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Let $X$ and $Y$ be topological spaces, and $\varphi:X\to Y$ a continuous map. Let $\mathbf{X}$ and $\mathbf{Y}$ be the categories corresponding to the open sets of $X$ and $Y$ with arrows given by reverse inclusion. Then a presheaf on $X$ is a functor $\mathcal{F}:\mathbf{X}\to\mathbf{Set}$, and the category $\mathbf{PShv}(\mathbf{X})$ of presheaves on $X$ is just the functor category $\mathbf{Set}^\mathbf{X}$.

We obtain a functor $\Phi:\mathbf{Y}\to\mathbf{X}$ by setting $\Phi(U)=\varphi^{-1}(U)$. Then the direct image functor $I:\mathbf{PShv}(\mathbf{X})\to\mathbf{PShv}(\mathbf{Y})$ is given by setting $I(\mathcal{F})=\mathcal{F}\circ\Phi$.

However I can't seem to give such a clean construction of the inverse image functor $J:\mathbf{PShv}(\mathbf{Y})\to\mathbf{PShv}(\mathbf{X})$. I know that for a presheaf $\mathcal{G}$ on $Y$ and an open $U\subseteq X$ we set $$\varphi^{-1}\mathcal{G}(U)=\mathop{\lim_{\longrightarrow}}_{V\supseteq\varphi(U)}\mathcal{G}(V)$$ I was hoping to find a more "categorical" description though, since, at least to me, the functoriality of $J$ is not obvious from this in the same way that it is with $I$. I know that $J$ is the left adjoint of $I$, so I feel like there should be another construction which makes this clearer.

Wikipedia says that the functoriality follows from the universal property of colimits. I think the colimit is taken over the images under $\mathcal{G}$ of the $V\subseteq Y$ such that there is an arrow $\Phi(V)\to U$ in $\mathbf{X}$, but I'm struggling to see how this shows $J$ is a functor.

Any help would be much appreciated.

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Ah, I just realized: $\mathcal F \overset{\mathrm{def}}{=} \mathcal G$.

You basically want to construct a morphism $\Gamma(V, \varphi^{-1} \mathcal F) = \varinjlim_\limits{U \supseteq \varphi(V)} \Gamma(U,\mathcal F) \to \varinjlim_\limits{U' \supseteq \varphi(V')}\Gamma(U', \mathcal F) = \Gamma(V', \varphi^{-1}\mathcal F)$ for any inclusion $V' \subseteq V$ of open subsets.

Now the universal property of the colimit used to compute $\Gamma(V,\varphi^{-1}\mathcal F)$ tells you how to obtain morphism starting at $\Gamma(V,\varphi^{-1}\mathcal F)$.

Namely: You have to give a morphism $\Gamma(U,\mathcal F) \to \varinjlim_\limits{U' \supseteq \varphi(V')}\Gamma(U', \mathcal F) = \Gamma(V', \varphi^{-1}\mathcal F)$ for any open $U \supseteq \varphi(V)$ compatible with the transition maps (which are of course given via restriction).

Now just note that since $V' \subseteq V$ one certainly has $U \supseteq \varphi(V) \supseteq \varphi(V')$ so that $\Gamma(U,\mathcal F)$ is a part of the second colimit, i.e. comes with a canonical map into $\Gamma(U,\mathcal F) \to \Gamma(V',\varphi^{-1}\mathcal F)$ as desired.

Then you have to check that these maps are compatible with restrictions so that they induce the desired morphism $\Gamma(V,\varphi^{-1}\mathcal F) \to \Gamma(V', \varphi^{-1} \mathcal F)$.

$\textit{Uniqueness}$ of the morphism induced on the colimit then tells you that this construction behaves well with respect to inclusions $V'' \subseteq V' \subseteq V$, i.e. is functorial.

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  • $\begingroup$ Ah just in case you haven't yet seen the notation: $\Gamma(U,\mathcal F) \overset{\mathrm{def}}{=} \mathcal F(U)$. $\endgroup$
    – lush
    Jul 14 '19 at 16:12
  • $\begingroup$ Also regarding the "naturality" of this construction: Think of the colimit as "We try to find a "smallest" open environment in $Y$ containing $\varphi(U)$. This will not always be possible though (if it is, the colimit will "degenerate" and just be the value of sections on this minimal open subset though!). $\endgroup$
    – lush
    Jul 14 '19 at 16:18
  • $\begingroup$ So instead we are taking all sections above some open subset containing $\varphi(U)$ into account and identify two such sections if they coincide on a smaller open subset still containing $\varphi(U)$. So this will result in the kind of "best approximation from the outside" of $\varphi(U)$ by sections over open sets. $\endgroup$
    – lush
    Jul 14 '19 at 16:18
  • $\begingroup$ Many thanks for your explanation, this has definitely helped me to see how universal properties come into this. I think maybe I should have been clearer, but when I mentioned functoriality, I meant as a functor between the presheaf categories, so that if we have a morphism $N:\mathcal{G}_1\to\mathcal{G_2}$ in $\mathbf{PShv}(\mathbf{Y})$ (which I think is just a natural transformation of the functors), then we have a map $\varphi^{-1}N:\varphi^{-1}\mathcal{G}_1\to\varphi^{-1}\mathcal{G}_2$... $\endgroup$
    – Dave
    Jul 14 '19 at 16:53
  • $\begingroup$ ... and that these behave well under composition etc. Maybe I misinterpreted the Wikipedia article, but is this also a consequence of the universal property? $\endgroup$
    – Dave
    Jul 14 '19 at 16:54

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