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19
Operads In HigherDimensional Category Theory
, 2004
"... The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n <= 2 ..."
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Cited by 32 (2 self)
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The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n <= 2. Generalized operads and multicategories play other parts in higherdimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to ncategories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures.
On the BreenBaezDolan stabilization hypothesis for Tamsamani’s weak ncategories
, 1998
"... In [2] Baez and Dolan established their stabilization hypothesis as one of a list of the key properties that a good theory of higher categories should have. It is the analogue for ncategories of the wellknown stabilization theorems in homotopy theory. To explain the statement, recall that BaezDol ..."
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Cited by 9 (0 self)
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In [2] Baez and Dolan established their stabilization hypothesis as one of a list of the key properties that a good theory of higher categories should have. It is the analogue for ncategories of the wellknown stabilization theorems in homotopy theory. To explain the statement, recall that BaezDolan introduce the notion of kuply monoidal ncategory which is an n + kcategory having only one imorphism for all i < k. This includes the notions previously defined and examined by many authors, of monoidal (resp. braided monoidal, symmetric monoidal) category (resp. 2category) and so forth, as is explained in [2] [4]. See the bibliographies of those preprints as well as that of the the recent preprint [9] for many references concerning these types of objects. In the case where the ncategory in question is an ngroupoid, this notion is—except for truncation at n—the same thing as the notion of kfold iterated loop space, or “Ekspace ” which appears in Dunn [10] (see also some anterior references from there). The fully stabilized notion of kuply monoidal ncategories for k ≫ n is what Grothendieck calls Picard ncategories in [12]. The stabilization hypothesis [2] states that for n + 2 ≤ k ≤ k ′ , the kuply monoidal
Iterated distributive laws
, 2007
"... We give a framework for combining n monads on the same category via distributive laws satisfying YangBaxter equations, extending the classical result of Barr and Wells which combines two monads via one distributive law. We show that this corresponds to iterating ntimes the process of taking the 2 ..."
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We give a framework for combining n monads on the same category via distributive laws satisfying YangBaxter equations, extending the classical result of Barr and Wells which combines two monads via one distributive law. We show that this corresponds to iterating ntimes the process of taking the 2category of monads in a 2category, extending the result of Street characterising distributive laws. We show that this framework can be used to construct the free strict ncategory monad on ndimensional globular sets; we first construct for each i a monad for composition along bounding icells, and then we show that the interchange laws define distributive laws between these monads, satisfying the necessary YangBaxter equations.
Limits in ncategories
, 1997
"... One of the main notions in category theory is the notion of limit. Similarly, one of the most commonly used techniques in homotopy theory is the notion of “homotopy limit” commonly called “holim ” for short. The purpose of the this paper is to begin to develop ..."
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One of the main notions in category theory is the notion of limit. Similarly, one of the most commonly used techniques in homotopy theory is the notion of “homotopy limit” commonly called “holim ” for short. The purpose of the this paper is to begin to develop
Effective generalized SeifertVan Kampen: how to calculate ΩX, preprint available at qalg/9710011
"... A central concept in algebraic topology since the 1970’s has been that of delooping machine [4] [23] [29]. Such a “machine ” corresponds to a notion of Hspace, or space with a multiplication satisfying associativity, unity and inverse properties up to homotopy in an appropriate way, including highe ..."
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A central concept in algebraic topology since the 1970’s has been that of delooping machine [4] [23] [29]. Such a “machine ” corresponds to a notion of Hspace, or space with a multiplication satisfying associativity, unity and inverse properties up to homotopy in an appropriate way, including higher order coherences as first investigated in [33]. A delooping machine is a specification of the extra homotopical structure carried by the loop space ΩX of a connected basepointed topological space X, exactly the structure allowing recovery of X by a “classifying space ” construction. The first level of structure is that the component set π0(ΩX) has a structure of group π1(X, x). Classically the SeifertVan Kampen theorem states that a pushout diagram of connected spaces gives rise to a pushout diagram of groups π1. The loop space construction ΩX with its delooping structure being the higherorder “topologized ” generalization of π1, an obvious question is whether a similar SeifertVan Kampen statement holds for ΩX. The aim of this paper is to describe the operation underlying pushout of spaces with loop space structure, answering the above question by giving a SeifertVan Kampen statement for delooping machinery. We work with Segal’s machine [28] [36]. Our SeifertVan
Flexible sheaves
, 1996
"... This is an unfinished explanation of the notion of “flexible sheaf”, that is a homotopical notion of sheaf of topological spaces (homotopy types) over a site. See “Homotopy over the complex numbers and generalized de Rham cohomology ” (submitted to proceedings of the Taniguchi conference on vector b ..."
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This is an unfinished explanation of the notion of “flexible sheaf”, that is a homotopical notion of sheaf of topological spaces (homotopy types) over a site. See “Homotopy over the complex numbers and generalized de Rham cohomology ” (submitted to proceedings of the Taniguchi conference on vector bundles, preprint of Toulouse 3, and also available at my homepage 2) for a more detailed introduction, and also for a further development of the ideas presented here. The present paper was finished in December 1993 while I was visiting MIT. Since writing this, I have realized that the theory sketched here is essentially equivalent to JardineIllusie’s theory of presheaves of topological spaces (although they talk about presheaves of simplicial sets which is the same thing). This is the point of view adopted in “Homotopy over the complex numbers and generalized de Rham cohomology”. 1 Added in August 1996: I am finally sending this to “Duke eprints ” because it now seems that there may be several useful features of the treatment given here. First of all, the direct construction of the homotopysheafification (by doing a certain operation n+2 times) seems to be useful, for example I need
A characterization of fibrant Segal categories
"... Abstract. In this note we prove that Reedy fibrant Segal categories are fibrant objects in the model category structure SeCatc. Combining this result with a previous one, we thus have that the fibrant objects are precisely the Reedy fibrant Segal categories. We also show that the analogous result ho ..."
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Abstract. In this note we prove that Reedy fibrant Segal categories are fibrant objects in the model category structure SeCatc. Combining this result with a previous one, we thus have that the fibrant objects are precisely the Reedy fibrant Segal categories. We also show that the analogous result holds for Segal categories which are fibrant in the projective model structure on simplicial spaces, considered as objects in the model structure SeCatf. 1.
A SURVEY OF (∞, 1)CATEGORIES
, 2006
"... Abstract. In this paper we give a summary of the comparisons between different definitions of socalled (∞,1)categories, which are considered to be models for ∞categories whose nmorphisms are all invertible for n> 1. They are also, from the viewpoint of homotopy theory, models for the homotopy th ..."
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Abstract. In this paper we give a summary of the comparisons between different definitions of socalled (∞,1)categories, which are considered to be models for ∞categories whose nmorphisms are all invertible for n> 1. They are also, from the viewpoint of homotopy theory, models for the homotopy theory of homotopy theories. The four different structures, all of which are equivalent, are simplicial categories, Segal categories, complete Segal spaces, and quasicategories. 1.