Contractions and extractions on twisted bialgebras and coloured Fock functors

We introduce a notion of extraction-contraction coproduct on twisted bialgebras, that is to say bialgebras in the category of linear species. If $P$ is a twisted bialgebra, a contraction-extraction coproduct sends $P[X]$ to $P[X\mathrm{/}\sim ]\otimes P[X]$ for any finite set $X$ and any equivalence relation $\sim$ on $X$, with a coassociativity constraint and compatibilities with the product and coproduct of $P$. We prove that if $P$ is a twisted bialgebra with an extraction-contraction coproduct, then $P\circ Com$ is a bialgebra in the category of coalgebraic species, that is to say species in the category of coalgebras.We then introduce a coloured version of the bosonic Fock functor. This induces a bifunctor which associates to any bialgebra $(V,\cdot ,{\delta }_V)$ and to any twisted bialgebra $P$ with an extraction-contraction coproduct a comodule-bialgebra $F_V[P]$: this object inherits a product $m$ and two coproducts $\mathrm{\Delta }$ and $\delta$, such that $(F_V[P],m,\mathrm{\Delta })$ is a bialgebra in the category of right $(F_V[P],m,\delta )$-comodules.As an example, this is applied to the twisted bialgebra of graphs. The coloured Fock functors then allow to extend the construction of the double bialgebra of graphs to double bialgebras of graphs which vertices are decorated by elements of any bialgebra $V$. Other examples (on mixed graphs, hypergraphs, noncrossing partitions…) will be given in a series of forthcoming papers.