In this article we show that every functor has a factorization into an initial functor followed by a discrete O-fibration and that this factorization is functorial. Size considerations will be ignored but may be easily filled in; we assume the existence of a category of sets large enough to dwarf any given finite number of categories. There is an analogy between the category Set of sets and the...

The final coalgebra for the finite power-set functor was described by Worrell who also proved that the final chain converges in ω + ω steps. We describe the step ω as the set of saturated trees, a concept equivalent to the modally saturated trees introduced by K. Fine in the 1970s in his study of modal logic. And for the bounded power-set functors Pλ, where λ is an infinite regular cardinal, we...

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 ...

As a practical foundation for a homotopy theory of abstract spacetime, we propose a convenient category S , which we show to extend a category of certain compact partially ordered spaces. In particular, we show that S ′ is Cartesian closed and that the forgetful functor S →T ′ to the category T ′ of compactly generated spaces creates all limits and colimits.

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...

Recall that for a triangulated category T , a Bousfield localization is an exact functor L : T → T which is coaugmented (there is a natural transformation Id → L; sometimes L is referred to as a pointed endofunctor) and idempotent (there is a natural isomorphism Lη = ηL : L → LL). The kernel ker(L) is the collection of objects X such that LX = 0. If T is closed under coproducts, it’s a localizi...