Context: Stacks of Prismatic Forms What we do here We give a 2023-exposition on synthetic spectra, using the recent result, proposition 3.6 in https://arxiv.org/pdf/2304.04685.pdf.
Idea
Synthetic spectra exhibits the category as a degeneration of the category .
That is, there’s a category living over , with zero fiber is , and generic fiber .
Construction (over )
Let denote the -category of even spectra, we’ll define
to be the subcategory spanned by those functors preserving exact sequences of spectra.
Warning Exact sequences here is NOT a categorical notion in the source catergoy Even, because is not even stable. The definition of doesn’t merely depend on the category , but also the embedding .
More generally, if is a finitely-generated stable -category, choose a generator , and define to be the smallest subcategory under retracts, we define
Construction (for compactly generated stable -categories) those spectral presheaves preserving exact sequences in . Let , we’ll write a shorthand
Observe that by Morita Theory (Schwede-Shipley theorem), all finitely generated stable -categories are of this form. For example, by choosing a generator as above and taking .
Remark Turns out the embedding of into spectral presheaves is a localization - that it’s a “stable -topos”. Furthermore, this localization is a topological one (which is not-automatic in topoi!). It turns out there even is an explicit way to write a Grothendieck topology that exhibits as spectral sheaves on some site.
Basic objects: Synthetic Realizations of Objects
- There’s a functor , given by sheaffification of the spectral Yoneda embedding. is said to be the synthetic realization of an object . (If the site is sufficiently nice we won’t need sheaffification, need to check this)
- We’ll write . The two indexes will respectively index shifts to the sphere prior to and applying synthetic realization: This indexing is so that can be read as a “total degree”.
Basic structure (The -endofunctor)
The following cofiber diagram of affines
\begin{CD} Z @>{}>> 0;\\ @VVV @VVV \\ 0 @>{}>> 0\cup_Z0; \end{CD}$$ is a fiber diagram in $Even^{op}$, applying a functor $X \in Syn$ then yields a comparison map $$X(0\cup_Z0) \to \Omega X(Z)$$ As a convention we'll shift it down by 1 to define the map\tau_X(Z): \Sigma X(0 \cup_Z 0) \to X
This lifts to an endofunctor $\tau \in End(Syn)$. Now, it will turn out that $$\Sigma X(0 \cup_Z 0) \simeq (S^{0, -1}\otimes_{syn}X)(Z)$$ where we recall the RHS is $\Sigma \nu (\Omega \mathbb S) \otimes_{syn}X$ . It turns out furthermore this map arises from an endomorphism of the synthetic sphere. $$\tau:S^{0, -1} \to \Sigma\nu(\Omega\mathbb S)\to\nu(\mathbb S):= S^{0, 0}$$ where the map is adjoint to a $$\nu(\Omega\mathbb S)\to\Omega \nu(\mathbb S)$$ Let's spell out the above in a universal example: $$Even \xrightarrow{\nu(\mathbb S)} Sp$$ sends an even spectrum $Z$ to the mapping spectrum $map_{Sp}(Z, \mathbb S)$, we get a mapmap(0 \cup_Z0, \mathbb S) \to \Omega map(Z, \mathbb S)
This endomorphism of $S^{0, 0}$ gives rise to an automorphism of the entire category $$S^{-1, 0} \otimes_{S^{0, 0}} X \xrightarrow{\tau \otimes} Id $$ There's an automorphism of the synthetic motivic sphere **Question A** Can we phrase eveness, as in the appearance of $\Sigma^2V$ above, in terms of the Sheering/Waldhausen-rotation map? **Remark** Here's a way to think about what all this $Syn(V)$ reconstructs $V$ out of - Even objects: $Even(V)$ - "Cofiber Data" **Guess to answer to question A** $Syn(V)$ is describing $V$ in terms of of the sheering $U(1)$ symmetry ($\Sigma^2(-)$). Perhaps we can even say $Syn(V)$ is (co)descent data along homotopy orbits. What would such a statement tell us about "what formal groups laws are".