190 research outputs found
Direct products and the contravariant hom-functor
We prove in ZFC that if is a (right) -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 -modules then G=0. The
result is valid even we restrict to families such that for all
A nonstandard construction of direct limit group actions
Manevitz and Weinberger proved that the existence of faithful -Lipschitz
-actions implies the existence of faithful
-Lipschitz -actions. The
-actions were constructed from suitable actions of a
sufficiently large hyperfinite cyclic group
in the sense of nonstandard
analysis. In this paper, we modify their construction, and prove that the
existence of -faithful -Lipschitz -actions implies
the existence of -faithful -Lipschitz -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
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
The Heyneman-Radford Theorem for Monoidal Categories
We prove Heyneman-Radford Theorem in the framework of Monoidal Categories
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