Construction of synthetic spectraThe category of synthetic spectra

# 2 The category of synthetic spectra

Following our previous outline, to construct the category of synthetic spectra, we start with $P_\Sigma ({\mathrm{Sp}}^\omega )$, and then for every fiber sequence $A \to B \to C$ such that the second map is $E_*$ surjective, we force the image in $P_\Sigma ({\mathrm{Sp}}^\omega )$ to be a fiber sequence.

In practice, there are some modifications we want to perform. Firstly, we want the category of synthetic spectra to be stable. This can be fixed by simply stabilizing $P_\Sigma ({\mathrm{Sp}}^\omega )$, and we have the following result:

Theorem 2.1

Let ${\mathcal{C}}$ be a small $\infty$-category with finite coproducts. Let $P_\Sigma ^{\mathrm{Sp}}({\mathcal{C}})$ be the full subcategory of $\operatorname{Fun}({\mathcal{C}}^{\mathrm{op}}, {\mathrm{Sp}})$ consisting of product-preserving functors. Then $P_\Sigma ^{\mathrm{Sp}}({\mathcal{C}})$ is the stabilization of $P_\Sigma ({\mathcal{C}})$.

To impose our second condition, we have to confront ourselves with the unfortunate fact that $E_*$ surjections are not closed under tensor products. For example, the map $S \to S/2$ is $(H{\mathbb {Z}})_*$-surjective, but it is not after tensoring with $S/2$. This will cause the resulting category to not have a symmetric monoidal structure. To avoid this, we make the following definition.

Definition 2.2

Let $E$ be a homotopy ring spectrum. We let ${\mathrm{Sp}}_E^{fp} \subseteq {\mathrm{Sp}}^\omega$ be the full subcategory of spectra $P$ such that $E_*P$ is a projective $E_*$-module.

If $P \in {\mathrm{Sp}}_E^{fp}$, then for any other $Y$, we have $E_*(P \otimes Y) = E_* P \otimes _{E_*} E_* Y$. So we learn that

1. ${\mathrm{Sp}}_E^{fp}$ is closed under tensor products; and

2. $E_*$-surjections are closed under tensor products in ${\mathrm{Sp}}_E^{fp}$.

Replacing ${\mathrm{Sp}}^\omega$ with ${\mathrm{Sp}}_E^{fp}$ should not be seen as a big change. In the case $E = H{\mathbb {F}}_p$, the two categories are equal, so there is literally no difference. In general, ${\mathrm{Sp}}_E^{fp}$ importantly contains the spheres, from which we can build all other finite spectra.

Thus, our starting category is $P_\Sigma ^{\mathrm{Sp}}({\mathrm{Sp}}_E^{fp})$. We impose our epimorphism condition as follows:

Definition 2.3

We define $\operatorname{Syn}_E$ to be the full subcategory of $P_\Sigma ^{\mathrm{Sp}}({\mathrm{Sp}}_E^{fp})$ consisting of functors $X \colon ({\mathrm{Sp}}_E^{fp})^{\mathrm{op}}\to {\mathrm{Sp}}$ such that for any cofiber sequence

$A \to B \to C$

of spectra living in ${\mathrm{Sp}}_E^{fp}$ that induces a short exact sequence on $E_*$-homology, the induced sequence

$X(C) \to X(B) \to X(A)$

is a fiber sequence of spectra.

Remark 2.4

Since ${\mathrm{Sp}}$ is stable, this is equivalent to requiring that $X(C) \to X(B) \to X(A)$ is a cofiber sequence. However, if we work with the non-stabilized version $P_\Sigma ({\mathrm{Sp}}_E^{fp})$, being a fiber sequence is the correct condition.

Remark 2.5

We can turn ${\mathrm{Sp}}_E^{fp}$ into a site by declaring coverings to be generated by $E_*$ surjections. Then $\operatorname{Syn}_E$ is exactly the presheaves that are sheaves under this topology. In particular, $\operatorname{Syn}_E$ is an accessible left exact localization of $P_\Sigma ^{\mathrm{Sp}}({\mathrm{Sp}}_E^{fp})$.

