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

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


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


Zbl 0415.54007
Full Text: DOI


[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
[6] Herrlich H., Canad. J. Math. 31 pp 1059– (1979) · Zbl 0435.18003
[7] Marny Th., Gen. Top. Appl. 10 pp 175– (1979) · Zbl 0415.54007
[8] Preuss G., Quaest. Math. 2 pp 297– (1977)
[9] Preuss G., Categorical Topology, Lecture Notes in Math 719 pp 293– (1979)
[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)
[13] Spanier E. H., Algebraic Topology (1981)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.