geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
The Albanese variety $Alb(X)$ of a projective algebraic variety $X$ with a chosen basepoint is the universal way of turning this pointed variety into an abelian variety. Moreover, the Albanese variety of the Albanese variety is the Albanese variety. Thus, taking the Albanese variety defines an idempotent monad on the category of pointed projective algebraic varieties.
By ‘variety’ let us mean a connected complete algebraic variety over an algebraically closed field. Given any variety $X$ with a chosen basepoint there is an abelian variety called the Albanese variety $Alb(X)$. This is defined by the following universal property: there is a map of pointed varieties called the Albanese map
such that any map of pointed varieties $f: X \to A$ where $A$ is abelian factors uniquely as $i_X$ followed by a map of abelian varieties (in particular, a group homomorphism):
That is:
This process defines a functor
from pointed varieties to abelian varieties which has a right adjoint
sending any abelian variety to its underlying pointed variety. The right adjoint $U$ is faithful, but more remarkably it is also full: any basepoint-preserving map of varieties between abelian varieties is automatically a group homomorphism. (A proof of this fact is outlined in the article abelian variety.) Moreover, $U$ is monadic. As a consequence the composite functor
is an idempotent monad on $Var_*$, and its algebras are the abelian varieties.
It follows that the Albanese map $i_X \colon X \to A(X)$ is the unit of the monad $T$, and $Alb(Alb(X)) \cong Alb(X)$. For more details, see the nCafé discussion Two miracles in algebraic geometry.
The Albanese variety of $X$ is dual, as an abelian variety, to its Picard variety.
For $X$ a suitably well behaved (smooth complex, projective) algebraic variety of dimension $dim(X)$, its Albanese variety is the intermediate Jacobian in degree $2 dim(X)-1$:
Is forming the Albanese variety a monad?, MathOverflow.
Patrick Walls, Intermediate Jacobians and Abel-Jacobi maps, 2012 (pdf)
<http://mathoverflow.net/questions/2548/albanese-schemes-when-does-an-initial-abelian-scheme-exist-under-a-given-sch>
nLab page on Albanese variety