# Construction of Inverse Image Functor

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.

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.

• Ah just in case you haven't yet seen the notation: $\Gamma(U,\mathcal F) \overset{\mathrm{def}}{=} \mathcal F(U)$.
– lush
Jul 14 '19 at 16:12
• 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!).
– lush
Jul 14 '19 at 16:18
• 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.
– lush
Jul 14 '19 at 16:18
• 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 $\varphi^{-1}N:\varphi^{-1}\mathcal{G}_1\to\varphi^{-1}\mathcal{G}_2$...
– Dave
Jul 14 '19 at 16:53
• ... and that these behave well under composition etc. Maybe I misinterpreted the Wikipedia article, but is this also a consequence of the universal property?
– Dave
Jul 14 '19 at 16:54