arXiv subjects

Louis Carlier

Publications and source records attributed to Louis Carlier.

4 recordsLinked to original sources

arXiv2019

Incidence bicomodules, Möbius inversion, and a Rota formula for infinity adjunctions

In the same way decomposition spaces, also known as unital 2-Segal spaces, have incidence (co)algebras, and certain relative decomposition spaces have incidence (co)modules, we identify the structures that have incidence bi(co)modules: they are certain augmented double Segal spaces subject to some exactness conditions. We establish a Möbius inversion principle for (co)modules, and a Rota formula for certain more involved structures called Möbius bicomodule configurations. The most important instance of the latter notion arises as mapping cylinders of infinity adjunctions, or more generally of adjunctions between Möbius decomposition spaces, in the spirit of Rota's original formula.

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arXiv2019

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.

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arXiv2018

Möbius functions of directed restriction species and free operads, via the generalised Rota formula

We present some tools for providing situations where the generalised Rota formula of arXiv:1801.07504 applies. As an example of this, we compute the Möbius function of the incidence algebra of any directed restriction species, free operad, or more generally free monad on a finitary polynomial monad.

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arXiv2018

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 $μ= ζ\circ S$, just as in Hopf algebras. At the level of decomposition spaces, the weak antipode takes the form of a formal difference of linear endofunctors $S_{\textrm{even}} - S_{\textrm{odd}}$, and it is a refinement of the general Möbius inversion construction of Gálvez-Kock-Tonks, but exploiting the monoidal structure.

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