2 The category of synthetic spectra
Following our previous outline, to construct the category of synthetic spectra, we start with , and then for every fiber sequence such that the second map is surjective, we force the image in 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 , and we have the following result:
Let be a small -category with finite coproducts. Let be the full subcategory of consisting of product-preserving functors. Then is the stabilization of .
To impose our second condition, we have to confront ourselves with the unfortunate fact that surjections are not closed under tensor products. For example, the map is -surjective, but it is not after tensoring with . This will cause the resulting category to not have a symmetric monoidal structure. To avoid this, we make the following definition.
Let be a homotopy ring spectrum. We let be the full subcategory of spectra such that is a projective -module.
If , then for any other , we have . So we learn that
is closed under tensor products; and
-surjections are closed under tensor products in .
Replacing with should not be seen as a big change. In the case , the two categories are equal, so there is literally no difference. In general, importantly contains the spheres, from which we can build all other finite spectra.
Thus, our starting category is . We impose our epimorphism condition as follows:
We define to be the full subcategory of consisting of functors such that for any cofiber sequence
of spectra living in that induces a short exact sequence on -homology, the induced sequence
is a fiber sequence of spectra.
Since is stable, this is equivalent to requiring that is a cofiber sequence. However, if we work with the non-stabilized version , being a fiber sequence is the correct condition.
We can turn into a site by declaring coverings to be generated by surjections. Then is exactly the presheaves that are sheaves under this topology. In particular, is an accessible left exact localization of .
We can write down some examples of synthetic spectra. Define the spectral Yoneda embedding by
Then for any , we see that is in fact a sheaf (i.e. in ). We should think of this as a bad thing. Since we didn't use anything about to conclude that is a sheaf, it cannot possibly contain much information about the -based Adams spectral sequence.
This turns out to be the less useful version of the Yoneda embedding. Instead, we define by
We should think of this as of the usual Yoneda embedding, characterized by the fact that it takes values in connective spectra and . In fact, Yoneda's lemma implies that if , then
Crucially, is not always a sheaf! Given a cofiber sequence
in , the induced sequence
is a cofiber sequence if and only if the map is surjective.
There is one case where this is in fact a sheaf. If is -injective, then the map is given by
Since is an injective -comodule and is an injective map, it follows that this map is in fact surjective. So
If is -injective, then is a sheaf.
In general, we define
For , we define to be the sheafification of .
Since sheafification is left adjoint to the inclusion, for and , we have
If is a cofiber sequence of spectra that induces a short exact sequence on -homology, then
is a cofiber sequence.
Combining these two results, what we learn is that to compute for any , we resolve by -injectives as in the Adams resolution, and then apply to this resolution. This remains a resolution in (barring convergence issues), and of -injectives are simply given by the connective Yoneda embedding. This is what makes much more interesting than .
Since the tensor product preserves sums and -epimorphisms, we find that
is a symmetric monoidal category, and is symmetric monoidal. In particular, is the unit.