  The physical world is fundamentally: (1) field-theoretic, (2) smooth, (3)
local, (4) gauged, (5) containing fermions, and last but not least: (6)
non-perturbative. Tautologous as this may sound, it is remarkable that the
mathematical notion of geometry which reflects all of these aspects -- namely,
as we will explain: "supergeometric homotopy theory" -- has received little
attention even by mathematicians and remains unknown to most physicists.
Elaborate algebraic machinery is known for perturbative field theories, but in
order to tackle the deep open questions of the subject, these will need to be
lifted to a global geometry of physics.
  Our aim in this series is, first, to introduce inclined physicists to this
theory, second to fill mathematical gaps in the existing literature, and
finally to rigorously develop the full power of supergeometric homotopy theory
and apply it to the analysis of fermionic (not necessarily super-symmetric)
field theories.
  To warm up, in this first part we explain how classical bosonic Lagrangian
field theory (variational Euler-Lagrange theory) finds a natural home in the
"topos of smooth sets", thereby neatly setting the scene for the higher
supergeometry discussed in later parts of the series. This introductory
material will be largely known to a few experts but has never been
comprehensively laid out before. A key technical point we make is to regard jet
bundle geometry systematically in smooth sets instead of just its subcategories
of diffeological spaces or even Fr\'echet manifolds -- or worse simply as a
formal object. Besides being more transparent and powerful, it is only on this
backdrop that a reasonable supergeometric jet geometry exists, needed for
satisfactory discussion of any field theory with fermions.
