Limits of small functors

For a small category K enriched over a suitable monoidal category V, the free completion of K under colimits is the presheaf category [K*,V]. If K is large, its free completion under colimits is the V-category PK of small presheaves on K, where a presheaf is small if it is a left Kan extension of some presheaf with small domain. We study the existence of limits and of monoidal closed structures on PK.Comment: 17 page

Quantitative Algebras and a Classification of Metric Monads

Quantitative algebras are $\Sigma$-algebras acting on metric spaces, where operations are nonexpanding. We prove that for finitary signatures $\Sigma$ there is a bijective correspondence between varieties of quantitative algebras and strongly finitary monads on the category $\mathsf{Met}$ of metric spaces. For uncountable cardinals $\lambda$ there is an analogous bijection between varieties of $\lambda$-ary quantitative algebras and strongly $\lambda$-accessible monads. Moreover, we present a bijective correspondence between $\lambda$-varieties of $\Sigma$-algebras as introduced by Mardare, Panangaden and Plotkin and enriched, surjections-preserving $\lambda$-accesible monads on $\mathsf{Met}$. Finally, a bijective correspondence between generalized $\lambda$-ary varieties and enriched $\lambda$-accessible monads on $\mathsf{Met}$ in general is presented

Locally class-presentable and class-accessible categories

We generalize the concepts of locally presentable and accessible categories. Our framework includes such categories as small presheaves over large categories and ind-categories. This generalization is intended for applications in the abstract homotopy theory

The fundamental pro-groupoid of an affine 2-scheme

A natural question in the theory of Tannakian categories is: What if you don't remember \Forget? Working over an arbitrary commutative ring $R$, we prove that an answer to this question is given by the functor represented by the \'etale fundamental groupoid \pi_1(\spec(R)), i.e.\ the separable absolute Galois group of $R$ when it is a field. This gives a new definition for \'etale \pi_1(\spec(R)) in terms of the category of $R$-modules rather than the category of \'etale covers. More generally, we introduce a new notion of "commutative 2-ring" that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of $\pi_1$ for the corresponding "affine 2-schemes." These results help to simplify and clarify some of the peculiarities of the \'etale fundamental group. For example, \'etale fundamental groups are not "true" groups but only profinite groups, and one cannot hope to recover more: the "Tannakian" functor represented by the \'etale fundamental group of a scheme preserves finite products but not all products.Comment: 46 pages + bibliography. Diagrams drawn in Tik

Long-term Test Arrangement for Externally Strengthened Reinforced Concrete Elements

Methods for external strengthening of concrete use elements of very high tensional strength glued on to its tensioned surface. These elements may be of metal, carbon fibers (CFRP), glass fibers or others, usually having very good mechanical properties. However, these high-strength elements are normally attached to concrete by epoxy resins. Epoxy resins have a low Young`s modulus and therefore a higher rate of creep may have an influence on the long-term behavior of such external strengthening. In order to verify this idea experimentally a special space-saving arrangement of tests is described in this paper. Panels act as loaded beams but simultaneously as a load for the other panels in a stand. The different load magnitude acting on a different layer of panels should make it possible to study the long-term influence of the degree of shear force on the glue creep. Certainly, the glue creep may be dependent on the type of epoxy resin; therefore several epoxy resin types are included in the tests

Interaction and observation: categorical semantics of reactive systems trough dialgebras

We use dialgebras, generalising both algebras and coalgebras, as a complement of the standard coalgebraic framework, aimed at describing the semantics of an interactive system by the means of reaction rules. In this model, interaction is built-in, and semantic equivalence arises from it, instead of being determined by a (possibly difficult) understanding of the side effects of a component in isolation. Behavioural equivalence in dialgebras is determined by how a given process interacts with the others, and the obtained observations. We develop a technique to inter-define categories of dialgebras of different functors, that in particular permits us to compare a standard coalgebraic semantics and its dialgebraic counterpart. We exemplify the framework using the CCS and the pi-calculus. Remarkably, the dialgebra giving semantics to the pi-calculus does not require the use of presheaf categories

Minimization via duality

We show how to use duality theory to construct minimized versions of a wide class of automata. We work out three cases in detail: (a variant of) ordinary automata, weighted automata and probabilistic automata. The basic idea is that instead of constructing a maximal quotient we go to the dual and look for a minimal subalgebra and then return to the original category. Duality ensures that the minimal subobject becomes the maximally quotiented object