Tambara module



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In higher category theory



A Tambara module is a profunctor endowed with additional structure that makes it interact nicely with the action of a monoidal category. In other words, Tambara modules generalize profunctors from categories to actegories. Tambara modules are used in the theory of optics (in computer science).



Let (M,i,)(\mathbf M, i, \odot) be a monoidal category, let C\mathbf C and D\mathbf D be two left M \mathbf M -actegories (in other terminology, left M\mathbf M-modules). We denote M\mathbf M actions by ()()(-)\cdot (-).

A (left) Tambara module is a profunctor P:C op×DSetP \,\colon\, \mathbf C^{op} \times \mathbf D \to \mathbf{Set} equipped with a family of morphisms called (left) strength

s a,b,m:P(a,b)P(ma,mb) s_{a,b,m} \,\colon\, P(a,b) \longrightarrow P(m \cdot a, m \cdot b)

which is natural in aa and bb and dinatural in mm, and satisfies two coherence laws:

  1. s a,b,i=P(ρ a,ρ b 1)s_{a,b,i} = P(\rho_a, \rho_b^{-1}), where ρ\rho comes with the module structures of C\mathbf C and D\mathbf D.

  2. s a,b,mn=P(μ m,n,a 1,μ m,n,b)s a,b,ms a,b,ns_{a,b,m \odot n} = P(\mu^{-1}_{m,n,a}, \mu_{m,n,b}) \circ s_{a,b,m} \circ s_{a,b,n} where μ\mu comes with the module structures of C\mathbf C and D\mathbf D.

This is same thing as (left) strong profunctors.

A right Tambara module is defined in the exact way except C\mathbf C and D\mathbf D are assumed to be right M\mathbf M-actegories and everything is correspondigly ‘done on the other side’.


Suppose now C\mathbf C and D\mathbf D have both left and right M\mathbf M-actegories structures. A Tambara bimodule is a profunctor P:C op×DSetP : \mathbf C^{op} \times \mathbf D \to \mathbf{Set} equipped with compatible left and right Tambara module structures ss and ss', i.e. such that they satisfy

s ma,mb,ns a,b,m=s an,bn,ms a,b,n s'_{m \cdot a, m \cdot b, n} \circ s_{a,b,m} = s_{a \cdot n, b \cdot n, m} \circ s'_{a,b,n}


Let P,Q:C op×DSetP, Q: \mathbf C^{op} \times \mathbf D \to \mathbf{Set} be (left) Tambara modules. A morphism of Tambara modules α:PQ\alpha : P \to Q is a natural transformation PQP \Rightarrow Q that commutes with PP and QQ‘s strenghts:

α a,bs a,b,m P=s a,b,m Qα ma,mb \alpha_{a,b} \circ s^P_{a,b,m} = s^Q_{a,b,m} \circ \alpha_{m \cdot a,m \cdot b}


Tambara modules are named after Daisuke Tambara who introduced them in Tamb06 to prove that for a V \mathbf V -enriched category A\mathbf A, Z(A op,V)Tamb(A,A)Z(\mathbf A^{op}, \mathbf V) \cong \mathbf{Tamb}(\mathbf A, \mathbf A).


Both Tamb06 and PS07 define Tambara module to mean Tambara bimodule. In the optical literature, however, left Tambara modules are called simply modules and they are the one used. This doesn’t pose any problem when M\mathbf M is symmetric monoidal since it’s evident how, in that case, any left module structure can be made into a right module structure, so that any left/right module is automatically a bimodule.


Last revised on October 6, 2021 at 13:37:13. See the history of this page for a list of all contributions to it.