We can write down some examples of synthetic spectra. Define the spectral Yoneda embedding $Y \colon {\mathrm{Sp}}\to \operatorname{Syn}_E$ by

$Y(X)(P) = F(P, X).$

Then for any $X \in {\mathrm{Sp}}$, we see that $Y(X)$ is in fact a sheaf (i.e. in $\operatorname{Syn}_E$). We should think of this as a bad thing. Since we didn't use anything about $E$ to conclude that $Y(X)$ is a sheaf, it cannot possibly contain much information about the $E$-based Adams spectral sequence.

This turns out to be the less useful version of the Yoneda embedding. Instead, we define $y \colon {\mathrm{Sp}}\to P_\Sigma ^{\mathrm{Sp}}({\mathrm{Sp}}_E^{fp})$ by

$y(X)(P) = \tau _{\geq 0} F(P, X).$

We should think of this as $\Sigma ^\infty$ of the usual Yoneda embedding, characterized by the fact that it takes values in connective spectra and $\Omega ^\infty y(X)(P) = {\mathrm{Sp}}(P, X)$. In fact, Yoneda's lemma implies that if $P \in {\mathrm{Sp}}_E^{fp}$, then

$P_\Sigma ^{\mathrm{Sp}}({\mathrm{Sp}}_E^{fp})(y(P), Z) = \Omega ^\infty Z(P).$

Crucially, $y(X)$ is not always a sheaf! Given a cofiber sequence

$A \to B \to C$

in ${\mathrm{Sp}}_E^{fp}$, the induced sequence

$\tau _{\geq 0} F(C, X) \to \tau _{\geq 0} F(B, X) \to \tau _{\geq 0} F(A, X)$

is a cofiber sequence if and only if the map $[B, X] \to [A, X]$ is surjective.

There is one case where this is in fact a sheaf. If $X$ is $E$-injective, then the map $[B, X] \to [A, X]$ is given by

$\operatorname{Hom}_{E_*E} (E_* B, E_* X) \to \operatorname{Hom}_{E_* E} (E_*A, E_* X).$

Since $E_* X$ is an injective $E_*E$-comodule and $E_* A \to E_*B$ is an injective map, it follows that this map is in fact surjective. So

Theorem 2.6

If $X$ is $E$-injective, then $y(X)$ is a sheaf.

In general, we define

Definition 2.7

For $X \in {\mathrm{Sp}}$, we define $\nu X$ to be the sheafification of $y(X)$.

Since sheafification is left adjoint to the inclusion, for $P \in {\mathrm{Sp}}_E^{fp}$ and $X \in \operatorname{Syn}_E$, we have

$\operatorname{Syn}_E(\nu P, X) = \Omega ^\infty X(P).$

Lemma 2.8 ([3, Lemma 4.23])

If $A \to B \to C$ is a cofiber sequence of spectra that induces a short exact sequence on $E_*$-homology, then

$\nu A \to \nu B \to \nu C$

is a cofiber sequence.

If these spectra are in ${\mathrm{Sp}}_E^{fp}$, then this follows from the definition of a sheaf plus the identification $\operatorname{Syn}_E(\nu P, X) = \Omega ^\infty X(P)$. The general case requires more work, but is still true nonetheless.

Combining these two results, what we learn is that to compute $\nu X$ for any $X$, we resolve $X$ by $E$-injectives as in the Adams resolution, and then apply $\nu$ to this resolution. This remains a resolution in $\operatorname{Syn}_E$ (barring convergence issues), and $\nu$ of $E$-injectives are simply given by the connective Yoneda embedding. This is what makes $\nu$ much more interesting than $Y$.

Since the tensor product preserves sums and $E_*$-epimorphisms, we find that

Theorem 2.9

$\operatorname{Syn}_E$ is a symmetric monoidal category, and $\nu \colon {\mathrm{Sp}}_E^{fp} \to \operatorname{Syn}_E$ is symmetric monoidal. In particular, ${\mathbb {S}}\equiv \nu S$ is the unit.