Abstract:We summarize several results about the regular coverings and the fundamental groupoids of Alexandroff spaces. In particular, we show that the fundamental groupoid of an Alexandroff space X is naturally isomorphic to the localization, at its set of morphisms, of the thin category associated to the set X considered as a preordered set with the specialization preorder. We also show that the regular coverings of an Alexandroff space X are represented by certain morphism-inverting functors with domain X, extending … Show more
“…Recall that a local coefficient system on a topological space B is a functor Π(B) → Ab where Π(B) is the fundamental groupoid of B. If B is a poset, then Π(B) can be canonically identified with the localization of B with respect to all its morphisms ( [1]) and therefore every morphism-inverting functor B → Ab can be factored uniquely as B → Π(B) → Ab. Hence there is a one-to-one correspondence between morphism-inverting functors from a poset B to Ab and local coefficient systems on B.…”
We study homology groups of posets with functor coefficients and apply our results to give a novel approach to study Khovanov homology of knots and related homology theories.
“…Recall that a local coefficient system on a topological space B is a functor Π(B) → Ab where Π(B) is the fundamental groupoid of B. If B is a poset, then Π(B) can be canonically identified with the localization of B with respect to all its morphisms ( [1]) and therefore every morphism-inverting functor B → Ab can be factored uniquely as B → Π(B) → Ab. Hence there is a one-to-one correspondence between morphism-inverting functors from a poset B to Ab and local coefficient systems on B.…”
We study homology groups of posets with functor coefficients and apply our results to give a novel approach to study Khovanov homology of knots and related homology theories.
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