##
**Iterated monoidal categories.**
*(English)*
Zbl 1030.18006

The most fundamental space associated to a small category is its classifying space. This space is of fundamental interest in algebraic topology and has significant homotopy theoretic applications. By “enriching” the category, it is possible to obtain better properties for its classifying space. In this context, it has long been observed that a monoidal category gives rise to a space with multiplication and in fact to a loop space. This observation due to Stasheff has been an inexhaustible source of inspiration to homotopy theorists. It has also brought significant new understanding to the coherence theory of categories as pursued by MacLane in the early sixties.

The next step forward, taken mainly by Segal, was to see that a symmetric monoidal category has classifying space an infinite loop space (after group completion). This was instrumental to Quillen’s definition of higher algebraic \(K\)-theory for example. Fiedorowicz, by relaxing the symmetry condition (\(\tau^2=1\)) in the monoidal category and using suitable “braiding” instead, showed that the newly obtained “braided” monoidal category has classifying space a second-fold loop space. Such a criterion has been useful for example in showing that the group completion of certain monoids of mapping class groups of surfaces are second-fold loop spaces.

The paper under review takes the daunting task of providing an analog to Stasheff and Fiedorowicz’s results for all \(n\)-fold loop spaces, \(n>2\), and then giving a categorical analog of the Boardman and Vogt little \(n\)-cube operad acting on \(n\)-fold loop spaces.

First of all the authors introduce the notion of an \(n\)-fold monoidal category which is one which supports \(n\) distinct strictly associative multiplications, each with a strict unit and affording a list of compatibility relations. They then show that the group completion of the nerve of such a category is precisely an \(n\)-fold loop space. The proof rests on techniques of Thomason and Street, and, similar to a construction of Dunn, on an iteration of Segal’s method for obtaining a single loop space out of a “special \(\Delta\)-space”. This part constitutes about a fifth of the paper.

Next, the authors introduce free \(n\)-fold monoidal categories, defined on small categories, and then construct an associated operad which acts on nerves of the \(n\)-fold monoidal categories they defined earlier. This operad \({\mathbf M}_n:=\{M_n(k)\}_{k\geq 0}\) has a simple combinatorial description whereby each component \(M_n(k)\) can be realized as some full subcategory of a free \(n\)-monoidal category on a finite set of \(k\) elements (viewed as a category with trivial morphisms). The monads on this operad are precisely the free \(n\)-fold monoidal categories. An emphasis is put on comparing the (nerve) operad (also denoted by \({\mathbf M}_n\)) to the preoperad of permutahedra introduced by Milgram in the sixties (or slight variants of it), and to showing that both are equivalent through an inclusion of preoperads.

The next main theorem this paper proves is that the operad \({\mathbf M}_n\) is equivalent through a chain of operad maps to the little \(n\)-cube operad. Since \({\mathbf M}_n\) acts on \(n\)-fold monoidal categories, this theorem gives a definite way of showing that the group completion of the nerves of such categories are \(n\)-fold loop spaces.

A coherence theorem for \(n\)-fold monoidal categories is also stated and proved in this paper. It gives necessary and sufficient conditions for the existence of morphisms between objects of \({\mathbf M}_n\) and shows uniqueness when such a morphism exists. The proof runs through a mighty twenty pages and is fairly technical. The remaining and last part of the paper deals in details with Milgram’s preoperad construction and its relation to \(n\)-monoidal categories.

The next step forward, taken mainly by Segal, was to see that a symmetric monoidal category has classifying space an infinite loop space (after group completion). This was instrumental to Quillen’s definition of higher algebraic \(K\)-theory for example. Fiedorowicz, by relaxing the symmetry condition (\(\tau^2=1\)) in the monoidal category and using suitable “braiding” instead, showed that the newly obtained “braided” monoidal category has classifying space a second-fold loop space. Such a criterion has been useful for example in showing that the group completion of certain monoids of mapping class groups of surfaces are second-fold loop spaces.

The paper under review takes the daunting task of providing an analog to Stasheff and Fiedorowicz’s results for all \(n\)-fold loop spaces, \(n>2\), and then giving a categorical analog of the Boardman and Vogt little \(n\)-cube operad acting on \(n\)-fold loop spaces.

First of all the authors introduce the notion of an \(n\)-fold monoidal category which is one which supports \(n\) distinct strictly associative multiplications, each with a strict unit and affording a list of compatibility relations. They then show that the group completion of the nerve of such a category is precisely an \(n\)-fold loop space. The proof rests on techniques of Thomason and Street, and, similar to a construction of Dunn, on an iteration of Segal’s method for obtaining a single loop space out of a “special \(\Delta\)-space”. This part constitutes about a fifth of the paper.

Next, the authors introduce free \(n\)-fold monoidal categories, defined on small categories, and then construct an associated operad which acts on nerves of the \(n\)-fold monoidal categories they defined earlier. This operad \({\mathbf M}_n:=\{M_n(k)\}_{k\geq 0}\) has a simple combinatorial description whereby each component \(M_n(k)\) can be realized as some full subcategory of a free \(n\)-monoidal category on a finite set of \(k\) elements (viewed as a category with trivial morphisms). The monads on this operad are precisely the free \(n\)-fold monoidal categories. An emphasis is put on comparing the (nerve) operad (also denoted by \({\mathbf M}_n\)) to the preoperad of permutahedra introduced by Milgram in the sixties (or slight variants of it), and to showing that both are equivalent through an inclusion of preoperads.

The next main theorem this paper proves is that the operad \({\mathbf M}_n\) is equivalent through a chain of operad maps to the little \(n\)-cube operad. Since \({\mathbf M}_n\) acts on \(n\)-fold monoidal categories, this theorem gives a definite way of showing that the group completion of the nerves of such categories are \(n\)-fold loop spaces.

A coherence theorem for \(n\)-fold monoidal categories is also stated and proved in this paper. It gives necessary and sufficient conditions for the existence of morphisms between objects of \({\mathbf M}_n\) and shows uniqueness when such a morphism exists. The proof runs through a mighty twenty pages and is fairly technical. The remaining and last part of the paper deals in details with Milgram’s preoperad construction and its relation to \(n\)-monoidal categories.

Reviewer: Sadok Kallel (Villeneuve d’Asq)

### MSC:

18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |

18D50 | Operads (MSC2010) |

55P47 | Infinite loop spaces |

19D23 | Symmetric monoidal categories |

### Keywords:

monoidal categories; Milgram’s preoperad; \(n\)-fold loop spaces; classifying space; braiding; little \(n\)-cube operad; preoperad of permutahedra
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\textit{C. Balteanu} et al., Adv. Math. 176, No. 2, 277--349 (2003; Zbl 1030.18006)

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