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Homotopy theory of C ∗algebras
, 2008
"... In this work we construct from ground up a homotopy theory of C ∗algebras. This is achieved in parallel with the development of classical homotopy theory by first introducing an unstable model structure and second a stable model structure. The theory makes use of a full fledged import of homotopy t ..."
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In this work we construct from ground up a homotopy theory of C ∗algebras. This is achieved in parallel with the development of classical homotopy theory by first introducing an unstable model structure and second a stable model structure. The theory makes use of a full fledged import of homotopy theoretic techniques into the subject of C ∗algebras. The spaces in C ∗homotopy theory are certain hybrids of functors represented by C ∗algebras and spaces studied in classical homotopy theory. In particular, we employ both the topological circle and the C ∗algebra circle of complexvalued continuous functions on the real numbers which vanish at infinity. By using the inner workings of the theory, we may stabilize the spaces by forming spectra and bispectra with respect to either one of these circles or their tensor product. These stabilized spaces or spectra are the objects of study in stable C ∗homotopy theory. The stable homotopy category of C ∗algebras gives rise to invariants such as stable homotopy groups and bigraded cohomology and homology theories. We
GENERALIZED BROWN REPRESENTABILITY IN HOMOTOPY CATEGORIES: ERRATUM
"... Abstract. Propositions 4.2 and 4.3 of the author’s article (Theory Appl. Categ. 14 (2005), 451479) are not correct. We show that their use can be avoided and all remaining results remain correct 1. Propositions 4.2 and 4.3 of the author’s [3] are not correct and I am grateful to J. F. Jardine for p ..."
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Abstract. Propositions 4.2 and 4.3 of the author’s article (Theory Appl. Categ. 14 (2005), 451479) are not correct. We show that their use can be avoided and all remaining results remain correct 1. Propositions 4.2 and 4.3 of the author’s [3] are not correct and I am grateful to J. F. Jardine for pointing it out. In fact, consider the diagram D sending the one morphism category to the point ∆0 in the homotopy category Ho(SSet) of simplicial sets. The standard weak colimit of D is the standard weak coequalizer ∆0 id id �∆0 This weak coequalizer is the homotopy pushout
Representability theorems for presheaves of spectra
, 2010
"... The Brown representability theorem gives a list of conditions for the representablity of a setvalued contravariant functor which is defined on the classical stable homotopy category. It has had many uses through the years, and has ..."
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The Brown representability theorem gives a list of conditions for the representablity of a setvalued contravariant functor which is defined on the classical stable homotopy category. It has had many uses through the years, and has
BROWN REPRESENTABILITY FOLLOWS FROM ROSICKY
"... We prove that the dual of a well generated triangulated category satisfies Brown representability, as long as there is a combinatorial model. This settles the major open problem in [13]. We also prove that Brown representability holds for nondualized well generated categories, but that only amounts ..."
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We prove that the dual of a well generated triangulated category satisfies Brown representability, as long as there is a combinatorial model. This settles the major open problem in [13]. We also prove that Brown representability holds for nondualized well generated categories, but that only amounts to the fourth known proof of the fact. The proof depends crucially on a new result of Rosicky [14].
LEFT DETERMINED MODEL STRUCTURES FOR LOCALLY PRESENTABLE CATEGORIES
, 901
"... Abstract. We extend a result of Cisinski on the construction of cofibrantly generated model structures from (Grothendieck) toposes to locally presentable categories and from monomorphism to more general cofibrations. As in the original case, under additional conditions, the resulting model structure ..."
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Abstract. We extend a result of Cisinski on the construction of cofibrantly generated model structures from (Grothendieck) toposes to locally presentable categories and from monomorphism to more general cofibrations. As in the original case, under additional conditions, the resulting model structures are ”left determined ” in the sense of Rosick´y and Tholen. 1.
unknown title
, 906
"... Abstract. Every separable nondegenerate C ∗correspondence over a commutative C ∗algebra with discrete spectrum is isomorphic to a graph correspondence. Let E be a directed graph with vertex set V, edge set E1, and range and source maps r, s: E1 → V. The graph correspondence is the nondegenerate C∗ ..."
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Abstract. Every separable nondegenerate C ∗correspondence over a commutative C ∗algebra with discrete spectrum is isomorphic to a graph correspondence. Let E be a directed graph with vertex set V, edge set E1, and range and source maps r, s: E1 → V. The graph correspondence is the nondegenerate C∗correspondence XE over c0(V) defined by XE = { ξ: E 1 → C ∣ ∑ the map v ↦ → ξ(e)  2 is in c0(V) }, s(e)=v with module actions and c0(V)valued inner product given by (a · ξ · b)(e) = a(r(e)) ξ(e) b(s(e)) and 〈ξ, η〉(v) = ∑ ξ(e)η(e) s(e)=v for a, b ∈ c0(V), ξ, η ∈ XE, e ∈ E 1, and v ∈ V. (We cite [3] as a general reference on graph algebras, and [3, Chapter 8] in particular for graph correspondences.) A graph correspondence XE contains at least as much information about E as the graph algebra C ∗ (E), since the CuntzPimsner algebra OXE is isomorphic to C ∗ (E) ([3, Example 8.13]; see also [1, Example 1, p. 4303]). Indeed, many properties of E are directly reflected in properties of XE: for example, XE is full in the sense that span〈XE, XE 〉 = c0(V) if and only if the graph E has no sinks; the homomorphism c0(V) → L(XE) associated to the left module operation maps into the compacts K(XE) if and only if no vertex receives infinitely many edges; c0(V) maps faithfully into L(XE) if and only if E has no sources. In this paper we further investigate this connection between graphs and C ∗correspondences. In Section 1, our main result (Theorem 1) expands and elaborates on a remark of Schweizer ([5, §1.6]; see also [2, Chapter 1, Example 2]) to the effect that every C ∗correspondence over C ∗algebras c0(X) and c0(Y) is unitarily equivalent to a correspondence
BROWN REPRESENTABILITY FOR SPACEVALUED FUNCTORS
, 707
"... Abstract. In this paper we prove two theorems which resemble the classical cohomological and homological Brown representability theorems. The main difference is that our results classify contravariant functors from spaces to spaces up to weak equivalence of functors. In more detail, we show that eve ..."
