1/9/2024 0 Comments Locally presentable categoryThe surprising result we find is that our homotopy theory of pro-spectra is Quillen equivalent to the opposite of the homotopy theory of spectra. 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. We introduce the concept of a class-cofibrantly generated model category, which is a model category generated by classes of cofibrations and trivial cofibrations satisfying some reasonable assumptions.Ĭofiltered diagrams of spectra, also called pro-spectra, have arisen in diverse areas, and to date have been treated in an ad hoc manner. The examples above suggest a natural extension of the framework of cofibrantly generated model categories. We also formulate a non-functorial version of the argument, which applies in two different model structures on the category of pro-spaces. Our current approach allows for construction of functorial factorizations and localizations in the equivariant model structures on diagrams of spaces and diagrams of chain complexes. Isaksen, A model structure on the category of pro-simplicial sets, Trans. Hovey, Quillen model structures for relative homological algebra, Math. Chorny, The model category of maps of spaces is not cofibrantly generated, Proc. The necessity of such a generalization arose with appearance of several important examples of model categories which were proven to be non-cofibrantly generated Papers in honour of the seventieth birthday of Professor Heinrich Kleisli (Fribourg, 2000) B. We generalize the small object argument in order to allow for its application to proper classes of maps (as opposed to sets of maps in Quillen's small object argument). Without domain constraints the model category is locally presentable. With inclusion dependencies, the instances are equivalence classes of the models. Without inclusion dependencies, the accessible category of models of the database sketch consists of all the satisfying instances and homomorphisms between them. The addition of domain constraints results in (finite limit, countable coproduct)-sketches. Such sketches include all the usual kinds of dependencies: functional, join and inclusion. Relational database schemata are shown to be sketched by finite limit sketches. Relational databases are given by database schemata, the syntax, and the class of sets of relations satisfying the syntactic constraints, called instances. As a motivating example, we consider relational databases. The Stone dualities for accessible categories and the subclass of Diers categories provide limit and colimit structuring principles for query languages and the associated database. This has been formalized in the monograph (AR1) where, instead of ''preceding'', the operations S are decom- posed into where each St -operation is required to be everywhere defined, and the definition domain of every operation s in Sp is determined by equations Def(s) over St. Locally presentable and locally generated categories It is an idea of Peter Freyd that for a description of locally presentable categories of Gabriel and Ulmer (GU) one can use essentially algebraic theories, i.e., theories of partial algebras where the definition domain of every operation is determined by equations in the ''preceding'' operations. And in the same vein Diers' locally multipresentable categories are characterized via essentially multialgebraic theories, and they are generalized to locally multigenerated categories. An analogous result for locally generated categories is obtained. A new proof of the fact that locally presentable categories precisely correspond to essentially algebraic theories is presented.
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