equality (definitional?, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
For $f : X \to Y$ a smooth function, a critical point is a point $x \in X$ at which the derivative $d X : T X \to T Y$ vanishes.
The collection of all critical points is also called the critical locus of $f$.
A formalization of this in cohesive geometry is at cohesive (infinity,1)-topos – infinitesimal cohesion – critical locus.
Critical loci are used to study topology in terms of Morse theory.
Critical loci of functionals on jet spaces are studied in variational calculus.