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A modern approach to derived functors, that has been shown to be useful in a number of different circumstances, is that of a derived category (see the book by Yakutieli, for example, here).

However, it has been remarked, for instance, in Weibel's book An introduction to homological algebra (in particular, 10.3.3, 10.3.6), and in Neeman's book Triangulated categories (Sections 2.2 and 2.3), that some non-trivial set theoretical considerations can appear. Neeman remarks that this point has not usually been addressed in the literature, but that this is usually harmless (which, I think, most mathematicians would agree upon), but, on the other hand, Weibel notes that in some cases, such as ex localisation of homology theories, these questions can become quite relevant.

Both authors address the question in their own way in their books, and I have seen discussions of these points in the set theory chapter of The Stacks project, Appendix A of the very nice book by Asashiba Categories and Representation Theory: With a Focus on 2-Categorical Covering Theory and 'en passant' in Shulman's great Set theorey for category theory.

However, it seems to me that all approaches to set theoretical intricacies differ among in these sources are (I may be wrong), and I also have the impression that in some cases (like Asashiba's), the developed theory, while adequate for other purposes, is more powerful than what is actually necessary for the basic derived category setting. Also, I have not seen an exposition aimed specifically at the set theoretical foundations of derived categories.

Considerations of size may actually matter, as they have been discussed, for example, in detail in this MO discussion.

Therefore, what I am looking in this question is for references or mathematical discussions regarding the question, which set-theoretical foundations are adequate to get derived categories 'off-the ground'.

If possible or relevant, I would like to see how these issues can be addressed, both, within a set theory with proper classes (like NBG) and in the (common) setting of Grothendieck Universes.

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    $\begingroup$ NBG is sufficient for derived categories and derived functors (just allow categories to have proper classes of maps between two objects). The only thing constraint is that, given two categories C and D, unless C is small, you cannot speak of the category Fun(C,D) of all functors of type C->D. Grothendieck universes are really needed if you start to think of subcategories of the category of all categories (e.g. the 2-category of topoi, or the one of presentable categories) and express their category-theoretic properties (e.g. existence of 2-(co)limits) in a global way. $\endgroup$ Commented Nov 10, 2024 at 11:36
  • $\begingroup$ I would say that it is just a fact of life that localisations (a certain type of 2-pushouts) do not always exist. In other words: a lot of category theory is based upon Yoneda lemma, and you need Homs to be sets (U-small sets in universe approach) to threat result of localisation (which always exists as "very big category" with class-sized Homs) as a honest category. $\endgroup$ Commented Nov 10, 2024 at 20:31
  • $\begingroup$ @D.-C.Cisinski thank you very much for your answer. But wouldn't class sized Hom sets create difficulties (such as the inability to talk about the function category) very soon? $\endgroup$ Commented Nov 10, 2024 at 21:33
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    $\begingroup$ @jg1896 You simply do not need functor categories in order to manipulate basic constructions concerning derived categories. To start seeing difficulties, you need to go much beyond the contents of Weibel's book. $\endgroup$ Commented Nov 11, 2024 at 2:08
  • $\begingroup$ As a complement to Zhen Lin's excellent answer, I would propose to meditate on the following fact. If you allow yourself to build mathematics using second order logic (and there are ways to this in a very controlled, even constructive, way, via dependent type theory), then the existence of a countable hierarchy of universes follows from Peano arithmetic (i.e. from the existence of a natural number object). The fact that universes look like a strong leap further from ZFC is a peculiarity of Set Theory and is not a difficulty we are supposed to carry with us for ever. $\endgroup$ Commented Nov 11, 2024 at 11:56

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Fundamentally, working in NBG is not much different from working in ZFC, except that you are allowed one level of freedom to form collections of sets that are not themselves sets. As such, you still have to encode the actual "elements" you want to work with as sets, and the encoding has to be reasonable enough that all the "operations" you wish to perform on "elements" can also be encoded. Conversely, if you can do such an encoding, then there is probably no obstacle to working in NBG.

Regarding the existence of localisations: there is basically no obstacle in NBG. Let $\mathcal{C}$ be a category in NBG, by which I mean we are given classes $\mathcal{C}_0, \mathcal{C}_1, \mathcal{C}_2$ and maps $d_0, d_1, d_2 : \mathcal{C}_2 \to \mathcal{C}_1$, $d_0, d_1 : \mathcal{C}_1 \to \mathcal{C}_0$, $s_0 : \mathcal{C}_0 \to \mathcal{C}_1$ satisfying the simplicial identities and the strict Segal condition. Let $\mathcal{W}$ be some subclass of $\mathcal{C}_1$; for simplicity, assume it contains all identities in $\mathcal{C}$ and is closed under composition. We can construct $\mathcal{D} = \mathcal{C} [\mathcal{W}^{-1}]$ as follows:

  • $\mathcal{D}_0 = \mathcal{C}_0$.

