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Quillen Closed Model Structures for Sheaves
, 1995
"... In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisim ..."
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In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisimplicial sheaves and the category of simplicial sheaves of groupoids. Subsequently, the homotopy theories of these categories are related to the homotopy theory of simplicial sheaves. 1 Introduction There are two ways of trying to generalize the well known closed model structure on the category of simplicial sets to the category of simplicial objects in a Grothendieck topos. One way is to concentrate on the local aspect, and to use the Kanfibrations as a starting point. In [14] Heller showed that for simplicial presheaves there is a local (there called right) closed model structure. In [2] K. Brown showed that for a topological space X the category of "locally fibrant" sheaves of spectra on ...
Pasting Schemes for the Monoidal Biclosed Structure on ωCat
, 1995
"... Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on ωcategories, which extends Gray's tensor product on 2categories and which is closely related to BrownHiggins's tensor product on ωgroupoids. Immediate consequences ..."
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Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on ωcategories, which extends Gray's tensor product on 2categories and which is closely related to BrownHiggins's tensor product on ωgroupoids. Immediate consequences are a general and uniform definition of higher dimensional lax natural transformations, and a nice and transparent description of the corresponding internal homs. Further consequences will be in the development of a theory for weak ncategories, since both tensor products and lax structures are crucial in this.
Sketches: Outline with references
, 1993
"... This package contains the original article, written in December, 1993, and this addendum, ..."
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This package contains the original article, written in December, 1993, and this addendum,
DISTRIBUTIVE LAWS IN PROGRAMMING STRUCTURES
, 2009
"... Generalised Distributive laws in Computer Science are rules governing the transformation of one programming structure into another. In programming, they are programs satisfying certain formal conditions. Their importance has been to date documented in several isolated cases by diverse formal approac ..."
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Generalised Distributive laws in Computer Science are rules governing the transformation of one programming structure into another. In programming, they are programs satisfying certain formal conditions. Their importance has been to date documented in several isolated cases by diverse formal approaches. These applications have always meant leaps in understanding the nature of the subject. However, distributive laws have not yet been given the attention they deserve. One of the reasons for this omission is certainly the lack of a formal notion of distributive laws in their full generality. This hinders the discovery and formal description of occurrences of distributive laws, which is the precursor of any formal manipulation. In this thesis, an approach to formalisation of distributive laws is presented based on the functorial approach to formal Category Theory pioneered by Lawvere and others, notably Gray. The proposed formalism discloses a rather simple nature of distributive laws of the kind found in programming structures based on lax 2naturality and Gray’s tensor product of 2categories. It generalises the existing more specific notions of distributive
Linear process algebra
"... Abstract. A linear process is a system of events and states related by an inner product, on which are defined the behaviorally motivated operations of tensor product or orthocurrence, sum or concurrence, sequence, and choice. Linear process algebra or LPA is the theory of this framework. LPA resemb ..."
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Abstract. A linear process is a system of events and states related by an inner product, on which are defined the behaviorally motivated operations of tensor product or orthocurrence, sum or concurrence, sequence, and choice. Linear process algebra or LPA is the theory of this framework. LPA resembles Girard’s linear logic with the differences attributable to its focus on behavior instead of proof. As with MLL the multiplicative part can be construed via the CurryHoward isomorphism as an enrichment of Boolean algebra. The additives cater for independent concurrency or parallel play. The traditional sequential operations of sequence and choice exploit processspecific state information catering for notions of transition and cancellation.1