Let $X$ be a variety over $\mathbb C$. Let $f\colon X \to \mathbb{A}^1$ be a regular function. I understand that there is an analytic nearby cycles functor, defined in terms of the exponential map. I also understand that there is an algebraic nearby cycles, defined in SGA 7.
Questions:
Any references would be much appreciated. Thanks!
I don't know a reference, but let me discuss a strategy for a proof of the comparison that I think should work.
By Artin's comparison, if we take a constant sheaf $\mathbb Z/\ell^n$ and then pull back to the cover where we invert $f$ and adjoin an $m$th root of $f$, pushforward to $X$, and then pull back to the zero locus of $f$, the resulting complex of sheaves will not depend on if we perform these operations in the worlds of analytic or étale sheaves.
We want to use this as a building block for the comparison. The way to do this to observe there is a natural map from the above complex for $m_1$ to the above complex for $m_2$ if $m_1$ divides $m_2$, and so we can take a colimit as $m$ goes to $\infty$. After this, we can take the limit as $n$ goes to $\infty$. This gives nearby cycles as a $\mathbb Z_\ell$-sheaf, and then we can tensor with $\mathbb Q_\ell$. If we do this in the étale world, we recover the usual construction of nearby cycles.
Indeed, the only difference between what I said and the usual construction is that the usual construction involves pulling back to the algebraic closure of some local field, i.e. (in characteristic zero) to the inverse limit of all these coverings (together with étale neighborhoods of zero in $\mathbb A^1$, which are irrelevant after taking $i^*$). But the pushforward from the inverse limit of coverings is the colimit of the pushforward from coverings by a general result (applied one open set at a time, since it's stated in terms of global sections rather than sheaves) with hypotheses that are easily checked in this case.
So it suffices to check that this construction agrees with the usual construction in the analytic world, where we adjoin a logarithm of $f$ instead of an $m$'th root, and where we work with $\mathbb Q$ coefficients. I would attempt this in two steps:
First, like in the étale world, check that the cohomology of the colimits of the finite covers agrees with the cohomology of the infinite cover. I would do this by expressing the finite cover as the quotient of the infinite cover by $\log f \mapsto \log f + 2\pi i m$ so the cohomology of the finite cover can be computed from the cohomology of the infinite cover and the action of the operator $\log f \mapsto \log f + 2 \pi i m$, i.e. the $m$th power of a generator of monodromy. Thinking about the action of this operator should give the statement.
Next, check that the $\mathbb Q_\ell$-cohomology constructed this way isjust the usual $\mathbb Q$-cohomology tensored with $\mathbb Q$. This should just require using the universal coefficient theorem to calculate the $\mathbb Z/\ell^n$ cohomology in terms of the $\mathbb Z$ cohomology and use that the $\mathbb Z$-cohomology is a finitely generated abelian group (surely that is known in this setting even with the logarithm map) to see that the inverse limit over $n$ agrees with the tensor product with $\mathbb Z_\ell$, then tensor both sides with $\mathbb Q$.
Everything I say should work with stacks but now finding references to individual steps will be more difficult.
In the infinite-dimensional case, it depends what you mean, but the étale side would usually be a limit of finite-dimensional things so the question is if the analytic side is as well.
I am not sure about the relation between the algebraic nearby cycles functors (at this point, I believe that the algebraic in your mind is the one in étale cohomology, which is somehow analogous but irrelevant here as the setting is $X \longrightarrow S$ with $S$ a strict local trait) and their analytic counterparts are compatible but there is such a compatibility in the motivic world. Let $X$ be a $\mathbb{C}$-variety and $X^{an}$ the complex points $X(\mathbb{C})$ endowed with the analytic topology. We denote by $\mathbf{SH}(X)$ the motivic stable homotopy category of Morel-Voevodsky and $\mathbf{D}(X^{an})$ the derived category of sheaves of abelian groups. Then Ayoub constructed a Betti realization $$\operatorname{Betti}_X \colon \mathbf{SH}(X) \longrightarrow \mathbf{D}(X^{an})$$ which is compatible with four operations $(f^*,f_*,f_!,f^!)$ and moreover, compatible with appropriate nearby cycles functors. More precisely, given a morphism $f \colon X \longrightarrow \mathbb{A}_{\mathbb{C}}^1$, and let $$\Psi_{f} \colon \mathbf{SH}(X_{\eta}) \longrightarrow \mathbf{SH}(X_{\sigma})$$ $$\Psi_{f^{an}} \colon \mathbf{D}(X_{\eta}^{an}) \longrightarrow \mathbf{D}(X_{\sigma}^{an})$$ be the motivic nearby cycles functor (constructed in Ayoub's thesis, it is more or less the same thing as you commented, and it is the tame one, not the unipotent) and the analytic nearby cycles functors (if I remember correctly, this one is in SGA7), Ayoub proved that there is an isomorphism $$\operatorname{Betti}_{X_{\eta}} \circ \Psi_f(A) \simeq \Psi_{f^{an}} \circ \operatorname{Betti}_{X_{\sigma}}(A)$$ when $A$ is a constructible motive. I do not know if we can replace $\mathbf{SH}$ with some other motivic theory, but in general, all reasonable theory of nearby cycles should be a non-virtual version of a virtual one coming from Grothendieck rings of varieties. More concrete, if we let $\mathscr{M}_X = K_0(\mathrm{Var}_X)[(\mathbb{A}_X^1)^{-1}]$ to be the Grothendieck ring of $X$-varieties, then Guibert-Loeser-Merle built a nearby cycles morphism $$\psi_f \colon \mathscr{M}_{X_{\eta}} \longrightarrow \mathscr{M}_{X_{\sigma}}$$ and there are commutative diagrams $\require{AMScd}$ \begin{CD} \mathscr{M}_{X_{\eta}} @>{\psi_f}>> \mathscr{M}_{X_{\sigma}} \\ @V{\chi_{X_{\eta},c}}VV @VV{\chi_{X_{\sigma},c}}V\\ K_0(H(X_{\eta})) @>{\Psi_f}>> K_0(H(X_{\sigma})) \end{CD} where $\chi$ denotes the Euler characteristic with compact support, and with $H$ being $MHM$ (mixed Hodge modules, by Guibert-Loeser-Merle), being $D^b_c$ (derived category of sheaves, by Denef-Loeser), being $\mathbf{SH}$ (by Ivorra-Sebag),...
The reason that all these theories look alike is that they all satisfy the so-called semi-stable reduction (in 'etale cohomology, it is the work of Rapoport-Zink), which roughly speaking, allows you to compute the nearby cycles functor when the special fiber is of semi-stable form (looks like a monomial, or a resolution of singularities if you prefer this terminology).
To answer your question, at least with $\mathbf{SH}(X)$, we do not require $X$ to be finitely dimensional, just quasi-compact and quasi-separated is already enough.
For stacks, I am aware of the work of Cisinski and Pippi, Étale tame vanishing cycles over $[\mathbb{A}^1_S/\mathbb{G}_{m,S}]$ but not read the details yet.
