# Does the inverse image sheaf functor has a left adjoint?

I am learning sheaf theory from Sheaves in Geometry and Logic by MacLane and Moerdijk, which I'll refer simply as [MM].

I am curious to know whether the inverse image sheaf functor has a left adjoint?

I think the answer is yes; however, I am not sure about it. Also, I couldn't find this in [MM], at least till the end of Chapter II. So given a continuous map $(X,\mathcal O_X) \xrightarrow f (Y,\mathcal O_Y)$, then my strategy is as follows:

• First use the ''change of base'' result (Theorem 4 on [page 59, MM]) on the category $\mathbf{Etale} (X)$ (i.e., the category of étalé bundles on $X$) to construct a left adjoint (denoted $\Sigma_f$) to the pullback functor $\mathbf{Etale}(Y) \xrightarrow {f^*} \mathbf{Etale}(X)$. Here, we note that the category $\mathbf {Etale}(X)$ has pullbacks (to prove this we need pullbacks from the category of topological spaces and the property that pullback of an étalé bundle along a continuous map is again an étalé bundle).

• Second use the functors $\mathbf {Top}/X \xrightarrow{\Gamma_X} \mathbf{Sets}^{\mathcal O_X^\text{op}}$ and $\mathbf{Sets}^{\mathcal O_X^\text{op}} \xrightarrow{\Lambda_X} \mathbf {Top}/X$ to define $\mathbf{Sh}(X) \xrightarrow{L_f} \mathbf{Sh}(Y)$ as $L_f=\Gamma_Y\Sigma_f\Lambda_X$.

So it remains to show that $L_f \dashv f^*$, where $\mathbf{Sh}(Y) \xrightarrow{f^*} \mathbf{Sh}(X)$ is given by the composition $\Gamma_Xf^*\Lambda_Y$. For this we derive,

\begin{align} \hom (\Gamma_Y \Sigma_f \Lambda_X F, G) \cong&\ \hom (\Gamma_Y \Sigma_f \Lambda_X F, \Gamma_Y \Lambda_Y G)\\ \cong&\ \hom (\Lambda_Y \Gamma_Y \Sigma_f \Lambda_X F, \Lambda_Y G)\\ \cong&\ \hom (\Sigma_f \Lambda_X F, \Lambda_Y G)\\ \cong&\ \hom (\Lambda_X F, f^* \Lambda_Y G)\\ \cong&\ \hom (F, \Gamma_X f^* \Lambda_Y G). \end{align}

Are the above steps valid? If yes, is there a special name to the functor $L_f$?

• The theories of sheaves of sets, sheaves of abelian groups, and sheaves of $\mathcal{O}$-modules all share some similarities, but they have differences as well. Left adjoints to a functor named $f^*$ tend to be named $f_!$. – Hurkyl Feb 12 '18 at 20:11
• Thank you for introducing this notation. So what I just wrote is just a standard well known concept. Can you also share some references where this connection is written out? – Hbeohar Feb 12 '18 at 20:22
• Can you expand a bit on your functor $\Gamma_X$ and $\Lambda_X$ ? As for the answer of your question, $f^{-1}$ almost never admit a left adjoint. It does however if $f$ is a local homeomorphism, for example the inclusion of an open subset, but it fails to have an adjoint for $f$ the inclusion of a point (aka stalk functor). If you are considering ringed spaces instead, the condition may be even more restrictive as it may even be non left exact. – Roland Feb 12 '18 at 21:37
• @Roland According to [MM], the functor $\Lambda_X$ assigns to each bundle the sheaf of cross-sections; while the functor $\Gamma_X$ assigns to each presheaf the bundle of germs. I will look into your suggested counterexample in more detail, however, can you point me to the fault(s) in my arguments. Thank you for your assistance. – Hbeohar Feb 12 '18 at 21:50
• @Hbeohar That is why I wanted to know more about $\Gamma_X$ and $\Lambda_X$. I don't have MacLane Moerdijk book and the notation are not standard. But I think I guess what was confusing me : what you denote $\mathcal{O}_X$ is not a sheaf of ring, but the lattice of open subset of $X$, right ? This is why Hurkyl and I were speaking of sheaves of $\mathcal{O}_X$-modules. – Roland Feb 12 '18 at 22:07