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We introduce a notion of extraction-contraction coproduct on twistedbialgebras, 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$P[X/\sim]\otimes P[X]$ for any finite set $X$ and any equivalence relation$\sim$ on $X$, with a coassociativity constraint and compatibilities with theproduct and coproduct of $P$. We prove that if $P$ is a twisted bialgebra withan extraction-contraction coproduct, then $P\circ Com$ is a bialgebra in thecategory of coalgebraic species, that is to say species in the category ofcoalgebras.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 acomodule-bialgebra $F_V[P]$: this object inherits a product $m$ and twocoproducts $\Delta$ and $\delta$, such that $(F_V[P],m,\Delta)$ is a bialgebrain the category of right $(F_V[P],m,\delta)$-comodules.As an example, this isapplied to the twisted bialgebra of graphs. The coloured Fock functors thenallow to extend the construction of the double bialgebra of graphs to doublebialgebras 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.

Contractions and extractions on twisted bialgebras and coloured Fock functors