Full and almost full embeddings of the category of graphs into the Abelian Groups and other categories -- motivated by localizations

Adam Przeździecki
Friday, July 26, 2013 - from 17:00 to 17:25
A localization is an endofunctor $L:\mathcal{C}\longrightarrow\mathcal{C}$ with a natural transformation $\eta_X:X\to LX$ which is idempotent in the sense that for every object $X$ we have a diagram $$ \xymatrix{ X \ar[r]^\eta \ar[d]_\eta & LX \ar[d]^{\eta_{LX}} \\ LX \ar[r]_{L\eta_X} & LLX } $$ where $\eta_{LX}=L_{\eta_X}$ are isomorphisms. The talk will survey a number of embeddings of the $Graphs$ into various categories $\mathcal{C}$ which are motivated by questions about the existence of arbitrary localizations in $\mathcal{C}$. The most surprising result is a functor $$G:Graphs\longrightarrow Abelian Groups$$ which, for any graphs $X$ and $Y$, induces isomorphisms $$G_{X,Y}:\mathbb{Z}[\Hom\nolimits_{Graphs}(X,Y)] \overset{\cong}{\longrightarrow}\Hom_{Abelian Groups}(GX,GY)$$ The $\mathbb{Z}[S]$ above is the free abelian group with basis $S=\Hom(X,Y)$. The existence of such $G$ implies a negative answer to an old question of Isbell, who asked whether every full subcategory of $Abelian Groups$, which is closed under limits, is an image of a localization. (Coauthor: R. Göbel)