nLab Frobenius reciprocity

Definition

The term Frobenius reciprocity has a meaning

• In representation theory

• In category theory.

(For different statements of a similar name see the disambiguation at Frobenius theorem.)

In representation theory

In representation theory, Frobenius reciprocity 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

$$(Ind \dashv Res) \;\colon\; Rep_G \stackrel{\overset{Ind}{\leftarrow}}{\underset{Red}{\longrightarrow}} Rep_H$$

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

$$Ind(Res(W) \otimes V) \cong W \otimes Ind(V)$$

is referred to as Frobenius reciprocity in representation theory (e.g. here on PlanetMath).

In cartesian categories

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 $f^* \colon \mathcal{Y} \to \mathcal{X}$ 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 $a,b$ of $\mathcal{Y}$.

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

(1)$$\pi \,\colon\, f_! (f^*a \times c) \longrightarrow a \times f_!(c)$$

is an isomorphism for each $a$ in $\mathcal{Y}$ and $c$ in $\mathcal{X}$.

This holds for instance for the base change between slice categories $\mathcal{C}_{/b}$, $\mathcal{C}_{/b'}$ of a finitely complete category $\mathcal{C}$ along a morphism $f \colon b' \to b$ – by the pasting law in $\mathcal{C}$:

(2)$$\array{ f^\ast a \times_{b'} c &\longrightarrow& f^\ast a &\longrightarrow& a \\ \big\downarrow && \big\downarrow && \big\downarrow \\ c &\underset{p_C}{\longrightarrow}& b' &\underset{f}{\longrightarrow}& b } \;\;\;\;\;\;\;\; \simeq \;\;\;\;\;\;\;\; \array{ a \times_b f_! c &\longrightarrow& a \\ \big\downarrow && \big\downarrow \\ c &\underset{ f \circ p_c }{\longrightarrow}& b \mathrlap{\,.} }$$

The condition (1) 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.

In six operations yoga

The projection formula plays a notable role in Grothendieck’s yoga of six operations. For example if an adjoint triple $(f_! \dashv f^\ast \dashv f_\ast)$ between symmetric closed monoidal categories is a Wirthmüller context (May 05), $f^\ast$ is a strong closed monoidal functor. This implies the projection formula, i.e. the existence of a natural isomorphism of the form

$$\pi \;\colon\; f_! \big( f^\ast a \otimes c \big) \overset{\sim}{\longrightarrow} a \otimes (f_! c) \,.$$

The projection formula also holds in a Grothendieck context or a Verdier-Grothendieck context (May 05).

Properties

Closed monoidal functors and the projection formula

The following result isolates the connection between closed functors and the projection formula. We begin with some context.

Recall that a monoidal category $\mathcal{Y}$ is left closed if each functor $a \otimes - \colon \mathcal{Y} \to \mathcal{Y}$ has a right adjoint $[a, -] \colon \mathcal{Y} \to \mathcal{Y}$, called the internal hom. We can similarly define right closed monoidal categories. A symmetric or even braided monoidal category is left closed if and only if it is right closed, and one then simply calls it closed, but for maximum generality we consider the merely monoidal case.

A functor $F$ between left closed monoidal categories is lax closed it if preserves the internal hom and the unit object up to a specified map

$$\hat{F} \;\colon\; F[a,b] \to [F a,F b], \qquad F_0 \colon I \to F(I) \,,$$

natural in both variables and obeying some coherence laws listed at closed functor. If these are natural isomorphisms we call the functor strong closed.

Any lax monoidal functor betweeen left closed monoidal categories is lax closed (for a sketch of the argument see closed functor), but a strong monoidal functor may not be strong closed.

Proposition

Suppose $f_! \dashv f^\ast$ is an adjunction between left closed monoidal categories. Then natural transformations

$$\phi \;\colon\; f^*[a,b] \to [f^*a, f^*b]$$

correspond bijectively to natural maps

$$\pi \;\colon\; f_! \big( (f^\ast a) \otimes c \big) \longrightarrow a \otimes (f_! c) \,.$$

Furthermore, $\phi$ is an isomorphism if and only if $\pi$ is, in which case we say the projection formula holds.

