The Bicategory of Bimodules, Tensor–Hom Adjunction, and Its Application to Frobenius Reciprocity
1 The bicategory of bimodules
The bicategory \(\mathsf{Bimod}\) consists of the following data:
- Objects: unital rings \(A,B,C,\ldots\).
- Hom-categories: \(\mathsf{Bimod}(A,B)\) is the category whose objects are \((B,A)\)-bimodules and whose morphisms are bimodule homomorphisms. Thus its objects are the \(1\)-morphisms \(A\to B\), and its morphisms are the \(2\)-morphisms.
- Horizontal composition: the functor \[ \mathsf{Bimod}(B,C)\times\mathsf{Bimod}(A,B) \longrightarrow\mathsf{Bimod}(A,C) \] sends \(({}_C N_B,{}_B M_A)\) to \({}_C(N\otimes_B M)_A\) and sends a pair of \(2\)-morphisms \((g,f)\) to \(g\otimes_B f\).
- Identity \(1\)-morphisms: the identity on \(A\) is the regular bimodule \({}_A A_A\).
- Associator: for composable bimodules \(M\), \(N\), and \(L\), there is a bimodule isomorphism \[ \alpha_{L,N,M}:(L\otimes_C N)\otimes_B M \xrightarrow{\sim}L\otimes_C(N\otimes_B M), \qquad (\ell\otimes n)\otimes m\longmapsto\ell\otimes(n\otimes m). \] Its components are natural in all three bimodules, so \(\alpha\) is a natural isomorphism between the two threefold-composition functors.
- Unitors: for each \({}_B M_A\), multiplication defines bimodule isomorphisms \[ \lambda_M:B\otimes_B M\xrightarrow{\sim}M,\quad b\otimes m\longmapsto b m, \qquad \rho_M:M\otimes_A A\xrightarrow{\sim}M,\quad m\otimes a\longmapsto m a. \] The families \(\lambda=(\lambda_M)_M\) and \(\rho=(\rho_M)_M\) are natural isomorphisms from composition with the appropriate identity \(1\)-morphism to the identity functor.
- Coherence: the associator and unitors are invertible \(2\)-morphisms satisfying the usual pentagon and triangle identities.
For four composable bimodules \(M\), \(N\), \(L\), and \(K\), the pentagon identity is the commutativity of the following diagram:
For composable bimodules \({}_C N_B\) and \({}_B M_A\), the triangle identity relates the associator to the left and right unitors:
Remark (Strictness and the role of \(2\)-morphisms). One of the main subtleties of \(\mathsf{Bimod}\) is that tensor product is not strictly associative. The bimodules \((L\otimes N)\otimes M\) and \(L\otimes(N\otimes M)\) are canonically isomorphic, but they are not literally equal. What does canonically mean here?
From a categorical point of view, it is not enough to choose an unrelated isomorphism for each triple \((L,N,M)\). The reassociation maps must form a natural isomorphism \[ \alpha:(-\otimes-)\otimes-\Longrightarrow-\otimes(-\otimes-) \] between the two threefold-composition functors. Likewise, the left and right unitors form natural isomorphisms from composition with an identity \(1\)-morphism to the identity functor. Their naturality says that these isomorphisms are compatible with every bimodule homomorphism.
This requires a second categorical layer. The source and target of each component \(\alpha_{L,N,M}\), \(\lambda_M\), or \(\rho_M\) are \(1\)-morphisms, namely bimodules. A map relating them is therefore a \(2\)-morphism, namely a bimodule homomorphism. The associator and unitors are invertible \(2\)-morphisms of this kind. This extra layer allows the associativity and unit laws for \(1\)-morphisms to hold weakly, through coherent natural isomorphisms, rather than as equalities.
A strict \(2\)-category, despite also having \(2\)-morphisms, requires the associativity and unit laws for \(1\)-morphisms to hold strictly. Its associator and unitors are therefore identity \(2\)-morphisms. \(\mathsf{Cat}\) is the standard example: its objects are categories, its \(1\)-morphisms are functors, and its \(2\)-morphisms are natural transformations.
