hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
symmetric monoidal (∞,1)-category of spectra
With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
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 $C$ 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,
(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.)
(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
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”.
The fact that “linear maps form themselves a vector space” is a general phenomenon for commutative monads on categories with equalizers. For instance,
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.
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.