An "almost" full embedding of the category of graphs into the category of groups

作者：Przezdziecki Adam J^*

来源：Advances in Mathematics, 2010, 225(4): 1893-1913.

DOI：10.1016/j.aim.2010.04.015

• Category of groups
• Localization
• Large cardinals

We construct a functor F :graphs -> groups which is faithful and "almost" full, in the sense that every nontrivial group homomorphism FX -> FY is a composition of an inner automorphism of FY and a homomorphism of the form Ff, for a unique map of graphs f: X -> Y. When F is composed with the Eilenberg-Mac Lane space construction K (FX, 1) we obtain an embedding of the category of graphs into the unpointed homotopy category which is full up to null-homotopic maps.
We provide several applications of this construction to localizations (i.e. idempotent functors); we show that the questions:
(1) Is every orthogonality class reflective?
(2) Is every orthogonality class a small-orthogonality class?
have the same answers in the category of groups as in the category of graphs. In other words they depend on set theory: (1) is equivalent to weak Vopenka's principle and (2) to Vopenka's principle. Additionally, the second question, considered in the homotopy category, is also equivalent to Vopenka's principle.

• 出版日期2010-11-10