John Casey's A treatise on the analytical geometry of the point, line, PDF

By John Casey

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Additional resources for A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples

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Covering projections. ([33], [91], [112], [158], [184], [215], [234], [247]). Definition. Let ???????? and ???? be two topological spaces. We say that a map ???? : ???????? → ???? is a topological covering projection (covering projection for short) of ???? if, for all ???? ∈ ????, there exists an open subset ???? of ???? such that ????−1 (????) is a disjoint union of open subsets ???????? of ???????? such that ????|???????? : ???????? → ???? is a homeomorphism for all ????. The space ???????? is said to be covering space. Definitions. – We say that a map between two topological spaces, ???? : ???????? → ????, is a local homeomorphism if, for all ???????? ∈ ???????? , there exists an open subset ???? of ???????? containing ???????? such that ????(????) is an open subset of ???? and ???? : ???? → ????(????) is a homeomorphism.

Analogously we define the distinguished triangles in ????∗ (A). With these families of distinguished triangles, ????∗ (A) and ????∗ (A) are triangulated categories. Definition. Let A and B be Abelian categories and let ???? : ????∗ (A) → ????(B) be a ????-functor. A right derived functor of ???? is a ????-functor ????∗ ???? : ????∗ (A) → ????(B) together with a morphism of functors from ????(A) to ????(B) ???? : ????B ∘ ???? → ????∗ ???? ∘ ????A with the following universal property: if is a ????-functor and ???? : ????∗ (A) → ????(B) ???? : ????B ∘ ???? → ???? ∘ ????A is a morphism of functors, then there exists a unique morphism ???? : ????∗ ???? → ???? such that ???? = (???? ∘ ????A ) ∘ ????.

Observe that, if ???????????? is a point of ???????? such that ????(???????????? ) = ???? and ???? is the image through ???? of a path from ???????????? to ???????? , then ????∗ (????1 (???????? , ???????????? )) = ????−1 ????????. Thus, in the assumptions of the theorem above, there exists a pointed covering homeomorphism between (???????? , ???????? ) and (???????? , ???????????? ) if and only if ????????????−1 = ????. In particular, if ???? is normal, the group of covering homeomorphisms of ???????? on ???? is transitive on ????−1 (????) and is isomorphic to ????1 (????, ????)/????. Note. In the case of two complex manifolds, “(ramified) covering projections” sometimes stands for holomorphic surjective maps between complex manifolds of the same dimension.

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A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples by John Casey


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