  • $\mathcal{D}_\rightsquigarrow$ is the class of finite sequences of elements of $\mathcal{C}_1$, say $(w_0, f_1, \ldots, f_n, w_n)$, where each $w_i$ is in $\mathcal{W}$, $d_1 (w_i) = d_1 (f_{i+1})$, and $d_0 (f_i) = d_0 (w_{i+1})$; in words, this is the class of zigzags where the backward-pointing arrows are in $\mathcal{W}$.

  • We define a binary relation on $\mathcal{D}_\rightsquigarrow$ relating two zigzags if there exists a small (i.e. set sized; actually, we could even assume finitely generated) subcategory $\mathcal{C}' \subseteq \mathcal{C}$ containing all the morphisms in the zigzags, such that the zigzags become equal in $\mathcal{C}' [(\mathcal{C}' \cap \mathcal{W})^{-1}]$.

  • It is easy to see the above is reflexive and symmetric; for transitivity, we can exploit the flexibility to choose the subcategory in the definition of the relation. Thus, we have an equivalence relation on $\mathcal{D}_\rightsquigarrow$. Using Scott's trick, or the axiom of global choice if you prefer, define $\mathcal{D}_1$ to be the class of representatives for this equivalence relation.

  • Define $\mathcal{D}_2$ to be the class of composable pairs of elements of $\mathcal{D}_1$, etc.

The point is that localisation is a finitary construction on categories, meaning that the functor $(\mathcal{C}, \mathcal{W}) \mapsto \mathcal{C} [\mathcal{W}^{-1}]$ preserves filtered colimits. Every category – morally, at least, even if our logical framework makes it extremely awkward to say exactly what we mean – is the filtered colimit of all its small subcategories. In the definition of the equivalence relation on $\mathcal{D}_\rightsquigarrow$ above, I exploited this fact to avoid giving an explicit construction (which can be done – see the papers of Dwyer and Kan!) and reduce to the case of localisation of small categories.

In the case of the derived category of an abelian category, there is a somewhat simpler description, owing to the fact that the category of chain complexes is a category of fibrant objects: morphisms in the homotopy category of a category of fibrant objects can always be represented as $f \circ w^{-1}$ for a fibration $f$ and a trivial fibration $w$ (i.e. a cocycle in the sense of Jardine), and composition can be represented by composition of spans. But as everyone has already said, this is not enough to make the class of morphisms between two objects small enough to be a set.

Personally, I find it quite awkward not being able to form collections of classes. It is possible to iterate the ZFC to NBG construction but it seems not well studied, let alone actually used. It is much easier to work with one or more Grothendieck universes in an ambient set theory. One interesting thing is that set theory in a Grothendieck universe can be stronger than the ambient set theory – for example, even in Mac Lane set theory, a Grothendieck universe is a model of full ZFC (and more; for example, it also satisfies the axiom of global choice, $\textrm{Con} (\textrm{ZFC})$, etc.).

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    $\begingroup$ Another point of due is strengthening your foundations using Morse-Kelly. But, at the end of the day, Grothendieck universes are simpler that iterating the NBG procedure, at the cost of assuming the existence of inaccesible cardinals. $\endgroup$ Commented Nov 11, 2024 at 11:28
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    $\begingroup$ What a fantastic answer! $\endgroup$ Commented Nov 11, 2024 at 19:44
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If you start with an abelian category $\mathcal{A}$ with sets of morphisms between its objects, the categories $\mathbf{C}(\mathcal{A})$ (of complexes of objects of $\mathcal{A}$) and $\mathbf{K}(\mathcal{A})$ (the category of complexes of objects of $\mathcal{A}$ up to homotopy) has sets of morphisms between its objects. However, $\mathbf{D}(\mathcal{A})$, defined as a localization of $\mathbf{K}(\mathcal{A})$ may have classes (as opposed to sets) of morphisms. In universes language, it lives in the next universe.

But if there are appropriate resolutions in $\mathbf{K}(\mathcal{A})$, then $\mathbf{D}(\mathcal{A})$ has sets of morphisms, (i.e. it lives in the same universe). This happens in the important case when $\mathcal{A}$ is Grothendieck ($\mathcal{A}$ has a generator and exact directed colimits) due to the existence of K-injective resolutions.

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  • $\begingroup$ thank you. So, if we were dealing with a set theory with classes and an axiom of limitation of size (every class can be well-ordered), then Ob(A) would be a proper class but D(A) would have a bigger proper class of morphisms? $\endgroup$ Commented Nov 10, 2024 at 21:30
  • $\begingroup$ The axiom of limitation of size says all proper classes are the same size. If you are asking whether you need to go beyond even that, then the answer is no. $\endgroup$ Commented Nov 10, 2024 at 22:16
  • $\begingroup$ @jg1896 It's a matter of taste. In general the morphisms between two objects in D(A) form a proper class. One way to deal with this is to consider then as subsets of the next universe. In this case you would be able take limits indexed by all ordinals and the problems on infinite constructions become trivial. In practice you want to work with well behaved categories that guarantee the existence of resolutions (so you don't need to change the universe) and you won't take limits indexed by classes. $\endgroup$ Commented Nov 10, 2024 at 22:24

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