Results 1  10
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12
New Model Categories From Old
 J. Pure Appl. Algebra
, 1995
"... . We review Quillen's concept of a model category as the proper setting for defining derived functors in nonabelian settings, explain how one can transport a model structure from one category to another by mean of adjoint functors (under suitable assumptions), and define such structures for ca ..."
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. We review Quillen's concept of a model category as the proper setting for defining derived functors in nonabelian settings, explain how one can transport a model structure from one category to another by mean of adjoint functors (under suitable assumptions), and define such structures for categories of cosimplicial coalgebras. 1. Introduction Model categories, first introduced by Quillen in [Q1], have proved useful in a number of areas  most notably in his treatment of rational homotopy in [Q2], and in defining homology and other derived functors in nonabelian categories (see [Q3]; also [BoF, BlS, DwHK, DwK, DwS, Goe, ScV]). From a homotopy theorist's point of view, one interesting example of such nonabelian derived functors is the E 2 term of the mod p unstable Adams spectral sequence of Bousfield and Kan. They identify this E 2 term as a sort of Ext in the category CA of unstable coalgebras over the mod p Steenrod algebra (see x7.4). The original purpose of this note w...
Functorial Factorization, Wellpointedness and Separability
"... A functorial treatment of factorization structures is presented, under extensive use of wellpointed endofunctors. Actually, socalled weak factorization systems are interpreted as pointed lax indexed endofunctors, and this sheds new light on the correspondence between reflective subcategories and f ..."
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A functorial treatment of factorization structures is presented, under extensive use of wellpointed endofunctors. Actually, socalled weak factorization systems are interpreted as pointed lax indexed endofunctors, and this sheds new light on the correspondence between reflective subcategories and factorization systems. The second part of the paper presents two important factorization structures in the context of pointed endofunctors: concordantdissonant and inseparableseparable.
Stable Homotopy of Algebraic Theories
 Topology
, 2001
"... The simplicial objects in an algebraic category admit an abstract homotopy theory via a Quillen model category structure. We show that the associated stable homotopy theory is completely determined by a ring spectrum functorially associated with the algebraic theory. For several familiar algebraic t ..."
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Cited by 12 (1 self)
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The simplicial objects in an algebraic category admit an abstract homotopy theory via a Quillen model category structure. We show that the associated stable homotopy theory is completely determined by a ring spectrum functorially associated with the algebraic theory. For several familiar algebraic theories we can identify the parameterizing ring spectrum; for other theories we obtain new examples of ring spectra. For the theory of commutative algebras we obtain a ring spectrum which is related to AndreH}Quillen homology via certain spectral sequences. We show that the (co)homology of an algebraic theory is isomorphic to the topological Hochschild (co)homology of the parameterizing ring spectrum. # 2000 Elsevier Science Ltd. All rights reserved. MSC: 55U35; 18C10 Keywords: Algebraic theories; Ring spectra; AndreH}Quillen homology; #spaces The original motivation for this paper came from the attempt to generalize a rational result about the homotopy theory of commutative rings. For...
Free Products of Higher Operad Algebras
, 909
"... Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product of 2categories. In this paper we continue the developments of [3] and [2] by understanding the natural generalisations of Gray’s little brother, ..."
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Cited by 2 (2 self)
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Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product of 2categories. In this paper we continue the developments of [3] and [2] by understanding the natural generalisations of Gray’s little brother, the funny tensor product of categories. In fact we exhibit for any higher categorical structure definable by an noperad in the sense of Batanin [1], an analogous tensor product which forms a symmetric monoidal closed structure
Ktheory and generalized free products of Salgebras: Localization methods
, 1999
"... A generalized free product diagram of Salgebras is a generalization and stabilization of the diagram of group rings arising from a Seifertvan Kampen situation. Our eventual goal is to obtain a description of the algebraic Ktheory of the \large" algebra in a generalized free product diagram i ..."
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A generalized free product diagram of Salgebras is a generalization and stabilization of the diagram of group rings arising from a Seifertvan Kampen situation. Our eventual goal is to obtain a description of the algebraic Ktheory of the \large" algebra in a generalized free product diagram in terms of the Ktheories of the three smaller algebras. We rst provide foundational material on generalized free product diagrams of Salgebras and associated categories of MayerVietoris presentations. We show that the categories of MayerVietoris presentations are categories with cobrations, weak equivalences, and mapping cylinders. In particular, the hypotheses of the \generic bration theorem" of Waldhausen (Algebraic Ktheory of spaces, Lecture Notes in Math. 1126(1985), 318419) are satised for two fundamental notions of weak equivalence, and there is, therefore, a threeterm bration sequence up to homotopy in which the Ktheory of MayerVietoris presentations with respect to the ne ...
