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20
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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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...
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.
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...
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).
On coalgebras over algebras
 In ”Proceedings of the Tenth Workshop on Coalgebraic Methods in Computer Science (CMCS 2010)”, Electr. Notes
"... We extend Barr’s wellknown characterization of the final coalgebra of a Setendofunctor H as the completion of its initial algebra to the EilenbergMoore category Alg(M) of algebras associated to a Setmonad M, if H can be lifted to Alg(M). As further analysis, we introduce the notion of commuting ..."
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We extend Barr’s wellknown characterization of the final coalgebra of a Setendofunctor H as the completion of its initial algebra to the EilenbergMoore category Alg(M) of algebras associated to a Setmonad M, if H can be lifted to Alg(M). As further analysis, we introduce the notion of commuting pair of endofunctors (T,H) with respect to a monad M and show that under reasonable assumptions, the final Hcoalgebra can be obtained as the completion of the free Malgebra on the initial Talgebra.
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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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
The enriched Vietoris monad on representable spaces
, 2012
"... Abstract. Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the downset monad becomes the filter monad, cocomplete ordered set translate ..."
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Abstract. Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the downset monad becomes the filter monad, cocomplete ordered set translates to continuous lattice, distributivity means disconnectedness, and so on. Curiously, the dual(?) notion of completeness does not behave as the mirror image of the one of cocompleteness; and in this paper we have a closer look at complete spaces. In particular, we construct the “upset monad ” on representable spaces (in the sense of L. Nachbin for topological spaces, respectively C. Hermida for multicategories); we show that this monad is of KockZöberlein type; we introduce and study a notion of weighted limit similar to the classical notion for enriched categories; and we describe the Kleisli category of our “upset monad”. We emphasize that these generic categorical notions and results can be indeed connected to more “classical ” topology: for topological spaces, the “upset monad ” becomes the lower Vietoris monad, and the statement “X is totally cocomplete if and only if Xop is totally complete” specialises to O. Wyler’s characterisation of the algebras of the Vietoris monad on compact Hausdorff spaces.
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