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We define a right Cartan-Eilenberg structure on the category of Kan's combinatorial spectra, and the category of sheaves of such spectra, assuming some conditions. In both structures, we use the geometric concept of homotopy equivalence as the strong equivalence. In the case of sheaves, we use local equivalence as the weak equivalence. This paper is the first step in a larger-scale program of investigating sheaves of spectra from a geometric viewpoint., Comment: To appear in Theory and Applications of Categories

A colored graph is a complete graph in which a color has been assigned to each edge, and a colorful cycle is a cycle in which each edge has a different color. We first show that a colored graph lacks colorful cycles iff it is Gallai, i.e., lacks colorful triangles. We then show that, under the operation $m\circ n\equiv m+n-2$, the omitted lengths of colorful cycles in a colored graph form a monoid isomorphic to a submonoid of the natural numbers which contains all integers past some point. We prove that several but not all such monoids are realized. We then characterize exact Gallai graphs, i.e., graphs in which every triangle has edges of exactly two colors. We show that these are precisely the graphs which can be iteratively built up from three simple colored graphs, having $2$, $4$, and $5$ vertices, respectively. We then characterize in two different ways the monochromes, i.e., the connected components of maximal monochromatic subgraphs, of exact Gallai graphs. The first characterization is in terms of their reduced form, a notion which hinges on the important idea of a full homomorphism. The second characterization is by means of a homomorphism duality.

The theory of integration over infinite-dimensional spaces is known to encounter serious difficulties. Categorical ideas seem to arise naturally on the path to a remedy. Such an approach was suggested and initiated by Segal in his pioneering article \cite{segal}. In our paper we follow his ideas from a different perspective, slightly more categorical, and strongly inspired by the point-free topology. First, we develop a general (point-free) concept of measurability (extending the standard Lebesgue integration when applying to the classical $\sigma$-algebra). Second (and here we have a major difference from the classical theory), we prove that every finite-additive function $\mu$ with values in $[0,1]$ can be extended to a measure on an abstract $\sigma$-algebra; this correspondence is functorial and yields uniqueness. As an example we show that the Segal space can be characterized by completely canonical data. Furthermore, from our results it follows that a satisfactory point-free integration arises everywhere where we have a finite-additive probability function on a Boolean algebra., Comment: Accepted for publication in Applied Categorical Structures

We characterize vertex algebras (in a suitable sense) as algebras over a certain graded co-operad. We also discuss some examples and categorical implications of this characterization., Comment: To appear in the Journal of Algebra

We extract the abstract core of finite homomorphism dualities using the techniques of Heyting algebras and (combinatorial) categories., Comment: 17 pages; v2: minor corrections