Simplicial sets in topology, category theory, and beyond
This is a motivated introduction to simplicial sets (ssets) with a view toward applications in higher category theory. It is suitable for students, or experts in distant branches, wishing to make a first acquaintance with the topic and to receive some pointers to current research. As such, the author begins with simplicial complexes, a more directly visual precursor to ssets, as models for topological spaces. The move to ssets is then made in two steps, by solving compatibility issues with limits and colimits, respectively. Namely, products of simplicial complexes and topological spaces are shown to be different, which motivates oriented simplicial complexes; then the latter are argued to model quotients poorly, which motivates ssets. A detailed and fully motivated exposition of ssets follows. The author then gives a quick introduction to basic category theory that reaches the definition of an sset as a presheaf on the simplex category. The singular-chains-geometric-realisation Quillen equivalence, and the fact that the singular complex of a space is a Kan complex, are discussed intuitively in low dimensions, without going into details. The paper then takes another categorical turn, embedding category theory into sset theory via the nerve construction. The definition of a quasi-category is motivated by the characterisation of nerves of categories by the version of the weak Kan property with unique lifts. The author ends by briefly mentioning some other developments that use simplicial objects. Overall, this is a worthwhile and timely addition to the literature as a well-written and undaunting source that builds up from lines and triangles to a well-motivated definition of quasi-categories, including a discussion of the topological as well as categorical aspects of the weak Kan property and its variants.
scientific article; zbMATH DE number 7629584