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The term Frobenius reciprocity has a meaning
(For different statements of a similar name see the disambiguation at Frobenius theorem.)
In representation theory, Frobenius reciprocity (sometimes Frobenious) is the statement that the induction functor for representations of groups (or in some other algebraic categories) is left adjoint to the restriction functor. Sometimes it is used for a decategorified version of this statement as well, on characters.
Specifically for $H \hookrightarrow G$ an subgroup inclusion, there is an adjunction
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 projection formula
is referred to as Frobenius reciprocity in representation theory (e.g. here on PlanetMath). See below the general discussion in Wirthmüller contexts.
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 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” morphism
is an isomorphism for each $A$ in $Y$ and $C$ in $X$.
This clearly makes sense also if the categories are cartesian but not necessarily 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:
When $f^*$ and $f_!$ are the 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 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 $\psi$.
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-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 (see also open geometric morphism). This usage of “Frobenius reciprocity” is sometimes also extended to the dual situation of proper maps of locales and topoi.
Generally, an adjoint triple $(f_! \dashv f^\ast \dashv f_\ast)$ between symmetric closed monoidal categories is called a Wirthmüller context (May 05) of six operations yoga, if $f^\ast$ is a strong closed monoidal functor.
In a Wirthmüller context, the projection formula/Frobenius reciprocity holds as a natural equivalence
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
By naturality in $A$ and by the Yoneda lemma this shows that $\overline{\pi}$ is an equivalence precisey if $f^\ast$ is strong closed.
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
mated to the coassociativity isomorphism
and the Frobenius law here is the assumption that the 2-cell $\phi$ is an isomorphism. (There are two Frobenius laws actually; the other is that a similar canonical map
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
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
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
and $\exists_f \dashv f^\ast$ because $f^\dagger$ is 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
can then be restated for the hyperdoctrine $\hom_B(i-, c)$; it takes the form
for any map $f: a \to b$ and predicates $q \in \hom(a, c)$, $r \in \hom(b, c)$.
Meanwhile, recall that a bicategory of relations is a (locally posetal) cartesian bicategory in which the Frobenius laws hold.
Frobenius reciprocity holds in each hyperdoctrine $\hom_B(i-, c)$ associated with a bicategory of relations.
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
and the counit by
Using this duality, each relation $r: b \to c$ has an opposite relation $r^{op} \colon c \to b$ given by
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
but in fact this is just a special case of the more general modular law, which holds in a bicategory of relations as shown here in a blog post by Walters. The modular law in turn depends crucially upon the Frobenius laws.
Thus, in this instance, Frobenius reciprocity follows from the Frobenius laws.
In a locally posetal cartesian bicategory, the Frobenius laws follow from Frobenius reciprocity.
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
is an equality. One (and therefore both) of the Frobenius laws will follow by taking the following choices for $f$, $q$, and $r$:
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 implies the Frobenius law in a cartesian bicategory.
Generally, for $\mathbf{H}$ a topos and $f \;\colon\; X \longrightarrow Y$ any morphism, then the induced base change etale geometric morphism
has inverse image $f^\ast$ a cartesian closed functor and hence (see there) exhibits Frobenius reciprocity.
The term ‘Frobenius reciprocity’, in the context of hyperdoctrines, was introduced by Lawvere in
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
General discussion in the context of projection formulas in monoidal categories (not necessarily cartesian) is in
Manifestations of the Frobenius reciprocity formula, in the sense of category theory, recur throughout mathematics in various forms (push-pull formula, projection formula); see for example this Math Overflow post:
Further MO discussion includes