Symmetric bimonoidal intermuting categories and $ω\timesω$ reduced bar constructions

Petric, Z.; Trimble, Todd

doi:10.48550/arxiv.0906.2954

Cited by 2 publications

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“…Hence, our Theorem 8.5 covers all the related results concerning these categories. Also, the correctness of the reduced bar construction of [19,Lemma 7.1] follows from this theorem.…”

Section: The N-fold Monoidal Categoriesmentioning

confidence: 85%

“…We discuss these matters in more details at the end of Section 9. Also, this gives a positive answer to the second question of [19,Section 8].…”

Section: Introductionmentioning

confidence: 90%

“…Here we will only give a definition of the reduced bar construction based on a strict monoidal category. We refer to [19,Section 6] for the complete analysis of this construction.…”

Section: The Reduced Bar Constructionmentioning

confidence: 99%

“…The idea of [8] and [19] was to laxify the interchanges for units as much as the coherence allows. Trimble and the second author showed that a coherence result for pseudocommutative pseudomonoids, for which some structural constraints are invertible, in a 2-category of symmetric monoidal categories, lax symmetric monoidal functors, and monoidal transformations is sufficient for the reduced bar construction.…”

Section: Introductionmentioning

confidence: 99%

“…To conclude, we mention that the interchanges between the monoidal structures required for n-fold monoidal categories are usually brought about by braiding and symmetry. It is pointed out in [8] and [19] that a bicartesian structure (a category with all finite coproducts and products) brings the desired interchanges but the corresponding coherence result required some unusual properties of such a category-a coproduct of terminal objects should be terminal. Our results show that this coherence is not necessary anymore and that every bicartesian category, for every n, may be conceived as an n-fold monoidal category in n + 1 different ways.…”

Section: Introductionmentioning

confidence: 99%

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The n-fold reduced bar construction

Petric

2017

J. Homotopy Relat. Struct.

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This paper is about a correspondence between monoidal structures in categories and n-fold loop spaces. We developed a new syntactical technique whose role is to substitute the coherence results, which were the main ingredients in the proof that the Segal-Thomason bar construction provides an appropriate simplicial space. The results we present here enable more common categories to enter this delooping machine. For example, such as the category of finite sets with two monoidal structures brought by the disjoint union and Cartesian product. (2010): 18D10, 57T30, 55P47, 55P48 to obtain a lax functor WM from (∆ op ) n , the nth power of the opposite of the simplicial category, to the category Cat, of categories and functors, such that Mathematics Subject ClassificationThe main result of this paper states that for every n ≥ 2, the n-fold reduced bar construction delivers a certain lax functor. This is what we mean by correctness of the reduced bar construction. We prove this result gradually-the cases n = 2, n = 3 and n ≥ 3 are dealt with respectively in Theorems 4.5, 6.5 and 8.5.Following the ideas of [3, Section 2], we show in Section 9, that the lax functor WM satisfies some additional conditions. Roughly speaking, some particular arrows of (∆ op ) n , which are built out of face maps corresponding to projections, have to be mapped by WM to identities. Such a lax functor is called Segal's in [20].By applying Street's rectification to WM (see [24]) one obtains a functor V , with the same source and target as WM. From [20, Corollary 4.4], when B is the classifying space functor, it follows that B • V is a multisimplicial space with some properties guaranteing that, up to group completion (see [22] and [18]), the realization of this multisimplicial space is an n-fold delooping of BM (see [20, Theorem 5.1]). A thorough survey of results concerning these matters is given in [20] and the case n = 2 is considered separately in [21].This paper is strongly influenced by [3]. One can find the main ideas followed by us in Sections 0, 1 and 2 of that paper. Also, the reader should consider [10] as an earlier source of these ideas. The notions of two, three and n-fold monoidal categories used in [3] and the corresponding notions used in this paper are compared in Sections 2, 5 and 7. The case n = 2 is studied systematically in Section 2.A definition of n-fold monoidal category is usually inductive as one starts with the 2-category Cat whose monoidal structure is given by 2-products. The 0-cells of a 2-category M on(Cat) are pseudomonoids (or monoids) in Cat, i.e., monoidal (or strict monoidal) categories. Then one makes a choice what to consider to be the 1-cells of M on(Cat), i.e., how strictly they should preserve the monoidal structure. The monoidal structure of M on(Cat) is again given by 2-products. A pseudomonoid (or a monoid) in M on(Cat) is a (strict) twofold monoidal category and if we iterate this procedure with the same degree of strictness, we obtain one possible notion of n-fold monoidal category.Joyal and Street, ...

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“…Hence, our Theorem 8.5 covers all the related results concerning these categories. Also, the correctness of the reduced bar construction of [19,Lemma 7.1] follows from this theorem.…”

Section: The N-fold Monoidal Categoriesmentioning

confidence: 85%

“…We discuss these matters in more details at the end of Section 9. Also, this gives a positive answer to the second question of [19,Section 8].…”

Section: Introductionmentioning

confidence: 90%

“…Here we will only give a definition of the reduced bar construction based on a strict monoidal category. We refer to [19,Section 6] for the complete analysis of this construction.…”

Section: The Reduced Bar Constructionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The n-fold reduced bar construction

Petric

2017

J. Homotopy Relat. Struct.

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Monoids, Segal's condition and bisimplicial spaces

Petric

2015

Preprint

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A characterization of simplicial objects in categories with finite products obtained by the reduced bar construction is given. The condition that characterizes such simplicial objects is a strictification of Segal's condition guaranteeing that the loop space of the geometric realization of a simplicial space X and the space X1 are of the same homotopy type. A generalization of Segal's result appropriate for bisimplicial spaces is given. This generalization gives conditions guaranteing that the double loop space of the geometric realization of a bisimplicial space X and the space X11 are of the same homotopy type.

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Symmetric bimonoidal intermuting categories and $ω\timesω$ reduced bar constructions

Cited by 2 publications

References 7 publications

The n-fold reduced bar construction

The n-fold reduced bar construction

Monoids, Segal's condition and bisimplicial spaces

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