Iterated monoidal categories

Abstract

The external associativity condition gives the commutative diagramRemark 1.4. Notice that we have natural transformations η A,0,0,B : A✷ 1 B → A✷ 2 B and η 0,A,B,0 : A✷ 1 B → B✷ 2 A.If we had insisted a 2-fold monoidal category be a monoid in the category of monoidal categories and strictly monoidal functors, this would amount to requiring that η = id. In view of the above, this would imply A✷ 1 B = A✷ 2 B = B✷ 1 A and similarly for morphisms. Thus the nerve of such a category would be a commutative topological monoid and its group completion would be equivalent to a product of abelian Eilenberg-MacLane spaces.Remark 1.5. Recall that a braided monoidal category (also known as braided tensor category) is a category C together with a functor ✷ 1 : C × C → C which is strictly associative, has a strict 2-sided unit object 0 and with a natural commutativity isomorphism c A,B : A✷ 1 B −→ B✷ 1 A satisfying the following properties:Assuming that ✷ 1 = ✷ 2 and that the natural isomorphism η A,B,C,D satisfies η A,B,0,C = η A,0,B,C = id A✷1B✷1C , one proceeds as follows to show that we have a braided monoidal category. In the internal associativity diagram take V = W = 0 and obtain that η U,X,Y,Z = id U ✷ 1 η 0,X,Y,Z . Then take X = Y = 0 and obtain that η U,V,W,Z = η U,V,W,0 ✷ 1 id Z . Combining these two facts, one obtains thatwhere c B,C = η 0,B,C,0 . Then take U = Z = W = 0 in the internal associativity law to get the first associativity law for c, and take U = Z = X = 0 to get the other one. With the additional conditions we have here the external associativity law is superfluous. Conversely given a braided monoidal category, we can define a 2-fold monoidal structure by ( * ).Remark 1.6. Joyal and Street [10] considered a very similar concept to our notion of 2-fold monoidal category. They loosened our requirement that the two operations ✷ 1 and ✷ 2 be strictly associative with a strict unit by only requiring these conditions to hold up to coherent natural isomorphisms. More significantly they required the natural transformation η A,B,C,D to be an isomorphism. They then showed that such a category is naturally equivalent to a braided monoidal category. Briefly given such a category one obtains an equivalent braided monoidal category by discarding one of the two operations, say ✷ 2 , and defining the commutativity isomorphism for the remaining operation ✷ 1 to be the compositeOur requirement that the operations be strictly associative and unital are not significant restrictions and were adopted for convenience and simplicity. One can always replace categories with operations which are associative and unital up to coherent natural isomorphisms by equivalent categories with strictly associative and unital operations.

Manin products, Koszul duality, Loday algebras and Deligne conjecture

“…We call it 2-monoidal category in Section 1. This definition generalizes previous notions given by A. Joyal and R. Street in [JS93] in the framework of braided tensor categories and by C. Balteanu, Z. Fiedorowicz, R. Schwänzl and R. Vogt in [BFSV03] in the framework of iterated monoidal category and iterated loop spaces. All the examples given above are monoids in a 2-monoidal category.…”

“…The notion of 2-monoidal category given here is a lax and more general version of the one given by A. Joyal and R. Street in [JS93] and the one given by C. Balteanu, Z. Fiedorowicz, R. Schwänzl and R. Vogt in [BFSV03].…”

“…This last condition forces the two monoidal products to be isomorphic. In order to model n-fold loop spaces, C. Balteanu, Z. Fiedorowicz, R. Schwänzl and R. Vogt introduced in [BFSV03] the notions of n-fold monoidal category. Their notion of 2-fold monoidal category is, in some sense, a lax version of the one given by Joyal and Street since they do not assume the interchange law to be an isomorphism.…”

Understanding the Small Object Argument

“…4.9 By a two-fold monoidal category [3], we mean a category V equipped with two monoidal structures (⊗, I, α, λ, ρ) and (⊙, ⊥, α ′ , λ ′ , ρ ′ ) in such a way that the functors ⊙ : V × V → V and ⊥ : 1 → V, together with the natural transformations α ′ , λ ′ and ρ ′ , are lax monoidal with respect to the (⊗, I) monoidal structure.…”

On the multiplicative structure of topological Hochschild homology

“…We want to point out that results from [3] and [1] imply that jK B n j is an E n -operad and that the inclusion jK B n j jK n j is a †-equivalence.…”