  The usual language of algebraic geometry is not appropriate for Arithmetical
geometry: addition is singular at the real prime. We developed two languages
that overcome this problem: one replace rings by the collection of "vectors" or
by bi-operads and another based on "matrices" or props. These are the two
languages of [Har17], but we omit the involutions which brings considerable
simplifications. Once one understands the delicate commutativity condition one
can proceed following Grothendieck footsteps exactly. The square matrices, when
viewed up to conjugation, give us new commutative rings with Frobenius
endomorphisms.
