In this survey, the author gives a concise overview of the definition and most important properties of the invariants mentioned in its title. It is intended for a reader unfamiliar with these notions. After presenting the definition and basic properties of sectional categories of fibrations, the author discusses Lusternik-Schnirelmann (LS) categories. He carries out their basic properties, including their lower bounds by cup lengths and the more recent notion of category weights, and provides computations of LS categories for some basic examples like spheres and tori. Subsequently, the author turns his focus to topological complexity, including a discussion of higher or sequential topological complexities. He again gives an overview over their main properties and provides explicit values for basic examples as well. In the end, the author discusses the behaviour of the sequences \(\{\mathrm{TC}_n(X)\}_n\) of higher topological complexities of a space and provides some open questions about them that are current topics of research.
The survey is written in a very concise and well-organized style which should help the reader in understanding the key points of the theory. It gives an overview over basic notions and presents some connections to more recent results. The author generally omits proofs and only occasionally indicates lines of argument, but provides a reference to a proof of each assertion.
It is worth pointing out that the author pays attention to historical accuracy and provides references to the original sources of several classical results like the lower bound of LS category by cup length.
For the entire collection see [Zbl 1511.55001].