nLab action of a monoidal category

Contents

Context

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Category theory

Contents

Idea

In vertical categorification of how monoids/monoid objects AA may act on other objects NN (action objects, module objects) inside an ambient monoidal category by maps

AβŠ—N⟢N. A \otimes N \longrightarrow N \,.

satisfying the action property, so a monoidal category π’œ\mathcal{A} may act on other categories 𝒩\mathcal{N} by functors

⊘:π’œΓ—π’©βŸΆπ’© \oslash \;\colon\; \mathcal{A} \times \mathcal{N} \longrightarrow \mathcal{N}

subject to associators, unitors and coherence conditions for action objects coherently internalized into the 2-category Cat, analogous to the laws in a monoidal category.

At least if some linear structure is present and respected (such as when π’œ\mathcal{A} qualifies as a 2-ring) it is natural to speak of module categories over π’œ\mathcal{A} (see also at n n -module).

Similarly, compatible actions from both sides, such as for a bimodule, give a notion of bimodule category.

Note that the term actegory, introduced by McCrudden (2000)1, is often used in the literature for this concept, and consequently biactegory for the two-sided case. However, since β€œactegory” is a single transposition away from β€œcategory”, we prefer to use the explicit terminology on this page and elsewhere.

Definition

For any category AA, the category of endofunctors End(A)End(A) is monoidal with respect to the (horizontal) composition (the composition of functors and the Godement product for natural transformations).

Given a monoidal category (C,βŠ—,I,l,r,a)(C,\otimes,I,l,r,a) a (left or right) CC-module category is a category AA together with a (left or right) coherent action of CC on AA. Depending on author and context, the left coherent action of CC on AA is a morphism of monoidal categories Cβ†’End(A)C\to End(A) in the lax, colax, pseudo or strict sense (most often in pseudo-sense) or, in another terminology, a monoidal, comonoidal, strong monoidal or strict monoidal functor. Right coherent actions correspond to the monoidal functors into the category End(A)End(A) with the opposite tensor product.

CC-module categories, colax CC-equivariant functors and natural transformations of colax CC-equivariant functors form a strict 2-category CAct c_C Act^c. A monad in CAct c_C Act^c amounts to a pair of a monad in CatCat and a distributive law between the monad and an action of CC.

The notion of CC-action (hence a CC-module category) is easily extendable to bicategories (see BakoviΔ‡β€˜s thesis).

Definition

A (left) π’ž\mathcal{C}-module category is

  1. a category π’œ\mathcal{A};
  2. a functor ⊘:π’žΓ—π’œβ†’π’œ\oslash : \mathcal{C} \times \mathcal{A} \to \mathcal{A} called the action;
  3. a natural isomorphism Ξ» a:aβ†’I⊘a\lambda_a : a \to I \oslash a called the unitor;
  4. a natural isomorphism Ξ± c,d,a:c⊘(d⊘a)β†’(cβŠ—d)⊘a\alpha_{c,d,a} : c \oslash (d \oslash a) \to (c \otimes d) \oslash a called the actor;

satisfying a pentagonal and two triangular laws (see KJ01, diagg. (1.1)-(1.3)) that witness the coherence of Ξ»\lambda and Ξ±\alpha with the unitors and associators of π’ž\mathcal{C}.

Connection with enrichment

If a category DD is enriched in CC with copowers, then the copower structure forms a module category on the ordinary category underlying DD.

Conversely, if module category is such that the functor (βˆ’)⊘d:Cβ†’D(-)\oslash d:C\to D has a right adjoint for all objects dd of DD, then the right adjoints D(d,βˆ’):Dβ†’CD(d,-):D\to C provide an enrichment of DD in CC for which the action is a copower. See KJ01.

More generally, supposing CC is small, the following data are equivalent: (1) a CC-module category DD; (2) an enrichment of DD in the category C^\hat C of presheaves on CC (i.e. a locally graded category), with copowers by representables. Here we regard C^\hat C as a monoidal category with the Day convolution.

Starting from a CC-module category DD, consider the enrichment given by D(d,dβ€²)=hom(βˆ’βŠ˜d,dβ€²):C opβ†’SetD(d,d')=hom(-\oslash d,d'):C^{op}\to Set. If these presheaves are representable, this is what it means to be enriched in CC for which the action is a copower. If they are not representable, it is still an enrichment, and the copowers by representables are the action.

For this reason, many concepts from enriched category theory make sense for module categories too.

References

  • Bodo Pareigis, Non-additive ring and module theory I. General theory of monoids, Publ. Math. Debrecen 24 (1977), 189–204. MR 56:8656; Non-additive ring and module theory II. C-categories, C-functors, and C-morphisms, Publ. Math. Debrecen 24 (351–361) 1977.

  • Max Kelly, George Janelidze, A note on actions of a monoidal category, Theory and Applications of Categories, Vol. 9, 2001, No. 4, pp 61–91 link

  • P. McCrudden, Categories of representations of coalgebroids, Advances in Mathematics 154 2 (2000) 299–332 [doi:10.1006/aima.2000.1926]

  • P. Schauenburg, Actions of monoidal categories and generalized Hopf smash products, J. Algebra 270 (2003) 521–563 (remark: actegories with action in the strong monoidal sense)

  • Zoran Ε koda, Distributive laws for actions of monoidal categories, arXiv:0406310, Equivariant monads and equivariant lifts versus a 2-category of distributive laws, arXiv:0707.1609

  • J. R. B. Cockett, Craig Pastro, The logic of message-passing arXiv:math/0703713.

A recent survey of many basic definitions and operations on actegories is

A generalisation from monoidal categories to bicategories is studied in, where actions are called representations:

  • Robert Gordon, and John Power. Enrichment through variation, Journal of Pure and Applied Algebra 120.2 (1997): 167-185.

  1. From Cockett & Pastro (2007): β€œThe term actegory is used to describe the situation of a monoidal category β€œacting” on a category. They first appeared (under a different name) in the work of BΓ©nabou as a simple example of a bicategory. B. Pareigis developed the theory of actegories (again under a different name) and showed there usefulness in the representation theory of monoids and comonoids. The word β€œactegory” was first suggested at the Australian Category Seminar and first appeared in print in the thesis of P. McCrudden where they were used to study categories of representations of coalgebroids.” ↩

Last revised on November 27, 2023 at 12:16:05. See the history of this page for a list of all contributions to it.