Results 11  20
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107
Weak Factorization Systems and Topological Functors
 Appl. Categ. Struct
, 2000
"... Weak factorization systems, important in homotopy theory, are related to injective objects in comma{categories. Our main result is that full functors and topological functors form a weak factorization system in the category of small categories, and that this is not cobrantly generated. We also prese ..."
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Cited by 16 (6 self)
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Weak factorization systems, important in homotopy theory, are related to injective objects in comma{categories. Our main result is that full functors and topological functors form a weak factorization system in the category of small categories, and that this is not cobrantly generated. We also present a weak factorization system on the category of posets which is not cobrantly generated. No such weak factorization systems were known until recently. This answers an open problem posed by M. Hovey.
Algebraic theories in homotopy theory
 Annals of Math
"... It is well known in homotopy theory that given a loop space X one can always find a simplicial group G weakly equivalent to X, such that the weak equivalence can be realized by maps preserving multiplication. It is also known that loop spaces are not the only class of spaces for which result of this ..."
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Cited by 13 (2 self)
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It is well known in homotopy theory that given a loop space X one can always find a simplicial group G weakly equivalent to X, such that the weak equivalence can be realized by maps preserving multiplication. It is also known that loop spaces are not the only class of spaces for which result of this kind holds; for example, any
Homotopy theory of comodules over a Hopf algebroid
, 2003
"... Given a good homology theory E and a topological space X, E∗X is not just an E∗module but also a comodule over the Hopf algebroid (E∗, E∗E). We establish a framework for studying the homological algebra of comodules over a wellbehaved Hopf algebroid (A, Γ). That is, we construct ..."
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Cited by 13 (3 self)
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Given a good homology theory E and a topological space X, E∗X is not just an E∗module but also a comodule over the Hopf algebroid (E∗, E∗E). We establish a framework for studying the homological algebra of comodules over a wellbehaved Hopf algebroid (A, Γ). That is, we construct
On The Duality Between Varieties And Algebraic Theories
"... . Every variety V of finitary algebras is proved to have an essentially unique algebraic theory Th(V) which is Cauchy complete, i.e. all idempotents split in Th(V). This defines a duality between varieties (and algebraically exact functors) and Cauchy complete theories (and theory morphisms). Algebr ..."
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Cited by 12 (2 self)
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. Every variety V of finitary algebras is proved to have an essentially unique algebraic theory Th(V) which is Cauchy complete, i.e. all idempotents split in Th(V). This defines a duality between varieties (and algebraically exact functors) and Cauchy complete theories (and theory morphisms). Algebraically exact functors are precisely the right adjoints preserving filtered colimits and regular epimorphisms; or, more succintly: the functors preserving limits and sifted colimits.
Presheaf Models for the piCalculus
, 1997
"... Recent work has shown that presheaf categories provide a general model of concurrency, with an inbuilt notion of bisimulation based on open maps. Here it is shown how this approach can also handle systems where the language of actions may change dynamically as a process evolves. The example is the p ..."
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Cited by 12 (4 self)
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Recent work has shown that presheaf categories provide a general model of concurrency, with an inbuilt notion of bisimulation based on open maps. Here it is shown how this approach can also handle systems where the language of actions may change dynamically as a process evolves. The example is the picalculus, a calculus for `mobile processes' whose communication topology varies as channels are created and discarded. A denotational semantics is described for the picalculus within an indexed category of profunctors; the model is fully abstract for bisimilarity, in the sense that bisimulation in the model, obtained from open maps, coincides with the usual bisimulation obtained from the operational semantics of the picalculus. While attention is concentrated on the `late' semantics of the picalculus, it is indicated how the `early' and other variants can also be captured.
Iterated wreath product of the simplex category and iterated loop spaces
 Adv. Math
"... Abstract. Generalising Segal’s approach to 1fold loop spaces, the homotopy theory of nfold loop spaces is shown to be equivalent to the homotopy theory of reduced Θnspaces, where Θn is an iterated wreath product of the simplex category ∆. A sequence of functors from Θn to Γ allows for an alternat ..."
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Cited by 11 (4 self)
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Abstract. Generalising Segal’s approach to 1fold loop spaces, the homotopy theory of nfold loop spaces is shown to be equivalent to the homotopy theory of reduced Θnspaces, where Θn is an iterated wreath product of the simplex category ∆. A sequence of functors from Θn to Γ allows for an alternative description of the Segal spectrum associated to a Γspace. In particular, each EilenbergMacLane space has a canonical reduced Θnset model. The number of (n + d)dimensional cells of the resulting CWcomplex of type K(Z/2Z, n) is a generalised Fibonacci number.
The microcosm principle and concurrency in coalgebras
 I. HASUO, B. JACOBS, AND A. SOKOLOVA
, 2008
"... Coalgebras are categorical presentations of statebased systems. In investigating parallel composition of coalgebras (realizing concurrency), we observe that the same algebraic theory is interpreted in two different domains in a nested manner, namely: in the category of coalgebras, and in the final ..."
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Cited by 11 (8 self)
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Coalgebras are categorical presentations of statebased systems. In investigating parallel composition of coalgebras (realizing concurrency), we observe that the same algebraic theory is interpreted in two different domains in a nested manner, namely: in the category of coalgebras, and in the final coalgebra as an object in it. This phenomenon is what Baez and Dolan have called the microcosm principle, a prototypical example of which is “a monoid in a monoidal category.” In this paper we obtain a formalization of the microcosm principle in which such a nested model is expressed categorically as a suitable lax natural transformation. An application of this account is a general compositionality result which supports modular verification of complex systems.
Flow does not model flows up to weak dihomotopy
 Applied Categorical Structures
, 2005
"... In particular, a new approach of dihomotopy involving simplicial presheaves over an appropriate small category is proposed. This small category is obtained by taking a full subcategory of a locally presentable ..."
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Cited by 10 (4 self)
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In particular, a new approach of dihomotopy involving simplicial presheaves over an appropriate small category is proposed. This small category is obtained by taking a full subcategory of a locally presentable
Quantum logic in dagger kernel categories
 Order
"... This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial inject ..."
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Cited by 9 (9 self)
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This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical/ordertheoretic properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, orthomodularity, atomicity and completeness. For instance, the Sasaki hook and andthen connectives are obtained, as adjoints, via the existentialpullback adjunction between fibres. 1