1.1 Internal and external hom constructions
\(\operatorname{Hom}_B(M,N)\) denotes homomorphisms \(M\to N\) compatible with the left \(B\)-actions. \(\operatorname{Hom}_B({}_B M_A,{}_B N_C)\) is an \((A,C)\)-bimodule, with \[ (a\cdot\varphi\cdot c)(m):=\varphi(m\cdot a)\cdot c. \]
\(\operatorname{Hom}(M,N)_A\) denotes homomorphisms compatible with the right \(A\)-actions. \(\operatorname{Hom}({}_B M_A,{}_C N_A)_A\) is a \((C,B)\)-bimodule, with \[ (c\cdot\varphi\cdot b)(m):=c\cdot\varphi(b\cdot m). \]
\(\operatorname{Hom}_B(M,N)_A\) denotes homomorphisms compatible with both actions. Taking bimodule homomorphisms defines a bifunctor \[ \operatorname{Hom}_B(-,-)_A: \bigl((B,A)\textsf{-Mod}\bigr)^{\mathrm{op}} \times (B,A)\textsf{-Mod}\longrightarrow\mathsf{Set}. \]
Remark (Slogan). Tensor products contract matching middle scalars. A left-linear \(\operatorname{Hom}_A(-,-)\) contracts LHS scalars, whereas a right-linear \(\operatorname{Hom}(-,-)_B\) contracts RHS scalars.
2 Tensor–hom adjunction
For \(M\in(B,A)\textsf{-Mod}\), \(N\in(C,B)\textsf{-Mod}\), and \(P\in(C,A)\textsf{-Mod}\), there are natural isomorphisms \[ \begin{matrix} \operatorname{Hom}_B(M, \operatorname{Hom}_C(N,P))_A &\cong& \operatorname{Hom}_C(N \otimes_B M, P)_A &\cong& \operatorname{Hom}_C(N, \operatorname{Hom}(M,P)_A)_B \\ (m \mapsto (n \mapsto p)) &\text{↤}& (n \otimes m \mapsto p) &\mapsto& (n \mapsto (m \mapsto p)) \end{matrix} \]
Equivalently, fixing one tensor factor gives the adjunctions \[ \begin{aligned} N\otimes_B- &\dashv \operatorname{Hom}_C(N,-) &&:\quad (B,A)\textsf{-Mod}\rightleftarrows(C,A)\textsf{-Mod},\\ -\otimes_B M &\dashv \operatorname{Hom}(M,-)_A &&:\quad (C,B)\textsf{-Mod}\rightleftarrows(C,A)\textsf{-Mod}. \end{aligned} \]
Thus, the three \(2\)-morphisms in the following diagram are naturally isomorphic:
These isomorphisms are the bimodule form of currying: a balanced bilinear map \((n,m)\mapsto p\) can be curried in either variable, yielding \(m\mapsto(n\mapsto p)\) or \(n\mapsto(m\mapsto p)\).
2.1 The commutative case
Recall that if \(A\) is commutative, every left \(A\)-module is canonically an \((A,A)\)-bimodule. Setting \(B=C=A\) above therefore recovers the familiar Tensor–Hom adjunction in \(A\textsf{-Mod}\): \[ \operatorname{Hom}_A\left(M,\operatorname{Hom}_A(N,P)\right) \cong \operatorname{Hom}_A\left(N\otimes_A M,P\right) \cong \operatorname{Hom}_A\left(N,\operatorname{Hom}_A(M,P)\right). \] Thus ordinary currying in the closed monoidal category \(A\textsf{-Mod}\) is the single-ring, commutative instance of the bimodule construction.
