Some categorical aspects of simplicial complexes. (English) Zbl 0535.18007

Preuß, Gerhard

Quaest. Math. 6, 313-322 (1983).

Let Simp be the category of all abstract simplicial complexes and all simplicial maps. In this paper the author proves that Simp is strongly topological, i.e. it is a Cartesian closed topological category and final epi-sinks in Simp are hereditary. By these facts one can explain the reason why Simp is more convenient than Top for algebraic topology. Furtheremore, for each natural number n, the full subcategory \(Simp_ n\) of Simp consisting of all simplicial complexes of dimension less than or equal to n is bicoreflective in Simp and is as convenient as Simp itself. It is remarked that Simp and \(Simp_ n\) (\(n\geq 1)\) are not universal in the sense of Th. Marny [Gen. Topology Appl. 10, 175-181 (1979; Zbl 0415.54007)].

Reviewer: R.Nakagawa

MSC:

18D15	Closed categories (closed monoidal and Cartesian closed categories, etc.)
18G30	Simplicial sets; simplicial objects in a category (MSC2010)
55U10	Simplicial sets and complexes in algebraic topology
18B30	Categories of topological spaces and continuous mappings (MSC2010)
18A40	Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
55U40	Topological categories, foundations of homotopy theory
54B30	Categorical methods in general topology

Keywords:

connection theory; concrete quasi-topoi; topological category; abstract simplicial complexes; strongly topological; Cartesian closed; bicoreflective; not universal

Citations:

Zbl 0415.54007

References:

[1]	Dubuc E. J., In: Applications of Sheaves, Lecture Notes in Math. 753 pp 239– (1979)
[2]	Godement R., Topologie algébrique et théorie des faisceaux (1958)
[3]	Herrlich H., Math. Coll. Univ. Cape Town 9 pp 1– (1974)
[4]	Herrlich H., Categorical topology 1971–1981 Gen. Top. and its Rel. to Mod. Analysis and Algebra V · Zbl 0215.51501
[5]	Herrlich H., Quaest. Math. 3 pp 189– (1979) · Zbl 0407.54006 · doi:10.1080/16073606.1979.9631571
[6]	Herrlich H., Canad. J. Math. 31 pp 1059– (1979) · Zbl 0435.18003 · doi:10.4153/CJM-1979-097-7
[7]	Marny Th., Gen. Top. Appl. 10 pp 175– (1979) · Zbl 0415.54007 · doi:10.1016/0016-660X(79)90006-0
[8]	Preuss G., Quaest. Math. 2 pp 297– (1977) · doi:10.1080/16073606.1977.9632549
[9]	Preuss G., Categorical Topology, Lecture Notes in Math 719 pp 293– (1979) · doi:10.1007/BFb0065281
[10]	Preuss G., In: Categorical Aspects of Topology and Analysis, Lecture Notes in Math. 915 pp 272– (1982)
[11]	Rhineghost Y. T., Top
[12]	Salicrup G., In: Categorical Topology, Lecture Notes in Math. 719 pp 326– (1979) · doi:10.1007/BFb0065284
[13]	Spanier E. H., Algebraic Topology (1981) · doi:10.1007/978-1-4684-9322-1