Pullbacks and pushforward of objects

Question

As we know the pullback, pushout in category theory is defined by filling the squares. Somehow I also learnt, actually before, pullback of a differential form on a manifold, pullback map on cohomology or pushforward measure... simply by precomposition or postcompostion. So how are these terms connected? Can I rewrite the definition pullback of a vector field as in category theory?

Answer 1

I tend to view "pullback" and "pushout" as sort of catch-all terms for constructions that behave similarly, without there necessarily being a formal definition that encompasses all usages of the terms. The pushforward of vector fields, and pushforward of measures, are similar: given $f : M \to N$ a map of manifolds, then $f_*$ defines a functor $\mathbf{Vect}_M \to \mathbf{Vect}_N$; and similarly, given a function $f : X \to Y$, the pushforward $f_*$ defines a function $\mathbf{Meas}_X \to \mathbf{Meas}_Y$. And furthermore, both are "functorial" in a sense on $f$: so we get a covariant functor $\mathbf{Vect} : \mathbf{Manifolds} \to \mathbf{Cat}$ and a covariant functor $\mathbf{Meas} : \mathbf{Sets} \to \mathbf{Sets}$. Or similarly, for pullbacks of $k$-forms on manifolds, we would get a contravariant functor $k\mathbf{-Forms} : \mathbf{Manifolds} \to \mathbf{Vect_{\mathbb{R}}}$.

The case of pullback or pushforward maps in cohomology might be a bit trickier to interpret in this way. Perhaps, you could define a "cohomology theory" on an abelian category $\mathbf{C}$ as being a family of covariant functors $H^i : \mathbf{C} \to \mathbf{Ab}$ along with the appropriate long exact sequences corresponding to short exact sequences in $\mathbf{C}$, possibly with some consistency conditions between the functorial structures of $H^i$ and the long exact sequences. Here, the pushforward maps would be the "covariant functor" parts of the definition.

Or similarly, for de Rham cohomology on manifolds, the pullback map associated to a morphism $f : M \to N$ gives a family of maps $f^* : H^i_{DR}(N) \to H^i_{DR}(M)$; and again, this is functorial in $f$, so that each $H^i_{DR}$ becomes a contravariant functor $\mathbf{Manifolds} \to \mathbf{Vect}_{\mathbb{R}}$. (And furthermore, in this case, it tends to be compatible with extra structure such as Mayer-Vietoris long exact sequences.)

As for the general categorical pullback construction? This might seem a bit artificial at first but becomes useful for example in the study of topos theory: if we have a category $\mathbf{C}$ with pullbacks and a morphism $f \in \operatorname{Hom}_{\mathbf{C}}(X, Y)$ where $X, Y \in \operatorname{Ob}(\mathbf{C})$, then the pullback construction induces a functor between slice categories $\mathbf{C}_{/Y} \to \mathbf{C}_{/X}$. (Here $\mathbf{C}_{/X}$ has as objects a pair of an object $Z$ in $\mathbf{C}$ and a morphism in $\operatorname{Hom}_{\mathbf{C}}(Z, X)$, and the morphisms from $(Z, g)$ to $(Z', g')$ are the morphisms $h \in \operatorname{Hom}_{\mathbf{C}}(Z, Z')$ such that $g' \circ h = g$.) And this functor $\mathbf{C}_{/Y} \to \mathbf{C}_{/X}$ is functorial in $f$, in the sense that $\mathbf{C}_{/-}$ roughly induces a contravariant functor $\mathbf{C} \to \mathbf{Cat}$. (Though due to the fact that the pullback is only unique up to unique isomorphism, we might need to say something along the lines that this is actually a 2-functor, rather than a regular functor. Come to think of it, the same probably also applies to $\mathbf{Vect} : \mathbf{Manifolds} \to \mathbf{Cat}$. What this means is: in these cases, instead of literally $(f\circ g)^* = g^* \circ f^*$, we instead have canonical isomorphisms of functors $(f \circ g)^* \simeq g^* \circ f^*$.)

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So, what I'd say is roughly the common theme of these examples: given a morphism $f : X \to Y$, a pullback construction is something taking objects associated in some way to $Y$ to objects associated in the same way to $X$; and usually, this should be (contravariant) functorial in $f$, so that $(f \circ g)^* = g^* \circ f^*$. Likewise, a pushforward construction is something taking objects associated in some way to $X$ to objects associated in the same way to $Y$, in a covariant functorial way, i.e. $(f \circ g)_* = f_* \circ g_*$.