Monads on Higher Monoidal Categories

Aguiar, Marcelo; Haim, Mariana; Franco, Ignacio López

doi:10.1007/s10485-017-9497-8

Cited by 4 publications

(12 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The case n = 0 of the product amounts to an object i ∈ A, and part of the colax structure involves maps a → ia and a → ai. For the connection between colax monoidal categories and multicategories, see [9] or [1]. For applications of lax monoidal structure to higher categories, see [22] and [2].…”

Section: Introductionmentioning

confidence: 99%

Skew monoidal categories and skew multicategories

Bourke

Lack

2018

Journal of Algebra

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 99%

Skew monoidal categories and skew multicategories

Bourke

Lack

2018

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…Recently in [1] a similar analysis to that of [25] was carried out for multimonoidal monads on multimonoidal categories. (In [1] the term higher monoidal was used.…”

Section: Introductionmentioning

confidence: 99%

“…Recently in [1] a similar analysis to that of [25] was carried out for multimonoidal monads on multimonoidal categories. (In [1] the term higher monoidal was used. However, we prefer to call the same thing multimonoidal and reserve the term higher to be used only for dimensionality of categorical structures.)…”

Section: Introductionmentioning

confidence: 99%

“…However, we prefer to call the same thing multimonoidal and reserve the term higher to be used only for dimensionality of categorical structures.) The aim of this paper is to extend the results of [1] to multimonoidal monads in symmetric strict monoidal 2-categories and, more importantly, derive them from a suitable 'formal theory'. With this we not only gain a deeper understanding of the origin of the formulae in [1] but, as a byproduct, also open a way to some practical applications.…”

Section: Introductionmentioning

confidence: 99%

“…The aim of this paper is to extend the results of [1] to multimonoidal monads in symmetric strict monoidal 2-categories and, more importantly, derive them from a suitable 'formal theory'. With this we not only gain a deeper understanding of the origin of the formulae in [1] but, as a byproduct, also open a way to some practical applications. For example, we obtain sufficient conditions under which the 2-category of pseudomonoids, monoidal 1-cells and monoidal 2-cells in a strict monoidal 2-category admits Eilenberg-Moore construction.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The formal theory of multimonoidal monads

Böhm

2018

Preprint

View full text Add to dashboard Cite

Certain aspects of Street's formal theory of monads in 2-categories are extended to multimonoidal monads in symmetric strict monoidal 2-categories. Namely, any symmetric strict monoidal 2-category M admits a symmetric strict monoidal 2-category of pseudomonoids, monoidal 1-cells and monoidal 2-cells in M. Dually, there is a symmetric strict monoidal 2-category of pseudomonoids, opmonoidal 1-cells and opmonoidal 2-cells in M. Extending a construction due to Aguiar and Mahajan for M = Cat, we may apply the first construction p-times and the second one q-times (in any order). It yields a 2-category M pq . A 0-cell therein is an object A of M together with p + q compatible pseudomonoid structures; it is termed a (p + q)-oidal object in M. A monad in M pq is called a (p, q)-oidal monad in M; it is a monad t on A in M together with p monoidal, and q opmonoidal structures in a compatible way. If M has monoidal Eilenberg-Moore construction, and certain (Linton type) stable coequalizers exist, then a (p + q)-oidal structure on the Eilenberg-Moore object A t of a (p, q)-oidal monad (A, t) is shown to arise via a symmetric strict monoidal double functor to Ehresmann's double category Sqr(M) of squares in M, from the double category of monads in Sqr(M) in the sense of Fiore, Gambino and Kock. While q ones of the pseudomonoid structures of A t are lifted along the 'forgetful' 1-cell A t → A, the other p ones are lifted along its left adjoint. In the particular example when M is an appropriate 2-subcategory of Cat, this yields a conceptually different proof of some recent results due to Aguiar, Haim and López Franco.

show abstract

The Gray Monoidal Product of Double Categories

Böhm

2019

Appl Categor Struct

View full text Add to dashboard Cite

The category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category A, the corresponding internal hom functor ⟦A, −⟧ sends a double category B to the double category whose 0-cells are the double functors A → B, whose horizontal and vertical 1-cells are the horizontal and vertical pseudotransformations, respectively, and whose 2cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure.

show abstract

Monads on Higher Monoidal Categories

Cited by 4 publications

References 36 publications

Skew monoidal categories and skew multicategories

Skew monoidal categories and skew multicategories

The formal theory of multimonoidal monads

The Gray Monoidal Product of Double Categories

Contact Info

Product

Resources

About