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 PΣ(Spω)P_\Sigma ({\mathrm{Sp}}^\omega ), and then for every fiber sequence ABCA \to B \to C such that the second map is EE_* surjective, we force the image in PΣ(Spω)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 PΣ(Spω)P_\Sigma ({\mathrm{Sp}}^\omega ), and we have the following result:

Theorem 2.1

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

To impose our second condition, we have to confront ourselves with the unfortunate fact that EE_* surjections are not closed under tensor products. For example, the map SS/2S \to S/2 is (HZ)(H{\mathbb {Z}})_*-surjective, but it is not after tensoring with S/2S/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 EE be a homotopy ring spectrum. We let SpEfpSpω{\mathrm{Sp}}_E^{fp} \subseteq {\mathrm{Sp}}^\omega be the full subcategory of spectra PP such that EPE_*P is a projective EE_*-module.

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

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

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

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

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

Definition 2.3

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

ABC A \to B \to C

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

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

is a fiber sequence of spectra.

Remark 2.4

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

Remark 2.5

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

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

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

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

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

y(X)(P)=τ0F(P,X). y(X)(P) = \tau _{\geq 0} F(P, X).

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

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

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

ABC A \to B \to C

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

τ0F(C,X)τ0F(B,X)τ0F(A,X) \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][A,X][B, X] \to [A, X] is surjective.

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

HomEE(EB,EX)HomEE(EA,EX). \operatorname{Hom}_{E_*E} (E_* B, E_* X) \to \operatorname{Hom}_{E_* E} (E_*A, E_* X).

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

Theorem 2.6

If XX is EE-injective, then y(X)y(X) is a sheaf.

In general, we define

Definition 2.7

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

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

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

Lemma 2.8 ([3, Lemma 4.23])

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

νAνBνC \nu A \to \nu B \to \nu C

is a cofiber sequence.

If these spectra are in SpEfp{\mathrm{Sp}}_E^{fp}, then this follows from the definition of a sheaf plus the identification SynE(νP,X)=ΩX(P)\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 νX\nu X for any XX, we resolve XX by EE-injectives as in the Adams resolution, and then apply ν\nu to this resolution. This remains a resolution in SynE\operatorname{Syn}_E (barring convergence issues), and ν\nu of EE-injectives are simply given by the connective Yoneda embedding. This is what makes ν\nu much more interesting than YY.

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

Theorem 2.9

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