if $n=0$ so that we have a [[continuous map]] of [[locales]], then a left adjoint $f_!$ satisfying Frobenius reciprocity makes it an [[open map]], and if $n=1$ so that we have 1-[[topoi]], then it is [[locally connected geometric morphism|locally connected]] (see also \emph{[[open geometric morphism]]}). This usage of ``Frobenius reciprocity'' is sometimes also extended to the dual situation of [[proper map]]s of locales and topoi.



\end{itemize}
\hypertarget{InWirthmuellerContexts}{}\subsubsection*{{In Wirthm\"u{}ller contexts of six-operations yoga}}\label{InWirthmuellerContexts}

Generally, an [[adjoint triple]] $(f_! \dashv f^\ast \dashv f_\ast)$ between [[symmetric monoidal category|symmetric]] [[closed monoidal categories]] is called a \emph{[[Wirthmüller context]]} (\hyperlink{May05}{May 05}) of \emph{[[six operations]]} yoga, if $f^\ast$ is a strong [[closed monoidal functor]].

\begin{prop}
\label{}\hypertarget{}{}
In a [[Wirthmüller context]], the projection formula/Frobenius reciprocity holds as a [[natural equivalence]]

\begin{displaymath}
\overline{\pi}
  \;\colon\;
  f_!(f^\ast(B) \otimes A) \stackrel{\simeq}{\longrightarrow} B \otimes f_! A
\end{displaymath}
\end{prop}
\begin{proof}
For all $A \in \mathcal{X}$ and $B,C \in \mathcal{Y}$ we have by the $(f_! \dashv f^\ast)$-[[adjunction]] and the tensor$\dashv$hom-adjunction a [[commuting diagram]] of the form

\begin{displaymath}
\itexarray{
    \mathcal{Y}(f_! ((f^\ast B) \otimes A),\, C)
     &
      \stackrel{
        \mathcal{Y}(\overline{\pi}(A,B), C)
      }{
        \longrightarrow
     }
     &
    \mathcal{Y}(B \otimes f_! A, \, C  )
    \\
    \downarrow^{\mathrlap{\simeq}}
     &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathcal{X}(A, [(f^\ast B), (f^\ast C)])
    &\stackrel{}{\longrightarrow}&
    \mathcal{X}(A, f^\ast [B,C])
  }
  \,.
\end{displaymath}
By naturality in $A$ and by the [[Yoneda lemma]] this shows that $\overline{\pi}$ is an equivalence precisey if $f^\ast$ is strong closed.

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

\hypertarget{relation_to_frobenius_laws_in_frobenius_algebras}{}\subsubsection*{{Relation to Frobenius laws (in Frobenius algebras)}}\label{relation_to_frobenius_laws_in_frobenius_algebras}

The name ``Frobenius'' is sometimes used to refer to other conditions on adjunctions, known as ``Frobenius laws''. The formal structure of the Frobenius law appears in the notion of [[Frobenius algebra]], in the axiom which relates multiplication to comultiplication, and recurs in another form isolated by Carboni and Walters in their studies of cartesian bicategories and bicategories of relations. Namely, if $\delta \colon 1 \to \otimes \Delta$ denotes the diagonal transformation on a cartesian bicategory (e.g., $Rel$), with right adjoint $\delta^\dagger$, then there is a canonical map

\begin{displaymath}
\delta \delta^\dagger \stackrel{\phi}{\to} (1 \otimes \delta^\dagger)(\delta \otimes 1)
\end{displaymath}
mated to the coassociativity isomorphism

\begin{displaymath}
(1 \otimes \delta)\delta \to (\delta \otimes 1)\delta
\end{displaymath}
and the \textbf{Frobenius law} here is the assumption that \emph{the 2-cell $\phi$ is an isomorphism}. (There are two Frobenius laws actually; the other is that a similar canonical map

\begin{displaymath}
\delta \delta^\dagger \stackrel{\phi'}{\to} (\delta^\dagger \otimes 1)(1 \otimes \delta),
\end{displaymath}
mated to the inverse coassociativity, is also an isomorphism. However, it may be shown that if one of the Frobenius laws holds, then so does the other; see the article [[bicategory of relations]].)

