190 research outputs found

    Direct products and the contravariant hom-functor

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    We prove in ZFC that if GG is a (right) RR-module such that the groups \Hom_R(\prod_{i\in I}G_i,G) and \prod_{i\in I}\Hom_R(G_i,G) are naturally isomorphic for all families of RR-modules (Gi)iI(G_i)_{i\in I} then G=0. The result is valid even we restrict to families such that GiGG_i\cong G for all iIi\in I

    A nonstandard construction of direct limit group actions

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    Manevitz and Weinberger proved that the existence of faithful KK-Lipschitz Z/nZ\mathbb{Z}/n\mathbb{Z}-actions implies the existence of faithful KK-Lipschitz Q/Z\mathbb{Q}/\mathbb{Z}-actions. The Q/Z\mathbb{Q}/\mathbb{Z}-actions were constructed from suitable actions of a sufficiently large hyperfinite cyclic group Z/γZ{}^{\ast}\mathbb{Z}/\gamma{}^{\ast}\mathbb{Z} in the sense of nonstandard analysis. In this paper, we modify their construction, and prove that the existence of ε\varepsilon-faithful KK-Lipschitz GλG_{\lambda}-actions implies the existence of ε\varepsilon-faithful KK-Lipschitz limGλ\varinjlim G_{\lambda}-actions. In a similar way, we generalise Manevitz and Weinberger's result to injective direct limits of torsion groups

    An embedding theorem for adhesive categories

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    Adhesive categories are categories which have pushouts with one leg a monomorphism, all pullbacks, and certain exactness conditions relating these pushouts and pullbacks. We give a new proof of the fact that every topos is adhesive. We also prove a converse: every small adhesive category has a fully faithful functor in a topos, with the functor preserving the all the structure. Combining these two results, we see that the exactness conditions in the definition of adhesive category are exactly the relationship between pushouts along monomorphisms and pullbacks which hold in any topos.Comment: 8 page
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