Certain combinatorics associated to geometry over finite fields (as first observed by Tits in the 50's), the connection of the stable homotopy groups of spheres with a "combinatorial" K-theory and, most ambitious, the Riemann hypothesis led to the philosophy of what a F1-geometry should be. In the last decade several approaches towards F1-schemes were taken by generalizing scheme theory from different aspects. In particular, it became clear that scheme theory can be mimicked to a far extent if rings are replaced by (multiplicative) monoids - the resulting objects are nowadays called M0-schemes and form the core of each F1-geometry in a certain sense. However, it became clear that a purely multiplicative approach is too restrictive.

In this talk, we introduce a new category of algebraic objects, called blueprints, that contains commutative monoids and commutative semi-rings as full subcategories. A scheme theory associated to these objects reproduces usual schemes as well as M0-schemes as full subcategories. We will give an introduction and an overview of connected problems. Besides giving a new answer to Tits' problem of Chevalley groups over F1 and other improvements concerning F1-geometry, this approach seems to be interesting beyond F1. Namely, blueprints combine well with Frederic Paugam’s viewpoint on analytic geometry. We hope that a theory of analytic blueprints ties up the connections between Huber adic spaces, Berkovich analytic spaces and tropical varieties.