  Our aim is to introduce and advocate non-$\Sigma$ (non-symmetric) modular
operads. While ordinary modular operads were inspired by the structure of the
moduli space of stable complex curves, non-$\Sigma$ modular operads model
surfaces with open strings outputs. An immediate application of our theory is a
short proof that the modular envelope of the associative operad is the
linearization of the terminal operad in the category of non-$\Sigma$ modular
operads. This gives a succinct description of this object that plays an
important role in open string field theory. We also sketch further perspectives
of the presented approach.

Algebraic Topology;Geometric Topology