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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.
A Monadic Approach to PolyCategories
 Theory Appl. Categ
, 2002
"... Polycategories form a rather natural generalization of multicategories. Besides the domains also the codomains of morphisms are allowed to be strings of objects. Multicategories are known to have an elegant global characterization as monads in a suitable bicategory of special spans with free m ..."
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Cited by 2 (0 self)
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Polycategories form a rather natural generalization of multicategories. Besides the domains also the codomains of morphisms are allowed to be strings of objects. Multicategories are known to have an elegant global characterization as monads in a suitable bicategory of special spans with free monoid as domains. To describe polycategories in similar terms, we investigate distributive laws in the sense of Beck between cartesian monads as tools for constructing new bicategories of modi ed spans. Three very simple such laws produce a bicategory in which the monads are precisely the planar polycategories (where composition only is de ned if the corresponding circuit diagram is planar). General polycategories, which only satisfy a local planarity condition, require a slightly more complicated construction.
A Convenient Category For Games And Interaction
, 1996
"... We present a simple construction of an orderenriched category gam that simultaneously dualizes and parallels the familiar construction of the category rel of relations. Objects of gam are sets, and arrows are games, viewed as special kinds of trees. The quest for identities for the composition of ..."
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Cited by 1 (1 self)
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We present a simple construction of an orderenriched category gam that simultaneously dualizes and parallels the familiar construction of the category rel of relations. Objects of gam are sets, and arrows are games, viewed as special kinds of trees. The quest for identities for the composition of trees naturally leads to the consideration of alternating sequences and games of a specific polarity. gam may be viewed as a canonical extension of rel , and just as for rel , the maps in gam admit a nice characterization. Disjoint union of sets induces a special tensor product on gam that allows us to recover the monoidal closed category of games and strategies of interest in game theory. If we allow games with explicit delay moves, the categorical description of the structure that leads to the monoidal closed category is even more satisfying. In particular, we then obtain an explicit involution. 0 Introduction People who study games from a mathematical perspective (e.g., Blass, Abramsky, ...
DUALITY FOR CCD LATTICES
"... Abstract. The 2category of constructively completely distributive lattices is shown to be bidual to a 2category of generalized orders that admits a monadic schizophrenic object biadjunction over the 2category of ordered sets. 1. ..."
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Abstract. The 2category of constructively completely distributive lattices is shown to be bidual to a 2category of generalized orders that admits a monadic schizophrenic object biadjunction over the 2category of ordered sets. 1.
A 2dimensional view of the Chuconstruction
, 2000
"... The cyclic Chuconstruction for closed bicategories, generalizing the original Chuconstruction for symmetric monoidal closed categories, turns out to have a noncyclic counterpart. Both constructions are based on socalled Chucells and can be generalized to chains of composable 1cells. This leads ..."
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The cyclic Chuconstruction for closed bicategories, generalizing the original Chuconstruction for symmetric monoidal closed categories, turns out to have a noncyclic counterpart. Both constructions are based on socalled Chucells and can be generalized to chains of composable 1cells. This leads to two hierarchies of closed bicategories for any closed bicategory with local pullbacks. Chucells in rel correspond to bipartite state transition systems. Even though their vertical composition may fail here due to the lack of pullbacks, basic gametheoretic constructions can be performed on cyclic Chucells. These generalize to all symmetric monoidal closed categories. If finite limits exist, the cyclic Chucells form the objects of a *autonomous category.