If A and B are Abelian groups, then Hom(A,B) is also an Abelian group under pointwise addition of functions. In this section we will see how Hom gives rise to classes of functors. Let A denote the category of Abelian groups. A covariant functor T : A → A associates to every Abelian group A an Abelian group T (A), and for every homomorphism f : A → B a homomorphism T (f) : T (A)→ T (B), such tha...

Proof: For f ∈ Hom(X,A), i ◦ f = 0 implies (i ◦ f)(x) = 0 for all x ∈ X, and then f(x) = 0 for all x since i is injective. Thus, Hom(X,A)→ Hom(X,B) is injective, giving exactness at the left joint. Since q ◦ i = 0, any f ∈ Hom(X,A) is mapped to 0 ∈ Hom(X,C) by f → q ◦ i ◦ f . That is, the image of i ◦ − is contained in the kernel of q ◦ −. On the other hand, when g ∈ Hom(X,B) is mapped to q ◦ g...

Let B be a stable continuous trace C*-algebra with spectrum Y . We prove that the natural suspension map S, : [Co(X), B ] -+ [Co(X)@ Co(R), B @ Co(R)] is a bijection, provided that both X and Y are locally compact connected spaces whose one-point compactifications have the homotopy type of a finite CW-complex and X is noncompact. This is used to compute the second homotopy group of 9 in terms o...

We exhibit a surprising connection between the following two concepts: Brown representability which arises in stable homotopy theory, and flat covers which arise in module theory. It is shown that Brown representability holds for a compactly generated triangulated category if and only if for every additive functor from the category of compact objects into the category of abelian groups a flat c...

Here HOM is the functor category, π∗ is induced by the natural projection π : hocolimI C −→ I, 0 is the category with only one map and id maps the only object of 0 to the identity functor. A reason for using the above definitions is that taking nerves one recovers the usual homotopy (co)limits for simplicial sets, up to homotopy in the case of hocolim ([T]) and up to isomorphism in the case of ...

In this note we consider different versions of coinduction functors between categories of comodules for corings induced by a morphism of corings. In particular we introduce a new version of the coinduction functor in the case of locally projective corings as a composition of suitable “Trace” and “Hom” functors and show how to derive it from a more general coinduction functor between categories ...

In this note we consider different versions of coinduction functors between categories of comodules for corings induced by a morphism of corings. In particular we introduce a new version of the coinduction functor in the case of locally projective corings as a composition of suitable “Trace” and “Hom” functors and show how to derive it from a more general coinduction functor between categories ...

Here HOM is the functor category, π∗ is induced by the natural projection π : hocolimI C −→ I, 0 is the category with only one map and id maps the only object of 0 to the identity functor. A reason for using the above definitions is that taking nerves one recovers the usual homotopy (co)limits for simplicial sets, up to homotopy in the case of hocolim ([T]) and up to isomorphism in the case of ...

Necessary and sufficient conditions are given for the EilenbergMoore comparison functor <1> arising from a functor U (having a left adjoint) to be a Galois connection in the sense of J. R. Isbell, in which case the functor U is said to be of subdescent type. These conditions, when applied to a contravariant hom-functor U = C(-, B) : C°p -» Set, read like a kind of functional completeness axiom ...

The object of this paper is to prove that the standard categories in which homotopy theory is done, such as topological spaces, simplicial sets, chain complexes of abelian groups, and any of the various good models for spectra, are all homotopically self-contained. The left half of this statement essentially means that any functor that looks like it could be a tensor product (or product, or sma...