nLab internal hom of algebras over a commutative monad

Idea

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.

Construction

Let $C$ be a closed category with equalizers, denote its unit by $1$ and its internal homs by $[X,Y]$. Let also $(T,\mu,\eta)$ be a commutative monad as defined for closed categories (here), with strength $t':[X,Y]\to [T X, T Y]$ and costrength $s':T[X,Y]\to [X, T Y]$.

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

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

(See here for the details.)

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

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

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

Theorem

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

• unit object given by the free algebra $T1$, where $1$ is the unit object of $C$;
• All the structure maps induced by those of $C$.

Interpretation

The internal hom $[A,B]$ of $A$ and $B$ in $C$ can be thought of as “containing all the morphisms $A\to B$ of $C$”. However, not all those morphisms are necessarily morphisms of $T$-algebras. A map $f:A\to B$ is a morphism of algebras if and only if $b\circ T f = f\circ a$. 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 internal hom of modules over a commutative ring as the most prominent example. Note that the ring has to be commutative: this corresponds to the commutativity of the monad.
• The vector space of linear maps between two given vector spaces is a particular example.

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 $C$ 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

• monoidal monad, commutative monad, strong monad, commutative algebraic theory
• tensor product of algebras over a commutative monad
• tensor product, tensor product of modules, multilinear map, representable multicategory
• monoidal category, closed category, closed monoidal category, internal hom

References

• 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)