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Abstract. In this paper we prove two theorems which resemble the classical cohomological and homological Brown representability theorems. The main difference is that our results classify contravariant functors from spaces to spaces up to weak equivalence of functors. In more detail, we show that every contravariant functor from spaces to spaces which takes coproducts to products up to homotopy, and takes homotopy pushouts to homotopy pullbacks is naturally weekly equivalent to a representable functor. The second representability theorem states: every contravariant continuous functor from the category of finite simplicial sets to simplicial sets taking homotopy pushouts to homotopy pullbacks is equivalent to the restriction of a representable functor. This theorem may be considered as a contravariant analog of Goodwillie’s classification of linear functors [15].
Accessible categories and . . .
, 2007
"... Accessible categories have recently turned out to be useful in homotopy theory. This text is prepared as notes for a series of lectures at ..."
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Accessible categories have recently turned out to be useful in homotopy theory. This text is prepared as notes for a series of lectures at
HOMOTOPICAL EQUIVALENCE OF COMBINATORIAL AND CATEGORICAL SEMANTICS OF PROCESS ALGEBRA
, 711
"... Abstract. It is possible to translate a modified version of K. Worytkiewicz’s combinatorial semantics of CCS (Milner’s Calculus of Communicating Systems) in terms of labelled precubical sets into a categorical semantics of CCS in terms of labelled flows using a geometric realization functor. It turn ..."
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Abstract. It is possible to translate a modified version of K. Worytkiewicz’s combinatorial semantics of CCS (Milner’s Calculus of Communicating Systems) in terms of labelled precubical sets into a categorical semantics of CCS in terms of labelled flows using a geometric realization functor. It turns out that a satisfactory semantics in terms of flows requires to work directly in their homotopy category since such a semantics requires noncanonical choices for constructing cofibrant replacements, homotopy limits and homotopy colimits. No geometric information is lost since two precubical sets are isomorphic if and only if the associated flows are weakly equivalent. The interest of the categorical semantics is that combinatorics totally disappears. Last but not least, a part of the categorical semantics of CCS goes down to a pure homotopical semantics of CCS using A. Heller’s privileged weak limits and colimits. These results can be easily adapted to any other process algebra for any synchronization algebra. Contents
Triangulated categories and applications
, 2011
"... The best way to think of triangulated categories is as a powerful technical tool which has had amazing applications over the last 30 years. Homological algebra was first introduced quite early in the 20th century, and by the 1950s the subject was viewed as fully developed: so much so that the author ..."
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The best way to think of triangulated categories is as a powerful technical tool which has had amazing applications over the last 30 years. Homological algebra was first introduced quite early in the 20th century, and by the 1950s the subject was viewed as fully developed: so much so that the authoritative book, by Cartan and Eilenberg, dates from this decade, and until recently no one bothered to write an update. Around 1960 Grothendieck and Verdier introduced a new twist (triangulated categories), a refinement no one expected. It took about 20 years before people started realizing the power of the method, and in the last 30 years the applications have reached corners of mathematics no one would have foreseen. In organizing the conference we tried to bring in people from diverse fields, who use triangulated categories in their work, to give them a chance to interact. Many of the participants told us it was the best conference they had been to in a long lime. 2 Recent Developments and Open Problems It’s hard to know where to begin, there has been so much recent progress and it’s been all over the place. Some of the progress has been on foundational questions, such as Brown representability [27, 11, 26, 5] and understanding the Balmer spectrum [1, 2, 4] of a tensor triangulated category. There has been considerable progress in understanding tstructures on various derived categories, for example the theory of Bridgeland stability conditions [8, 9, 10]. There are several theorems about obstructions to finding model structures for triangulated categories, examples of categories without model structures [25], and theorems telling us about the existence and uniqueness of model structures for large classes of triangulated categories [28, 23]. There has been major progress in computing the Balmer spectrum [3], as well as generalizations to categories with coproducts and applications in the specific cases [6, 7]. There has also been progress on applying triangulated categories to mathematical physics [12, 20, 21, 22] and to the study of C ∗ –algebras [24]. Every bit of progress opens up a myriad new questions, the very fact that we are making so much progress is evidence of how little we know. 3