nLab
Frobenius reciprocity

Definition

The term Frobenius reciprocity has a meaning

• In representation theory

• In category theory.

In representation theory

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.

In category theory

In category theory, Frobenius reciprocity is a condition on a pair of adjoint functors $f_{!}⊣f^{*}$. If both categories are cartesian closed, then the adjunction is said to satisfy Frobenius reciprocity if the right adjoint $f^{*}:Y\to X$ 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 morphism

$$f_{!}(C\times f^{*}B)\to (f_{!}C)\times B$$f_!(C \times f^*B) \to (f_! C) \times B

is an isomorphism for each $B$ 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 $P:S^{\mathrm{op}}\to \mathrm{Cat}$ is an $S$-indexed category such that each category $PX$ is cartesian closed and each functor $f^{*}=Pf$ has a left adjoint ${\exists }_f$ (also written $f_{!}$). Then $P$ is said to satisfy Frobenius reciprocity, or the Frobenius condition , if each of the adjunctions ${\exists }_f⊣f^{*}$ does. If the categories $PX$ 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 $\exists x.(\varphi \wedge \psi )$ is equivalent to $(\exists x.\varphi )\wedge \psi$ 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 $f_{!}⊣f^{*}$ 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.

Frobenius laws and Frobenius reciprocity

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 $\delta :1\to \otimes \Delta$ denotes the diagonal transformation on a cartesian bicategory (e.g., $\mathrm{Rel}$), with right adjoint ${\delta }^{†}$, then there is a canonical map

$$\delta {\delta }^{†}\stackrel{\varphi }{\to }(1\otimes {\delta }^{†})(\delta \otimes 1)$$\delta \delta^\dagger \stackrel{\phi}{\to} (1 \otimes \delta^\dagger)(\delta \otimes 1)

mated to the coassociativity isomorphism

$$(1\otimes \delta )\delta \to (\delta \otimes 1)\delta$$(1 \otimes \delta)\delta \to (\delta \otimes 1)\delta

and the Frobenius law here is the assumption that the 2-cell $\varphi$ is an isomorphism. (There are two Frobenius laws actually; the other is that a similar canonical map

$$\delta {\delta }^{†}\stackrel{\varphi \prime }{\to }({\delta }^{†}\otimes 1)(1\otimes \delta ),$$\delta \delta^\dagger \stackrel{\phi'}{\to} (\delta^\dagger \otimes 1)(1 \otimes \delta),

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

$${\mathrm{hom}}_B(i-,c):\mathrm{Map}(B{)}^{\mathrm{op}}\to \mathrm{Semilat}$$\hom_B(i-, c)\colon Map(B)^{op} \to Semilat

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

$$f^{*}r≔(a\stackrel{f}{\to }b\stackrel{r}{\to }1)$$f^\ast r \coloneqq (a \stackrel{f}{\to} b \stackrel{r}{\to} 1)

along $f$, and one may show that $f^{*}-$ preserves finite local meets. Indeed, the pushforward or quantification along $f$ takes $q:a\to 1$ to

$${\exists }_{f}q≔(b\stackrel{f^{†}}{\to }a\stackrel{q}{\to }1)$$\exists_f q \coloneqq (b \stackrel{f^\dagger}{\to} a \stackrel{q}{\to} 1)

and ${\exists }_f⊣f^{*}$ because $f^{†}$ is right adjoint to the map $f$. Because $f^{*}-$ is a right adjoint, it preserves local meets.

Frobenius reciprocity in this context, ordinarily written as

$$r\wedge {\exists }_{f}q={\exists }_f(f^{*}r\wedge q),$$r \wedge \exists_f q = \exists_f (f^\ast r \wedge q),

can then be restated for the hyperdoctrine ${\mathrm{hom}}_B(i-,c)$; it takes the form

$$r\wedge q{f}^{†}=(rf\wedge q)f^{†}$$r \wedge q f^\dagger = (r f \wedge q)f^\dagger

for any map $f:a\to b$ and predicates $q\in \mathrm{hom}(a,c)$, $r\in \mathrm{hom}(b,c)$.

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

Proposition

Frobenius reciprocity holds in each hyperdoctrine ${\mathrm{hom}}_B(i-,c)$ associated with a bicategory of relations.

Proof (sketch)

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

$${\eta }_b=(1\stackrel{{\epsilon }^{†}}{\to }b\stackrel{\delta }{\to }b\otimes b)$$\eta_b = (1 \stackrel{\varepsilon^\dagger}{\to} b \stackrel{\delta}{\to} b \otimes b)

and the counit by

$${\theta }_b=(b\otimes b\stackrel{{\delta }^{†}}{\to }b\stackrel{\epsilon }{\to }1).$$\theta_b = (b \otimes b \stackrel{\delta^\dagger}{\to} b \stackrel{\varepsilon}{\to} 1).

Using this duality, each relation $r:b\to c$ has an opposite relation $r^{\mathrm{op}}:c\to b$ given by

$$c\stackrel{c\otimes {\eta }_b}{\to }c\otimes b\otimes b\stackrel{1\otimes r\otimes 1}{\to }c\otimes c\otimes b\stackrel{{\theta }_c\otimes b}{\to }b.$$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.

It may further be shown that in a bicategory of relations, if $f:a\to b$ is a map, then its right adjoint $f^{†}$ equals the opposite $f^{\mathrm{op}}$. Therefore Frobenius reciprocity becomes the equation

$$r\wedge q{f}^{\mathrm{op}}=(rf\wedge q)f^{\mathrm{op}}$$r \wedge q f^{op} = (r f \wedge q)f^{op}

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.

Proposition

In a locally posetal cartesian bicategory, the Frobenius laws follow from Frobenius reciprocity.

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

$$(q\wedge rf)f^{†}\le q{f}^{†}\wedge r$$(q \wedge r f)f^\dagger \leq q f^\dagger \wedge r

is an equality. One (and therefore both) of the Frobenius laws will follow by taking the following choices for $f$, $q$, and $r$:

$$f={\delta }_x,q={\epsilon }_x^{†}\otimes 1_x,r={\epsilon }_x\otimes 1_x\otimes {\epsilon }_x^{†}$$f = \delta_x, \qquad q = \varepsilon_{x}^{\dagger} \otimes 1_x, \qquad r = \varepsilon_x \otimes 1_x \otimes \varepsilon_{x}^{\dagger}

where ${\delta }_x:x\to x\otimes x$ is the diagonal map and ${\epsilon }_x:x\to 1$ 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.

References

The term ‘Frobenius reciprocity’, in the context of hyperdoctrines, was introduced by Lawvere in

• F.W. Lawvere, Adjointness in Foundations, TAC Reprint, 2006. (link)

A textbook source is around lemma 1.5.8 in

• Peter Johnstone, Sketches of an Elephant

Manifestations of the Frobenius reciprocity formula, in the sense of this section, recur throughout mathematics in various forms (push-pull formula, projection formula); see for example this Math Overflow post:

• Andrea Ferretti, Ubiquity of the push-pull formula, MO Question 18799, March 20, 2010. (link)