Decomposition Space Theory and the Bing Shrinking Criterion

‘Decomposition Space Theory and the Bing Shrinking Criterion’ gives a proof of the central Bing shrinking criterion and then provides an introduction to the key notions of the field of decomposition space theory. The chapter begins by proving the Bing shrinking criterion, which characterizes when a given map between compact metric spaces is approximable by homeomorphisms. Next, it develops the elements of the theory of decomposition spaces. A key fact is that a decomposition space associated with an upper semi-continuous decomposition of a compact metric space is again a compact metric space. Decomposition spaces are key in the proof of the disc embedding theorem.

Skyscrapers Are Standard: The Details

‘Skyscrapers Are Standard: The Details’ provides a thorough and detailed proof that every skyscraper is homeomorphic to the standard 2-handle, relative to the attaching region. Results from decomposition space theory established in Part I and the constructive results from Part II are combined. The idea is to construct a subset of a skyscraper called the design, define an embedding of this subset into the standard 2-handle, and then consider the decomposition spaces obtained by quotienting out the connected components of the complement of this common subset. It is shown that the decomposition spaces are homeomorphic, and that both quotient maps are approximable by homeomorphisms. This chapter also shows that everything can be done fixing a neighbourhood of the attaching region. It is then deduced that skyscrapers are standard, as desired.

Hereditary Species as Monoidal Decomposition Spaces, Comodule Bialgebras, and Operadic Categories

We show that Schmitt’s hereditary species induce monoidal decomposition spaces and exhibit Schmitt’s bialgebra construction as an instance of the general bialgebra construction on a monoidal decomposition space. We show furthermore that this bialgebra structure coacts on the underlying restriction-species bialgebra structure so as to form a comodule bialgebra. Finally, we show that hereditary species induce a new family of examples of operadic categories in the sense of Batanin and Markl [ 4].

Embeddings of shearlet coorbit spaces into Sobolev spaces

We investigate the existence of embeddings of shearlet coorbit spaces associated to weighted mixed [Formula: see text]-spaces into classical Sobolev spaces in dimension three by using the description of coorbit spaces as decomposition spaces. This different perspective on these spaces enables the application of embedding results that allow the complete characterization of embeddings for certain integrability exponents, and thus provides access to a deeper understanding of the smoothness properties of coorbit spaces, and of the influence of the choice of shearlet groups on these properties. We give a detailed analysis, identifying which features of the dilation groups have an influence on the embedding behavior, and which do not. Our results also allow to comment on the validity of the interpretation of shearlet coorbit spaces as smoothness spaces.

Antipodes of monoidal decomposition spaces

We introduce a notion of antipode for monoidal (complete) decomposition spaces, inducing a notion of weak antipode for their incidence bialgebras. In the connected case, this recovers the usual notion of antipode in Hopf algebras. In the non-connected case, it expresses an inversion principle of more limited scope, but still sufficient to compute the Möbius function as [Formula: see text], just as in Hopf algebras. At the level of decomposition spaces, the weak antipode takes the form of a formal difference of linear endofunctors [Formula: see text], and it is a refinement of the general Möbius inversion construction of Gálvez–Kock–Tonks, but exploiting the monoidal structure.