An algebraic view of program composition
 Algebraic Methodology and Software Technology, number 1548 in Lect. Notes Comp. Sci
, 1998
"... Abstract. We propose a general categorical setting for modeling program composition in which the callbyvalue and callbyname disciplines fit as special cases. Other notions of composition arising in denotational semantics are captured in the same framework: our leading examples are nondeterminist ..."
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Abstract. We propose a general categorical setting for modeling program composition in which the callbyvalue and callbyname disciplines fit as special cases. Other notions of composition arising in denotational semantics are captured in the same framework: our leading examples are nondeterministic callbyneed programs and nonstrict functions with side effects. Composition of such functions is treated in our framework with the same degree of abstraction that Moggi’s categorical approach based on monads allows in the treatment of callbyvalue programs. By virtue of such abstraction, interesting program equivalences can be validated axiomatically in mathematical models obtained by means of modular constructions. 1
GENERALIZED HOPF MODULES FOR BIMONADS
"... Abstract. Bruguières, Lack and Virelizier have recently obtained a vast generalization of Sweedler’s Fundamental Theorem of Hopf modules, in which the role of the Hopf algebra is played by a bimonad. We present an extension of this result which involves, in addition to the bimonad, a comodulemonad ..."
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Abstract. Bruguières, Lack and Virelizier have recently obtained a vast generalization of Sweedler’s Fundamental Theorem of Hopf modules, in which the role of the Hopf algebra is played by a bimonad. We present an extension of this result which involves, in addition to the bimonad, a comodulemonad and a algebracomonoid over it. As an application we obtain a generalization of another classical theorem from the Hopf algebra literature, due to Schneider, which itself is an extension of Sweedler’s result (to the setting of Hopf Galois extensions).
TENSORS, MONADS AND ACTIONS Dedicated to the memory of Pawel Waszkiewicz
"... We exhibit sufficient conditions for a monoidal monad T on a monoidal ..."
M. PaulAndré MELLIES Rapporteur et Examinateur
, 2013
"... présentée et soutenue par Kruna SEGRT Morita theory in enriched context ..."
Categorical Term Rewriting:
, 1997
"... Abstract Term rewriting systems are widely used throughout computer science as they provide an abstract model of computation while retaining a comparatively simple syntax and semantics. In order to reason within large term rewriting systems, structuring operations are used to build large term rewrit ..."
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Abstract Term rewriting systems are widely used throughout computer science as they provide an abstract model of computation while retaining a comparatively simple syntax and semantics. In order to reason within large term rewriting systems, structuring operations are used to build large term rewriting systems from smaller ones. Of particular interest is whether key properties are modular, that is, if the components of a structured term rewriting system satisfy a property, then does the term rewriting system as a whole? A body of literature addresses this problem, but most of the results and proofs depend on strong syntactic conditions and do not easily generalize. Although many specific modularity results are known, a coherent framework which explains the underlying principles behind these results is lacking. This thesis posits that part of the problem is the usual, concrete and syntaxoriented semantics of term rewriting systems, and that a semantics is needed which on the one hand elides unnecessary syntactic details but on the other hand still possesses enough expressive power to model the key concepts arising from the term structure, such as substitutions, layers, redexes etc. Drawing on the concepts of category theory, such a semantics is proposed, based on the concept of a monad, generalising the very elegant treatment of equational presentations in category theory. The theoretical basis of this work is the theory of enriched monads. It is shown how structuring operations are modelled on the level of monads, and that the semantics is compositional (it preserves the structuring operations). Modularity results can now be obtained directly at the level of combining monads without recourse to the syntax at all. As an application and demonstration of the usefulness of this approach, two modularity results for the disjoint union of two term rewriting systems are proven, the modularity of confluence (Toyama's theorem) and the modularity of strong normalization for a particular class of term rewriting systems (noncollapsing term rewriting systems). The proofs in the categorical setting provide a mild generalisation of these results.