Abstract
Abstract We show that Schmitt’s restriction species (such as graphs, matroids, posets, etc.) naturally induce decomposition spaces (a.k.a. unital $2$-Segal spaces), and that their associated coalgebras are an instance of the general construction of incidence coalgebras of decomposition spaces. We introduce directed restriction species that subsume Schmitt’s restriction species and also induce decomposition spaces. Whereas ordinary restriction species are presheaves on the category of finite sets and injections, directed restriction species are presheaves on the category of finite posets and convex maps. We also introduce the notion of monoidal (directed) restriction species, which induce monoidal decomposition spaces and hence bialgebras, most often Hopf algebras. Examples of this notion include rooted forests, directed graphs, posets, double posets, and many related structures. A prominent instance of a resulting incidence bialgebra is the Butcher–Connes–Kreimer Hopf algebra of rooted trees. Both ordinary and directed restriction species are shown to be examples of a construction of decomposition spaces from certain cocartesian fibrations over the category of finite ordinals that are also cartesian over convex maps. The proofs rely on some beautiful simplicial combinatorics, where the notion of convexity plays a key role. The methods developed are of independent interest as techniques for constructing decomposition spaces.
Highlights
The notion of decomposition space was introduced in [18] as a very general framework for incidencealgebras and Mobius inversion
Let us briefly recount the abstraction steps that led to this notion, taking as starting point the classical theory of incidence algebras of locally finite posets
The second example, the Butcher–Connes–Kreimer Hopf algebra is an example of a new notion we introduce, directed restriction species, and the terminal such is the example of finite posets
Summary
The notion of decomposition space was introduced in [18] as a very general framework for incidence (co)algebras and Mobius inversion. At the same time it becomes clear that the algebraic structures can be defined and manipulated at the objective level, postponing the act of taking cardinality, and that structural phenomena can be seen at this level which are not visible at the usual ‘numerical’ level At this level of abstraction one can view the algebra of species under the Cauchy tensor product as the incidence algebra of the symmetric monoidal category of finite sets and bijections [21]. We show that every directed restriction species defines a decomposition space, and a coalgebra Instead of constructing these simplicial objects by hand, we found it worth taking a slight detour through some more abstract constructions. Monoidal directed restriction species naturally induce monoidal decomposition spaces and bialgebras Examples of this notion include rooted forests, directed graphs, posets, double posets, and many related structures. The relevant definitions and results from these papers (mostly [18]) are reviewed below as needed, to render the paper reasonably self-contained
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