MathOverflow Asked by leibnewtz on November 3, 2021
The category $text{AffSch}_S$ of affine schemes over some base affine scheme $S$ is not essentially small. This lends itself to certain set-theoretical difficulties when working with a category $Sh(text{AffSch}_S)$ of abelian sheaves on $text{AffSch}_S$ with respect to some Grothendieck topology. In fact, many definitions of the notion of category would not consider this a category at all.
Nevertheless, in some sense, such a category of sheaves should be somewhat like a presentable category; it should have a collection of generators indexed by $text{AffSch}_S$. As such, I would like to be able to use arguments involving theorems like the adjoint functor theorem. For example, I would like to show that for an affine scheme $c$, the evaluation functor $mathcal{F}_c: Sh(text{AffSch}_S) to Ab$ given by $mathcal{F}_c(F)=F(c)$ has a left-adjoint. If $Sh(text{AffSch}_S)$ were presentable, this would follow from the adjoint functor theorem for presentable categories.
Even though $text{AffSch}_S$ is not essentially small, can we still expect statements such as the adjoint functor theorem to hold for $Sh(text{AffSch}_S)$? I know that in many cases, one is able to restrict to some sufficiently large small subcategory of the category of affine schemes, but I’m not sure how to do it in this case. To make matters worse, certain “small” sites such as the small $text{fpqc}$ site over a scheme are not even essentially small, so when working with an arbitrary topos associated to $text{AffSch}_S$ it seems difficult to restrict to small subcategories for many purposes.
Let me start by discussing a bit the option of having a large class of generators. You might be interested in the notion of locally class-presentable.
To be precise here, I need to be a bit set-theoretical, thus, let me start with an informal comment.
Informal comment. Indeed your category is locally class-presentable, class-accessibility is a very strong weakening of the notion of accessibility. On one hand, it escapes the world of categories with a dense (small) generator, on the other, it still allows to build on the technical power of the small object argument via a large class of generators.
Locally class-presentable and class-accessible categories, B. Chorny and J. Rosický, J. Pure Appl. Alg. 216 (2012), 2113-2125.
discusses a part of the general theory of class-accessibility and class-local presentability. Unfortunately, the paper is designed towards a homotopical treatment and thus insists on weak factorization systems and injectivity but a lot of techniques coming from the classical theory can be recast in this setting.
Formal comment. To be mathematically precise, your category is locally large, while locally class-presentable categories will be locally small. Here there are two options, the first one is to study small sheaves $mathsf{Shv}_{text{small}}(text{AffSch})$, this is a full subcategory of the category of sheaves and contains many relevant sheaves you want to study. In the informal comment, this is the locally class-presentable one. Two relevant paper to mention on this topic are:
- Exact completions and small sheaves, M. Shulman, Theory and Applications of Categories, Vol. 27, 2012, No. 7, pp 97-173.
- Limits of small functors, B. J. Day and S. Lack, Journal of Pure and Applied Algebra, 210(3):651-683, 2007.
The other option is the one of being very careful with universes, in fact restricting to small presheaves might destroy sometimes your only chance of having a right adjoint. It would be too long to elaborate on this last observation here. As a general remark, small presheaves will give you the free completion under small colimits, while all presheaves will give you the free completion under large colimits, how big you need to go depends on the type of constructions that you need to perform.
Coming to the adjoint functor theorem, let me state the most general version that I know of. Since it is an if and only if, I hope it provides you with a good intuition on when one can expect a right adjoint to exists. The dual version is true for functors preserving limits.
Thm. (AFT) Let $f: mathsf{A} to mathsf{B}$ be a functor preserving colimits from a cocomplete category. The following are equivalent:
- For every $b in mathsf{B}$, $mathsf{B}(f_,b): mathsf{A}^circ to mathsf{Set}$ is a small presheaf.
- $f$ has a right adjoint.
This version of the AFT is designed for locally small categories and can be made enrichment sensitive - and thus work also for locally large categories - using the proper notion of smallness, or equivalently choosing the correct universe.
Unfortunately, I do not know a reference for this version of the AFT. Indeed it can be deduced by the too general Thm 3.25 in On the unicity of formal category theories by Loregian and myself, where it appears as a version of the very formal adjoint functor theorem by Street and Walters in the language of the preprint.
Finally, about to the evaluation functor, I am not an expert of the abelian world, but it looks to me that one can mimic the argument presented in the accepted answer to this question (at least if the topology is subcanonical). Thus, the left adjoint should indeed exist.
https://math.stackexchange.com/questions/2187846/adjoints-to-the-evaluation-functors
This answer is closely connected to this other.
Answered by Ivan Di Liberti on November 3, 2021
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