It is very easy to make a slip and call the Frobenius law ``Frobenius reciprocity'', perhaps all the more because there are close connections between the two. One example occurs in the context of bicategories of relations, as follows.

Given a locally posetal [[cartesian bicategory]] $B$ and any object $c$ of $B$, one may construct a hyperdoctrine of the form

\begin{displaymath}
\hom_B(i-, c)\colon Map(B)^{op} \to Semilat
\end{displaymath}
where $i: Map(B) \to B$ is the inclusion, and $Semilat$ is the 2-category of meet-semilattices. Here $r \in \hom(i b, c)$ is thought of as a relation from $b$ to $c$, and for a map $f: a \to b$, the relation $f^\ast r$ is the pulling back

\begin{displaymath}
f^\ast r \coloneqq (a \stackrel{f}{\to} b \stackrel{r}{\to} 1)
\end{displaymath}
along $f$, and one may show that $f^\ast-$ preserves finite local meets. Indeed, the pushforward or quantification along $f$ takes $q: a \to 1$ to

\begin{displaymath}
\exists_f q \coloneqq (b \stackrel{f^\dagger}{\to} a \stackrel{q}{\to} 1)
\end{displaymath}
and $\exists_f \dashv f^\ast$ because $f^\dagger$ is \emph{right} adjoint to the map $f$. Because $f^\ast-$ is a right adjoint, it preserves local meets.

Frobenius reciprocity in this context, ordinarily written as

\begin{displaymath}
r \wedge \exists_f q = \exists_f (f^\ast r \wedge q),
\end{displaymath}
can then be restated for the hyperdoctrine $\hom_B(i-, c)$; it takes the form

\begin{displaymath}
r \wedge q f^\dagger = (r f \wedge q)f^\dagger
\end{displaymath}
for any map $f: a \to b$ and predicates $q \in \hom(a, c)$, $r \in \hom(b, c)$.

Meanwhile, recall that a \textbf{bicategory of relations} is a (locally posetal) cartesian bicategory in which the Frobenius laws hold.

\begin{prop}
\label{FLtoFR}\hypertarget{FLtoFR}{}
Frobenius reciprocity holds in each hyperdoctrine $\hom_B(i-, c)$ associated with a bicategory of relations.

\end{prop}
\begin{proof}
One first proves that a bicategory of relations is a compact closed bicategory in which each object $b$ is self-dual. The unit here is given by

\begin{displaymath}
\eta_b = (1 \stackrel{\varepsilon^\dagger}{\to} b \stackrel{\delta}{\to} b \otimes b)
\end{displaymath}
and the counit by

\begin{displaymath}
\theta_b = (b \otimes b \stackrel{\delta^\dagger}{\to} b \stackrel{\varepsilon}{\to} 1).
\end{displaymath}
Using this duality, each relation $r: b \to c$ has an opposite relation $r^{op} \colon c \to b$ given by

\begin{displaymath}
c \stackrel{c \otimes \eta_b}{\to} c \otimes b \otimes b \stackrel{1 \otimes r \otimes 1}{\to} c \otimes c \otimes b \stackrel{\theta_c \otimes b}{\to} b.
\end{displaymath}
It may further be shown that in a bicategory of relations, if $f: a \to b$ is a map, then its right adjoint $f^\dagger$ equals the opposite $f^{op}$. Therefore Frobenius reciprocity becomes the equation

\begin{displaymath}
r \wedge q f^{op} = (r f \wedge q)f^{op}
\end{displaymath}
but in fact this is just a special case of the more general modular law, which holds in a bicategory of relations as shown \href{http://rfcwalters.blogspot.com/2009/10/categorical-algebras-of-relations.html}{here} in a blog post by Walters. The modular law in turn depends crucially upon the Frobenius laws.

