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.
Sometimes also the projection formula
In category theory, Frobenius reciprocity is a condition on a pair of adjoint functors . If both categories are cartesian closed, then the adjunction is said to satisfy Frobenius reciprocity if the right adjoint is a cartesian closed functor; that is, if the canonical map is an isomorphism for all objects of .
is an isomorphism for each in and in .
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 and are the inverse and direct image functors along a map in a hyperdoctrine. Here is a category and is an -indexed category such that each category is cartesian closed and each functor has a left adjoint (existential quantifier, also written ). Then is said to satisfy Frobenius reciprocity, or the Frobenius condition, if each of the adjunctions does. If the categories 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 is equivalent to if is not free in .
When is the inverse image part of a geometric morphism between (n,1)-topoi and is a left adjoint of it, if the adjunction satisfies Frobenius reciprocity, then the geometric morphism is called locally (n-1)-connected. In particular, if so that we have a continuous map of locales, then a left adjoint satisfying Frobenius reciprocity makes it an open map, and if 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.
By naturality in and by the Yoneda lemma this shows that is an equivalence precisey if 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 denotes the diagonal transformation on a cartesian bicategory (e.g., ), with right adjoint , then there is a canonical map
mated to the coassociativity isomorphism
and the Frobenius law here is the assumption that the 2-cell 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 and any object of , one may construct a hyperdoctrine of the form
where is the inclusion, and is the 2-category of meet-semilattices. Here is thought of as a relation from to , and for a map , the relation is the pulling back
along , and one may show that preserves finite local meets. Indeed, the pushforward or quantification along takes to
and because is right adjoint to the map . Because is a right adjoint, it preserves local meets.
Frobenius reciprocity in this context, ordinarily written as
can then be restated for the hyperdoctrine ; it takes the form
for any map and predicates , .
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 associated with a bicategory of relations.
One first proves that a bicategory of relations is a compact closed bicategory in which each object is self-dual. The unit here is given by
and the counit by
Using this duality, each relation has an opposite relation given by
It may further be shown that in a bicategory of relations, if is a map, then its right adjoint equals the opposite . 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 means that for any map and any two relations , , the canonical inclusion
is an equality. One (and therefore both) of the Frobenius laws will follow by taking the following choices for , , and :
where is the diagonal map and 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.
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