TITLE:

    Algebraic geometry over symmetric monoidal 
    categories (after To\"en and Vaqui\'e)

ABSTRACT:

I outline the approach of To\"en and Vaqui\'e to algebraic 
geometry relative to a symmetric monoidal category $C$.  Let 
$Aff_C$ denote the opposite of the category of commutative 
monoids in $C$, and define a notion of Zariski topology by 
taking as open immersions the flat monos of finite presentation.
Now a scheme relative to $C$ is a sheaf on $Aff_C$ that admits
a Zariski open cover by representables.  The construction is
functorial in $C$, and base change is an important aspect of
the theory.  When $C$ is the category of abelian groups, the 
usual notion of scheme results.  The interesting new cases lie 
below $Spec \mathbb{Z}$, escaping the realm of commutative rings.
When $C$ is the category of sets we get a version of the mythical
``schemes over the field with one element'' ($\mathbb{F}_1$-schemes), 
cf.~Soul\'e, which can be base changed to $\mathbb{Z}$ to get usual 
schemes.  Examples of $\mathbb{F}_1$-schemes are toric varieties.
There is a homotopical version of the theory, relative to a $C$ 
with a Quillen model structure.  This leads to different forms of
homotopical algebraic geometry, including ``brave new schemes''.
(Everything is from To\"en-Vaqui\'e, math.AG/0509684.)