nLab internal hom of algebras over a commutative monad



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Algebras over a commutative monad, in many cases, admit an internal hom analogous to the one of modules over a commutative ring.

On a monoidal closed category, this internal hom is in many cases right-adjoint to a tensor product, giving a hom-tensor adjunction for the algebras too. Again, this generalizes the case of modules.


Let CC be a closed category with equalizers, denote its unit by 11 and its internal homs by [X,Y][X,Y]. Let also (T,μ,η)(T,\mu,\eta) be a commutative monad as defined for closed categories (here), with strength t:[X,Y][TX,TY]t':[X,Y]\to [T X, T Y] and costrength s:T[X,Y][X,TY]s':T[X,Y]\to [X, T Y].

Let now (A,a)(A,a) and (B,b)(B,b) be TT-algebras. Thanks to the costrength, the internal hom [A,B][A,B] of CC has a canonical “pointwise” TT-algebra structure,

T[A,B][A,TB][A,B]. T[A,B] \to [A, T B] \to [A,B].

(See here for the details.)

The internal hom of AA and BB in C TC^T is defined to be the equalizer of the following parallel pair: where a *:[A,B][TA,B]a^*:[A,B]\to [T A,B] is the internal precomposition with a:TAAa:T A\to A, and b *:[TA,TB][TA,B]b_*:[T A,T B]\to [T A,B] is the internal postcomposition with b:TBBb:T B\to B.

Denote this object by [A,B] T[A,B]_T.

(See Brandenburg, Remark 6.4.1, as well as the original work Kock ‘71, Section 2.)


(Kock ‘71, Theorem 2.2) Let CC be a closed category with equalizers, and (T,μ,η)(T,\mu,\eta) a commutative monad on CC. Then [,] T[-,-]_T makes the Eilenberg-Moore category C TC^T itself a closed category, with

  • unit object given by the free algebra T1T1, where 11 is the unit object of CC;
  • All the structure maps induced by those of CC.


The internal hom [A,B][A,B] of AA and BB in CC can be thought of as “containing all the morphisms ABA\to B of CC”. However, not all those morphisms are necessarily morphisms of T T -algebras. A map f:ABf:A\to B is a morphism of algebras if and only if bTf=fab\circ T f = f\circ a. In terms of internal homs, this condition is exactly given by the parallel maps above. The equalizer of that pair can be thought of as of containing “all the maps satisfying that condition”.

Examples and implications

The fact that “linear maps form themselves a vector space” is a general phenomenon for commutative monads on categories with equalizers. For instance,

  • For the case of the distribution monad, this says that convex combinations of affine functions are affine;
  • For the case of the power set monad, this says that pointwise suprema of join-preserving maps between join-semilattices are join-preserving.

On monoidal closed categories

If the category CC is monoidal closed, under some condition the internal hom of algebras is part of a closed monoidal structure on the algebras themselves. See here for more information.

See also


  • Martin Brandenburg, Tensor categorical foundations of algebraic geometry (arXiv:1410.1716)

  • William Keigher, Symmetric monoidal closed categories generated by commutative adjoint monads, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 19 no. 3 (1978), p. 269-293 (NUMDAM, pdf)

  • Anders Kock, Monads on symmetric monoidal closed categories, Arch. Math. 21 (1970), 1–10.

  • Anders Kock, Strong functors and monoidal monads, Arhus Universitet, Various Publications Series No. 11 (1970). PDF.

  • Anders Kock, Closed categories generated by commutative monads, 1971 (pdf)

  • Gavin J. Seal, Tensors, monads and actions (arXiv:1205.0101)

Last revised on January 27, 2024 at 12:51:31. See the history of this page for a list of all contributions to it.