On a class of categorical topological spaces. (English. Russian original) Zbl 1186.18007

Skurikhin, E. E.

Russ. Math. Surv. 63, No. 1, 170-172 (2008); translation from Usp. Mat. Nauk. 63, No. 1, 167-168 (2008).

Summary: A categorical topological space is a generalized object of an arbitrary Grothendieck site together with the structure of its subobjects [E. E. Skurikhin, Proc. Steklov Inst. Math. 193, 187–191 (1993); translation from Tr. Mat. Inst. Steklova 193, 169–173 (1992; Zbl 0807.55006); Proc. Steklov Inst. Math. 239, 273–300 (2002); translation from Tr. Mat. Inst. Im. V. A. Steklova 239, 289–317 (2002; Zbl 1066.18010)]. In this note the results are applied to the case when a site is a lower semilattice and the Grothendieck topology is defined by a unique family. The lengths of a Chu space are determined, and their relations to the cohomological dimensions and, in particular, to the flabby dimension, are found. The cohomological characteristic of the dimension of a Noetherian space and, as a consequence, of an algebraic variety, is obtained.

MSC:

18F20	Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
18B30	Categories of topological spaces and continuous mappings (MSC2010)
55N30	Sheaf cohomology in algebraic topology
54B30	Categorical methods in general topology
54B40	Presheaves and sheaves in general topology

Citations:

Zbl 1066.18010; Zbl 0807.55006