algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
Given any kind of cohomology theory, it is contravariant functor on some category $\mathcal{C}$ of spaces of sorts. For example, for ordinary cohomology or Whitehead-generalized cohomology theories this functor goes from pointed CW-complexes $X$ (or more generally: CW-pairs) to graded abelian groups $E^\bullet(X)$.
The value of this functor on any morphism $f \;\colon\; X \to Y$ is called the pullback in $E$-cohomology $f^\ast \;\colon\; E^\bullet(Y) \to E^\bullet(Y)$.
Notice that, a priori, this is not related to the notion of pullback in the sense of a cospan-shaped limit in some category, though for good enough “geometric cycles” for $E$-cohomology the notions may actually agree. For example, pullback in $G$-non-abelian cohomology is given by forming the pullback bundles of the $G$-principal bundles which are classified by the given cohomology classes.
Notions of pullback:
pullback, fiber product (limit over a cospan)
lax pullback, comma object (lax limit over a cospan)
(∞,1)-pullback, homotopy pullback, ((∞,1)-limit over a cospan)