3 Frobenius reciprocity
Let \(H\leq G\) be groups, let \(k\) be a commutative ring, let \(V\) be a left \(k[H]\)-module, and let \(W\) be a left \(k[G]\)-module. For convenience, write \[ \operatorname{Hom}_G=\operatorname{Hom}_{k[G]},\qquad \operatorname{Hom}_H=\operatorname{Hom}_{k[H]},\qquad \otimes_G=\otimes_{k[G]},\qquad \otimes_H=\otimes_{k[H]}. \] Induction, restriction, and coinduction are given by \[ \operatorname{Ind}_H^G V=k[G]\otimes_H V,\qquad \operatorname{Res}_H^G W={}_{k[H]}W,\qquad \operatorname{Coind}_H^G V=\operatorname{Hom}_H(k[G],V). \] Here \(k[G]\) in the coinduction formula is an \((H,G)\)-bimodule, and the \(G\)-action on coinduction is \[ (g\cdot\varphi)(x)=\varphi(xg). \]
The two forms of Frobenius reciprocity are the adjunctions \[ \operatorname{Ind}_H^G\dashv\operatorname{Res}_H^G\dashv\operatorname{Coind}_H^G, \] or, equivalently, the natural isomorphisms \[ \operatorname{Hom}_G\left(\operatorname{Ind}_H^G V,W\right) \cong \operatorname{Hom}_H\left(V,\operatorname{Res}_H^G W\right) \] and \[ \operatorname{Hom}_H\left(\operatorname{Res}_H^G W,V\right) \cong \operatorname{Hom}_G\left(W,\operatorname{Coind}_H^G V\right). \]
Both are direct instances of the Tensor–Hom adjunction. For the first, \[ \begin{aligned} \operatorname{Hom}_G\left(k[G]\otimes_H V,W\right) &\cong\operatorname{Hom}_H\left(V,\operatorname{Hom}_G(k[G],W)\right)\\ &\cong\operatorname{Hom}_H\left(V,\operatorname{Res}_H^G W\right), \end{aligned} \] where \(\operatorname{Hom}_G(k[G],W)\cong\operatorname{Res}_H^G W\) by evaluation at \(1\). Concretely, \[ \begin{matrix} F &\mapsto& \left( v\mapsto F(1\otimes v) \right) \\ \left( g\otimes v\mapsto g\cdot f(v) \right) &\text{↤}& f \end{matrix} \] For the second, \[ \begin{aligned} \operatorname{Hom}_H\!\left(\operatorname{Res}_H^G W,V\right) &\cong\operatorname{Hom}_H\left(k[G]\otimes_G W,V\right)\\ &\cong\operatorname{Hom}_G\left(W,\operatorname{Hom}_H(k[G],V)\right)\\ &=\operatorname{Hom}_G\left(W,\operatorname{Coind}_H^G V\right). \end{aligned} \] Explicitly, \[ \begin{matrix} f &\mapsto& \left( w \mapsto \left( x\mapsto f(x\cdot w)\right) \right) \\ \left(w\mapsto\Phi(w)(1)\right) &\text{↤}& \Phi \end{matrix} \]
Remark (Coind = ind? (This part is under construction)). Fix some field \(k\). For any \(k\)-algebra \(A\), \(B\) and \((B,A)\)-module \(V\), let \(V^*\) denote \(\operatorname{Hom}_k(V, k)\), equipped with a \((A,B)\)-module struture of with \((a \cdot \varphi \cdot b) (v):= \varphi(b \cdot v \cdot a)\).
When \(G\) is finite, \(k\) is a field with \(|G|\) invertible inside, it turns out that we are enabled with two additional isomorphisms (TODO: are they natural?) (TODO: in what category?) \[ k[G] \cong k[G]^*,\qquad \operatorname{Hom}_G(V,W) \cong V^* \otimes_G W \] where the first one is the Frobenius self-duality of finite group (that is, the inner product), and the second can be found in finite group rep speedrun (slightly differs). Above two isomorphisms allow us to equalize induction and coinduction \[ \operatorname{Ind}_H^G V =k[G]\otimes_H V \cong\operatorname{Hom}_H(k[G]^*,V) \cong\operatorname{Hom}_H(k[G],V) =\operatorname{Coind}_H^G V. \] Substituting this identification into the second reciprocity isomorphism \[ \operatorname{Hom}_H\left(\operatorname{Res}_H^G W,V\right) \cong \operatorname{Hom}_G\left(W,\operatorname{Ind}_H^G V\right), \] so in this case, induction and restriction in this case are bi-directionally adjoint \[ \operatorname{Ind}_H^G\dashv\operatorname{Res}_H^G\dashv\operatorname{Ind}_H^G. \] (TODO: write the correspondence explicitly)