Proof

Suppose $\mathcal{X}$ and $\mathcal{Y}$ are left closed monoidal categories and $f_! \colon \mathcal{X} \to \mathcal{Y}$ is left adjoint to $f^* \colon \mathcal{Y} \to \mathcal{X}$. Suppose we have a natural map

$$\phi \colon f^*[a,b] \to [f^*a, f^*b]$$

for $a,b \in \mathcal{Y}$. Thus we obtain a natural map

$$\mathcal{X}(c, f^*[a,b]) \to \mathcal{X}(c, [f^*a, f^*b]),$$

for arbitrary $c \in \mathcal{X}$ (now natural in all three variables). By hom-tensor adjointness and the fact that $f_!$ is the left adjoint of $f^*$ we can rewrite this as

$$\mathcal{Y}(f_! c, [a,b]) \to \mathcal{X}(f^*a \otimes c,f^*b).$$

Using both these facts again we obtain

$$\mathcal{Y}(a \otimes f_! c, b) \to \mathcal{X}(f_!(f^\ast a \otimes c), b)\,.$$

By the Yoneda lemma this gives the desired natural map

$$\pi \colon\; f_!(f^\ast a \otimes c) \longrightarrow a \otimes f_! c\,.$$

By running through this calculation one can see that if $\phi$ is a invertible then all the other natural maps listed above are too, including $\pi$. Conversely, starting with $\pi$ we can run the argument backwards and get $\phi$, and if $\pi$ is invertible then so is $\phi$.

It follows that if $f^*$ is strong closed, the projection formula holds. Also if $f$ is strong monoidal and the projection formula holds, $f^*$ is strong closed.

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

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

mated to the coassociativity isomorphism

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

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

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

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

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

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

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

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

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

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

$$r \wedge \exists_f q = \exists_f (f^\ast r \wedge q),$$

can then be restated for the hyperdoctrine $\hom_B(i-, c)$; it takes the form

$$r \wedge q f^\dagger = (r f \wedge q)f^\dagger$$

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.

Proposition

Frobenius reciprocity holds in each hyperdoctrine $\hom_B(i-, c)$ 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

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

and the counit by

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

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

$$c \stackrel{c \otimes \eta_b}{\to} c \otimes b \otimes b \stackrel{1 \otimes r \otimes 1}{\to} c \otimes c \otimes b \stackrel{\theta_c \otimes b}{\to} b.$$

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

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

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

$$(q \wedge r f)f^\dagger \leq q f^\dagger \wedge r$$

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

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

where $\delta_x: x \to x \otimes x$ is the diagonal map and $\varepsilon_x: x \to 1$ 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.

Examples

Example

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

$$(f_! \dashv f^\ast \dashv f_\ast) \;\colon\; \mathbf{H}_{/X} \to \mathbf{H}_{/Y}$$

has inverse image $f^\ast$ a cartesian closed functor and hence (see there) exhibits Frobenius reciprocity.

Related concepts

• Beck-Chevalley condition

• Wirthmüller context

References

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

• Bill Lawvere, Equality in hyperdoctrines and comprehension schema as an adjoint functor, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970) 1-14 [pdf]

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).

Related discussion is in:

• Robert A. G. Seely, p. 511 of: Hyperdoctrines, Natural Deduction and the Beck Condition, Zeitschr. f. math. Logik und Grundlagen d. Math. 29 (1983) 505-542 [doi:10.1002/malq.19830291005, pdf]

• Duško Pavlović, p. 164 in: Maps II: Chasing Diagrams in Categorical Proof Theory, Logic Journal of the IGPL, 4 2 (1996) 159–194 [doi:10.1093/jigpal/4.2.159, pdf]

A textbook source is around lemma 1.5.8 in

• Peter Johnstone, Sketches of an Elephant

General discussion in the context of projection formulas in monoidal categories (not necessarily cartesian) is in

• Halvard Fausk, Po Hu, Peter May, Isomorphisms between left and right adjoints, Theory and Applications of Categories, 11 4 (2003) 107-131 [tac:11-04, pdf]

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:

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

Further MO discussion includes

• Wrong-way Frobenius reciprocity for finite groups representations