# On nearby cycle sheaves and a 2-fibered product of topoi

In SGA7 Exposé XIII, Deligne introduces an algebraic theory of nearby cycle sheaves and vanishing cycle sheaves on schemes over the $S$, where $S$ is the spectrum of a Henselian discrete valuation ring.

Let $i:s\hookrightarrow S$ denote the closed point of $S$. As $S$ is Henselian, pullback along $i$ induces an isomorphism from the finite étale covers of $s$ to finite étale covers of $S$. As every étale cover of $s$ is finite, we see this yields a functor $ex:s_\text{ét}=s_\text{fét}\rightarrow S_\text{fét}\hookrightarrow S_\text{ét}$. Precomposing with $ex$ yields a functor $sp_*:\widetilde{S_\text{ét}}\rightarrow\widetilde{s_\text{ét}}$ from étale sheaves on $S$ to étale sheaves on $s$ that retracts $i_*$. Using the Galois description of étale sheaves on $S$, one can show that $sp_*=i^*$.

Deligne uses $sp_*$ to construct the target space of his nearby cycles functor $\Psi$. For this, let $\eta$ denote the generic point of $S$, let $\bar\eta$ be a geometric point of $\eta$, and let $\bar{s}$ be the induced geometric point of $s$. Let $f:Y\rightarrow s$ be a scheme over $s$. In SGA7.XIII.1.2.3, it seems that Deligne introduces the 2-fibre product of topoi $\widetilde{Y_\text{ét}}\times_{\widetilde{s_\text{ét}}}\widetilde{S_\text{ét}}$, where the functor $\widetilde{S_\text{ét}}\rightarrow\widetilde{s_\text{ét}}$ is $f_*$ and the functor $\widetilde{S_\text{ét}}\rightarrow\widetilde{s_\text{ét}}$ is $sp_*$. This is presumably the category of triples $(\mathcal{G},\mathcal{H},\phi)$, where $\mathcal{G}$ is an étale sheaf over $Y$, $\mathcal{H}$ is an étale sheaf over $S$, and $\phi$ is an isomorphism $\phi:f_*\mathcal{G}\rightarrow sp_*\mathcal{H}=i^*\mathcal{H}$.

At this point, he then gives an alternate, Galois description of $\widetilde{Y_\text{ét}}\times_{\widetilde{s_\text{ét}}}\widetilde{S_\text{ét}}$ as the category of triples $(\mathcal{F}_s,\mathcal{F}_\eta,\varphi)$, where

1. $\mathcal{F}_s$ is an étale sheaf on $Y$ (equivalently, an étale sheaf $\mathcal{F}_\bar{s}$ on $Y_\bar{s}$ with an action of $\operatorname{Gal}(\kappa(\bar{s}),\kappa(s))$),
2. $\mathcal{F}_\eta$ is an étale sheaf $\mathcal{F}_\bar\eta$ on $Y_\bar{s}$ with an action of $\operatorname{Gal}(\kappa(\bar{\eta}),\kappa(\eta))$ through the canonical reduction map $\operatorname{Gal}(\kappa(\bar{\eta}),\kappa(\eta))\twoheadrightarrow\operatorname{Gal}(\kappa(\bar{s}),\kappa(s))$,
3. $\varphi$ is an Galois-equivariant map $\varphi:\mathcal{F}_\bar{s}\rightarrow\mathcal{F}_\bar\eta$.

Under these two descriptions, it's pretty clear that $\mathcal{G}$ should correspond to $\mathcal{F}_s$. But how do we translate between other parts of these two descriptions? Unless I'm misunderstanding what Deligne means when he constructs this 2-fibre product?

• I am not completely sure what the tilde '~' hats are; but I think they are giving toposes and the fibre products are over geometric morphisms. These 2-fibre products are given by describing the site of the 2-fibre product and aren't constructed using the triples you give (which decribe topos pushouts ... ). I think that's what is going on, but I am not familiar enough with this part of SGA to be sure. Hopefully that's of some use! – Christopher Townsend Feb 15 '17 at 7:51
• That makes sense—soon afterwards in SGA7.XIII.1.2.5, Deligne gives a description of the geometric points of $Y\times_sS$ in terms of those of $Y$ and $S$. How does one construct the site of the 2-fiber product? What's a good reference for this construction? Thank you! – Charles Denis Feb 16 '17 at 15:20
• E.g. Theorem C2.4.6 in 'Sketches of an Elephant A Topos Theory Compendium' by Peter Johnstone. However, I don't think the detail of how the construction is done informs you very much about what you are looking at; what you are looking at is determined by the universal property (the geometric points). – Christopher Townsend Feb 18 '17 at 19:07