In Awodey's Category Theory, p233 of 2nd ed. (or p205 of 1st ed.), he states:

Indeed, the UMP of pullbacks essentially states that composition along any function α is left adjoint to pullback along α.

However, what I could only prove, is that the adjunction holds in slices categories, whereas the pullback is still defined in the raw category. I wanted to find the adjunction with categories coherent with that of the pullback, but I failed.

Here is what I mean.

First, let name objects and arrows according to the figure below:

Let $\mathrm E$ be the category where objects belong. In this category, the pullback of $\alpha$ along $h$ is $\alpha'$ and $h'$ with the universal property of pullbacks.

To investigate how pullback's left adjoint can be $\alpha$ composition, we should search for the natural isomorphism:

$$ \begin{array}{r c l} \alpha \circ f & \rightarrow & h \\\\ \hline f & \rightarrow & pullback(\alpha, h) \\ \end{array} $$

where we saw that $pullback_{\mathrm E}(\alpha,h)$ is the couple of arrows of $\mathrm E^1$ $(\alpha',h')$. But to be consistent with the fact that $f$ is a sole arrow and not a couple, we shoud see $pullback_{\mathrm E}(\alpha,h)$ as the result of applying $\alpha$-pullback operation to $h$, so this is $pullback_{\mathrm E}(\alpha)(h)=h'$.

But, then, $\alpha \circ f$, $h$, $f$ and $h'$ are all arrows: really, if we search for arrows between those arrows, we are not in ${\mathrm E}$ anymore, we are in some kind of arrow category. However, both the couples $(\alpha \circ f, h)$ and $(f, pullback_{\mathrm E}(\alpha)(h))$ are couples where the two arrows have the same codomain. And it suffices to take slice-category definitions of arrow. Then we have $g : \alpha \circ f \rightarrow h$ and $u : f \rightarrow h'$, that will let the natural isomorphism fulfill the definition of the universal property for pullbacks.

So, finally, the natural isomorphism would be meant:

$Hom_{Slice_{C}}(\alpha\circ f, h) \cong_{(f,h)} Hom_{Slice_{A}}(f,pullback_{\mathbf E}(\alpha)(h))$

Somebody could help me see why the pullback and the adjunction do not live in the same categories? Is that what Awodey meant, or is there something wrong in my understanding?

  • 3
    $\begingroup$ Awodey definitely meant the adjunction between those functors defined on the slice categories. $\endgroup$ Nov 22 '15 at 4:22
  • $\begingroup$ @Omar Thanks for your comment! Then I wonder: as the pullback is not defined on a slice category (and the alpha composition neither), contrary to the adjunction, can we still speak about composition being "pullback's left adjoint"? $\endgroup$
    – Almeo Maus
    Nov 22 '15 at 4:48
  • 2
    $\begingroup$ The pullback along $\alpha$ can be thought of a functor $Slice_C \to Slice_A$ sending the object $h$ to $h'$. This is maybe not literally the same as the pullback that you are talking about, but it's so closely related you should forgive people for using the same name for it, @Almeo. $\endgroup$ Nov 22 '15 at 4:54
  • $\begingroup$ @Omar Ok, you cleared my doubts. Thank you so much! Do you want to write an answer? $\endgroup$
    – Almeo Maus
    Nov 22 '15 at 5:05

Awodey did mean the adjunction you described. composing with $\alpha$ can be thought of as a function sending morphisms with codomain $A$ to morphisms with codomain $C$, but can easily be upgraded to a functor $Slice_A \to Slice_C$ (by the way more common notation for these slices involves the name of the category, $\mathrm E$ in your example: $\mathrm E_{/A}$ or $\mathrm E \downarrow A$). The resulting functor is also called "composition with $\alpha$" without any embarrassment or resulting confusion.

Similarly, pullback along $\alpha$ can be made into a functor $Slice_C \to Slice_A$, whose function on objects sends $h$ to $h'$ in your diagram. Here too the functor is called "pullback along $\alpha$". As you figured out these two are the functors Awodey meant are adjoint. (A good exercise might be to fill in the details: what do these functors do to morphisms in the slice categories, for example?)


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