We give a precise characterization for when the models of the tensor product of sketches are structurally isomorphic to the models of either sketch in the models of the other. For each base category K call the just mentioned property (sketch) K-multilinearity. Say that two sketches are K-compatible with respect to base category K just in case in each K-model, the limits for each limit specifica...

Let X = ΣY be a suspension. A question in homotopy theory is how to decompose the n-fold self smash product X into a wedge of spaces. Consider the set of homotopy classes [X, X]. Let the symmetric group Sn act on X by permuting positions. So for each σ ∈ Sn there is a map σ : X → X. This gives a function θ : Sn → [X, X]. By assuming that X is a suspension, [X, X] is an abelian group and so ther...

The internal size of an objectM inside a given category is, roughly, the least infinite cardinal λ such that any morphism from M into the colimit of a λ+-directed system factors through one of the components of the system. The existence spectrum of a category is the class of cardinals λ such that the category has an object of internal size λ. We study the existence spectrum in μ-abstract elemen...

Cofiltered diagrams of spectra, also called pro-spectra, have arisen in diverse areas, and to date have been treated in an ad hoc manner. The purpose of this paper is to systematically develop a homotopy theory of pro-spectra and to study its relation to the usual homotopy theory of spectra, as a foundation for future applications. The surprising result we find is that our homotopy theory of pr...

Category theory provides an excellent foundation for studying structured speciica-tions and their composition. For example, theories can be structured together in a diagram, and their composition can be obtained as a colimit. There is, however, a growing awareness, both in theory and in speciication practice, that structured theories should not be viewed just as the \scaaolding" used to build u...

We prove that if M is a CW-complex, then the homotopy type of the skeletal filtration of M does not depend on the cell decomposition of M up to wedge products with n-disks D, when the later are given their natural CW-decomposition with unique cells of order 0, (n − 1) and n; a result resembling J.H.C. Whitehead’s work on simple homotopy types. From the Colimit Theorem for the Fundamental Crosse...

We study the behavior of the Nil-subgroups of K-groups under localization. As a consequence we obtain that the relative assembly map from the family of finite subgroups to the family of virtually cyclic subgroups is rationally an isomorphism. Combined with the equivariant Chern character we obtain a complete computation of the rationalized source of the K-theoretic assembly map in terms of grou...

Although numerous contributions from divers authors, over the past fifteen years or so, have brought enriched category theory to a developed state, there is still no connected account of the theory, or even of a substantial part of it. As the applications of the theory continue to expand - some recent examples are given below - the lack of such an account is the more acutely felt. The present b...

This paper shows how systems can be built from their component parts with specified sharing. Its principle contribution is a modular language for configuring systems. A configuration is a description in the new language of how a system is constructed hierarchically from specifications of its component parts. Category theory has been used to represent the composition of specifications that share...

Given a manifold X , the set of manifold structures on X × ∆ relative to the boundary can be viewed as the k-th homotopy group of a space S̃s(X). This space is called the block structure space of X . Free involutions on spheres are in one-to-one correspondence with manifold structures on real projective spaces. We generalize Wall’s join construction for free involutions on spheres to define a fu...