  Let $f:U\to{\mathbb A}^1$ be a regular function on a smooth scheme $U$ over a
field $\mathbb K$. Pantev, Toen, Vaquie and Vezzosi (arXiv:1111.3209,
arXiv:1109.5213) define the "derived critical locus" Crit$(f)$, an example of a
new class of spaces in derived algebraic geometry, which they call
"$-1$-shifted symplectic derived schemes".
  They show that intersections of algebraic Lagrangians in a smooth symplectic
$\mathbb K$-scheme, and stable moduli schemes of coherent sheaves on a
Calabi-Yau 3-fold over $\mathbb K$, are also $-1$-shifted symplectic derived
schemes. Thus, their theory may have applications in algebraic symplectic
geometry, and in Donaldson-Thomas theory of Calabi-Yau 3-folds.
  This paper defines and studies a new class of spaces we call "algebraic
d-critical loci", which should be regarded as classical truncations of
$-1$-shifted symplectic derived schemes. They are simpler than their derived
analogues. We also give a complex analytic version of the theory, "complex
analytic d-critical loci", and an extension to Artin stacks, "d-critical
stacks".
  In the sequels arXiv:1305.6302, arXiv:1211.3259, arXiv:1305.6428,
arXiv:1312.0090 we will define truncation functors from $-1$-shifted symplectic
derived schemes or stacks to algebraic d-critical loci or d-critical stacks,
and we will apply d-critical loci to motivic and categorified Donaldson-Thomas
theory, and to intersections of (derived) complex Lagrangians in complex
symplectic manifolds. We will show that the important structures one wants to
associate to a derived critical locus -- virtual cycles, perverse sheaves and
mixed Hodge modules of vanishing cycles, and motivic Milnor fibres -- can be
defined for oriented d-critical loci and oriented d-critical stacks.
