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 complex-valued 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

Abstract. Propositions 4.2 and 4.3 of the author’s article (Theory Appl. Categ. 14 (2005), 451-479) 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

The Brown representability theorem gives a list of conditions for the representablity of a set-valued contravariant functor which is defined on the classical stable homotopy category. It has had many uses through the years, and has

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].

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.

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 Cuntz-Pimsner 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

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].