Direct products and the contravariant hom-functor

We prove in ZFC that if $G$ is a (right) $R$-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 $R$-modules $(G_i)_{i\in I}$ then G=0. The result is valid even we restrict to families such that $G_i\cong G$ for all $i\in I$

A nonstandard construction of direct limit group actions

Manevitz and Weinberger proved that the existence of faithful $K$-Lipschitz $\mathbb{Z}/n\mathbb{Z}$-actions implies the existence of faithful $K$-Lipschitz $\mathbb{Q}/\mathbb{Z}$-actions. The $\mathbb{Q}/\mathbb{Z}$-actions were constructed from suitable actions of a sufficiently large hyperfinite cyclic group ${}^{\ast}\mathbb{Z}/\gamma{}^{\ast}\mathbb{Z}$ in the sense of nonstandard analysis. In this paper, we modify their construction, and prove that the existence of $\varepsilon$-faithful $K$-Lipschitz $G_{\lambda}$-actions implies the existence of $\varepsilon$-faithful $K$-Lipschitz $\varinjlim G_{\lambda}$-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

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