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Recent documents in Mathematical Sciences Faculty Researchen-usMon, 02 Oct 2017 16:53:02 PDT3600Higher Dimensional Enrichment
http://digitalscholarship.tnstate.edu/mathematics/6
http://digitalscholarship.tnstate.edu/mathematics/6Wed, 02 May 2012 14:29:35 PDT
Lyubashenko has described enriched 2–categories as categories enriched over V–Cat, the 2–category of categories enriched over a symmetric monoidal V. Here I generalize this to a k–fold monoidal V. The latter is defined as by Balteanu, Fiedorowicz, Schw¨anzl and Vogt but with the addition of making visible the coherent associators _i. The symmetric case can easily be recovered. The introduction of this paper proposes a recursive definition of V–n–categories and their morphisms. Then I consider the special case of V–2–categories and give the details of the proof that with their morphisms these form the structure of a 3–category.
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Stefan ForceyOperads In Iterated Monoidal Categories
http://digitalscholarship.tnstate.edu/mathematics/5
http://digitalscholarship.tnstate.edu/mathematics/5Wed, 02 May 2012 13:41:43 PDT
The structure of a k-fold monoidal category as introduced by Balteanu, Fiedorowicz, Schw¨anzl and Vogt in [2] can be seen as a weaker structure than a symmetric or even braided monoidal category. In this paper we show that it is still sufficient to permit a good definition of (n-fold) operads in a k-fold monoidal category which generalizes the definition of operads in a braided category. Furthermore, the inheritance of structure by the category of operads is actually an inheritance of iterated monoidal structure, decremented by at least two iterations. We prove that the category of n-fold operads in a k-fold monoidal category is itself a (k − n)-fold monoidal, strict 2-category, and show that n-fold operads are automatically (n − 1)-fold operads. We also introduce a family of simple examples of k-fold monoidal categories and classify operads in the example categories.
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Stefan Forcey et al.Enrichment As Categorical Delooping I: Enrichment Over Iterated Monoidal Categories
http://digitalscholarship.tnstate.edu/mathematics/4
http://digitalscholarship.tnstate.edu/mathematics/4Wed, 02 May 2012 11:41:54 PDT
Joyal and Street note in their paper on braided monoidal categories [10] that the 2–category V–Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. What is meant by “based upon” here will be made more clear in the present paper. The exception that they mention is the case in which V is symmetric, which leads to V–Cat being symmetric as well. The symmetry in V–Cat is based upon the symmetry of V. The motivation behind this paper is in part to describe how these facts relating V and V–Cat are in turn related to a categorical analogue of topological delooping first mentioned by Baez and Dolan in [1]. To do so I need to pass to a more general setting than braided and symmetric categories – in fact the k–fold monoidal categories of Balteanu et al in [3]. It seems that the analogy of loop spaces is a good guide for how to define the concept of enrichment over various types of monoidal objects, including k–fold monoidal categories and their higher dimensional counterparts. The main result is that for V a k–fold monoidal category, V–Cat becomes a (k − 1)–fold monoidal 2–category in a canonical way. I indicate how this process may be iterated by enriching over V–Cat, along the way defining the 3–category of categories enriched over V–Cat. In the next paper I hope to make precise the n–dimensional case and to show how the group completion of the nerve of V is related to the loop space of the group completion of the nerve of V–Cat.
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Stefan ForceyVertically Iterated Classical Enrichment
http://digitalscholarship.tnstate.edu/mathematics/3
http://digitalscholarship.tnstate.edu/mathematics/3Wed, 02 May 2012 11:28:19 PDT
Lyubashenko has described enriched 2-categories as categories enriched over V-Cat, the 2-category of categories enriched over a symmetric monoidal V. This construction is the strict analogue for V-functors in V-Cat of Brian Day’s probicategories for V-modules in V-Mod. Here I generalize the strict version to enriched n-categories for k-fold monoidal V. The latter is defined as by Balteanu, Fiedorowicz, Schw¨anzl and Vogt but with the addition of making visible the coherent associators αi. The symmetric case can easily be recovered. This paper proposes a recursive definition of V-n-categories and their morphisms. We show that for V k-fold monoidal the structure of a (k−n)-fold monoidal strict (n + 1)-category is possessed by V-n-Cat. This article is a completion of the work begun in [Forcey, 2003], and the initial sections duplicate the beginning of that paper.
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Stefan ForceyEnrichment Over Iterated Monoidal Categories
http://digitalscholarship.tnstate.edu/mathematics/2
http://digitalscholarship.tnstate.edu/mathematics/2Wed, 02 May 2012 10:21:46 PDT
Joyal and Street note in their paper on braided monoidal categories [Braided tensor categories, Advances in Math. 102(1993) 20-78] that the 2-category V-Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. The exception that they mention is the case in which V is symmetric, which leads to V-Cat being symmetric as well. The symmetry in V-Cat is based upon the symmetry of V. The motivation behind this paper is in part to describe how these facts relating V and V-Cat are in turn related to a categorical analogue of topological delooping. To do so I need to pass to a more general setting than braided and symmetric categories -- in fact the k-fold monoidal categories of Balteanu et al in [Iterated Monoidal Categories, Adv. Math. 176(2003) 277-349]. It seems that the analogy of loop spaces is a good guide for how to define the concept of enrichment over various types of monoidal objects, including k-fold monoidal categories and their higher dimensional counterparts. The main result is that for V a k-fold monoidal category, V-Cat becomes a (k-1)-fold monoidal 2-category in a canonical way. In the next paper I indicate how this process may be iterated by enriching over V-Cat, along the way defining the 3-category of categories enriched over V-Cat. In future work I plan to make precise the n-dimensional case and to show how the group completion of the nerve of V is related to the loop space of the group completion of the nerve of V-Cat.
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Stefan ForceyAssociative Structures Based Upon a Categorical Braiding
http://digitalscholarship.tnstate.edu/mathematics/1
http://digitalscholarship.tnstate.edu/mathematics/1Wed, 02 May 2012 10:07:40 PDT
It is well known that the existence of a braiding in a monoidal category V allows many structures to be built upon that foundation. These include a monoidal 2-category V-Cat of enriched categories and functors over V, a monoidal bicategory V-Mod of enriched categories and modules, a category of operads in V and a 2-fold monoidal category structure on V. We will begin by focusing our exposition on the first and last in this list due to their ability to shed light on a new question. We ask, given a braiding on V, what non-equal structures of a given kind in the list exist which are based upon the braiding. For instance, what non-equal monoidal structures are available on V-Cat, or what non-equal operad structures are available which base their associative structure on the braiding in V. We demonstrate alternative underlying braids that result in an infinite family of associative structures. The external and internal associativity diagrams in the axioms of a 2-fold monoidal category will provide us with several obstructions that can prevent a braid from underlying an associative structure.
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Stefan Forcey