\end{proof}
Thus, in this instance, \emph{Frobenius reciprocity follows from the Frobenius laws}.

\begin{prop}
\label{FRtoFL}\hypertarget{FRtoFL}{}
In a locally posetal cartesian bicategory, the Frobenius laws follow from Frobenius reciprocity.

\end{prop}
\begin{proof}
Again, Frobenius reciprocity in a (locally posetal) cartesian bicategory $B$ means that for any map $f: a \to b$ and any two relations $q \in B(a, c)$, $r \in B(b, c)$, the canonical inclusion

\begin{displaymath}
(q \wedge r f)f^\dagger \leq q f^\dagger \wedge r
\end{displaymath}
is an equality. One (and therefore both) of the Frobenius laws will follow by taking the following choices for $f$, $q$, and $r$:

\begin{displaymath}
f = \delta_x, \qquad q = \varepsilon_{x}^{\dagger} \otimes 1_x, \qquad r = \varepsilon_x \otimes 1_x \otimes \varepsilon_{x}^{\dagger}
\end{displaymath}
where $\delta_x: x \to x \otimes x$ is the diagonal map and $\varepsilon_x: x \to 1$ is the projection. The remainder of the proof is best exhibited by a string diagram calculation, which is given here: [[Frobenius-reciprocity.pdf:file]].

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

\begin{example}
\label{}\hypertarget{}{}
Generally, for $\mathbf{H}$ a [[topos]] and $f \;\colon\; X \longrightarrow Y$ any [[morphism]], then the induced [[base change]] [[etale geometric morphism]]

\begin{displaymath}
(f_! \dashv f^\ast \dashv f_\ast)
  \;\colon\;
  \mathbf{H}_{/X}
  \to 
  \mathbf{H}_{/Y}
\end{displaymath}
has [[inverse image]] $f^\ast$ a [[cartesian closed functor]] and hence (see there) exhibits Frobenius reciprocity.

\end{example}
\hypertarget{References}{}\subsection*{{References}}\label{References}

The term `Frobenius reciprocity', in the context of hyperdoctrines, was introduced by Lawvere in

\begin{itemize}%
\item F.W. Lawvere, \emph{Equality in hyperdoctrines and comprehension schema as an adjoint functor}, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14.

\end{itemize}
Lawvere defines Frobenius reciprocity by either of the two equivalent conditions (see ``Definition-Theorem'' on p.6), and notes that ``one of these kinds of identities is formally similar to, and reduces in particular to, the Frobenius reciprocity formula for permutation representations of groups'' (p.1).

A textbook source is around lemma 1.5.8 in

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]}

\end{itemize}
General discussion in the context of projection formulas in [[monoidal categories]] (not necessarily cartesian) is in

\begin{itemize}%
\item H. Fausk, P. Hu, [[Peter May]], \emph{Isomorphisms between left and right adjoints}, Theory and Applications of Categories , Vol. 11, 2003, No. 4, pp 107-131. (\href{http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html}{TAC}, \href{http://www.math.uiuc.edu/K-theory/0573/FormalFeb16.pdf}{pdf})

\end{itemize}
Manifestations of the Frobenius reciprocity formula, \hyperlink{InCartegoryTheory}{in the sense of category theory}, recur throughout mathematics in various forms (push-pull formula, projection formula); see for example this Math Overflow post:

\begin{itemize}%
\item Andrea Ferretti, Ubiquity of the push-pull formula, MO Question 18799, March 20, 2010. \href{http://mathoverflow.net/questions/18799/ubiquity-of-the-push-pull-formula}{(link)}

\end{itemize}
Further MO discussion includes

\begin{itemize}%
\item \href{http://mathoverflow.net/questions/132272/wrong-way-frobenius-reciprocity-for-finite-groups-representations}{Wrong-way Frobenius reciprocity for finite groups representations}

\end{itemize}
[[!redirects Frobenius reciprocity]] [[!redirects Frobenious reciprocity]] [[!redirects Frobenius law]] [[!redirects Frobenius condition]] [[!redirects Frobenius axiom]]



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