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Sets and Descent Sets and Descent Brice Halimi Université Paris Nanterre La Défense (IREPH)& SPHERE Abstract Algebraic Set Theory, a reconsideration of Zermelo-Fraenkel set theory (ZFC) in category-theoretic terms, has been built up in the mid-nineties by André Joyal and Ieke Moerdijk. Since then, it has developed into a whole research program. This paper gets back to the original formulation by Joyal & Moerdijk, and more specifically to its first three axioms. It explains in detail that these axioms set up a framework directly linked to descent theory, a theory having to do with the shift from local data to a global item in modern algebraic geometry. Fibered cat- egories, introduced by Grothendieck, provide a powerful framework for descent theory: They formalize in a very general way the consideration of local data of dif- ferent kinds over the objects of some base category. The paper shows that fibered categories fit Joyal & Moerdijk’s axiomatization in a natural way, since the latter actually aims to secure a descent condition. As a result, Algebraic Set Theory is shown to accomplish, not only an original and fruitful combination of set theory with category theory, but the genuine graft of a deeply geometric idea onto the usual setting of ZFC. Keywords: Algebraic Set Theory. Fibered categories. Descent theory. Several arguments have been put forward against category theory as a possible foundational theory for mathematics. But none is compelling. First, a central objection has been made to category theory, to the effect that the very notion of category (de- fined as a “collection” of objects together with a “collection” of arrows) presupposes the notion of set. Nothing prescribes, however, to conceive of an object or an arrow as being some element of a set in the sense of set theory. Otherwise, any formal language (based on some “collection” of symbols) would presuppose, if not formal set theory (ZFC, say), at least some prior theory of what a “collection” is. And this would apply to the language of that theory itself. In fact, category theory only presupposes a back- ground universe whose existence and set-theoretic constitution is implicit, but which has nothing to do with a model of ZFC. Set theory presupposes such a background universe as much as category does: Models of ZFC are lifted out of it as much as any category is. Most mathematical notions are defined in set-theoretic terms. However, thinking in those terms is but confusing a mathematical theory with its set models, on the basis of Gödel’s completeness theorem. The theory of fields, for instance, has nothing to do as such with set theory. So, even though the objects and arrows which compose a given 1 category can be taken to be set-theoretic constructions, this is but one available option. Functorial semantics, and sketch theory in particular, is a nice illustration of that: A mathematical object such as a field can be introduced as a functor from a diagrammatic presentation of the field structure itself, to some realization category which is usually the category Sets of all sets and maps, but which can be other than Sets. Moreover, it is a more and more acknowledged fact that there are non-concrete categories, a category C being said to be concrete in case there exists a faithful functor from C to Sets. For instance, the homotopy category of the category of topological spaces, Ho(Top), is not concrete, as proved by Peter Freyd.1 Higher categories, such as the category to which modern homotopy theory gives rise, provide other examples of non-concrete categories. It is a fact that the category Sets is pervasive throughout mathematics, and functo- rial semantics cannot get rid of it, not in the least. Moreover, the Yoneda lemma itself proves that any category, however abstract it is, can be embedded into a category of presheaves of sets.2 Nevertheless, Sets is brought up almost always merely as the “uni- versal” or “free” category with colimits generated by one object (any singleton set). So an object of Sets has not to be construed as a set in the sense of classical set theory, and Sets, as a universal category, is characterized up to categorical equivalence only, which is a way to keep track of the generality of most mathematical constructions. Moreover, if you fix a base topos S and work with S-toposes (toposes over S), the set theory con- stituted by the internal language of S becomes the standard set theory. So ZFC is but one particular case of set theory. This led William Lawvere to extract “an elementary theory of the category of sets” in the form of category-theoretic “elementary” axioms, free of any reference to sets.3 Tom Leinster, in a recent text, “Rethinking set theory,” goes as far as to say: “The approach described here is not a rival to set theory: it is set theory.”4 This paper, however, does not seek to elicit arguments against ZFC, in favor of its replacement by an alternative category-theoretic framework. It rather claims that one should go beyond the contest, at least beyond too naively framed a contest between set theory and category theory, and that some mathematicians already went beyond it. Not that the contest should be left to mathematical practice to be settled. Indeed, the rivalry between set theory and category theory is undoubtably also a historical and sociologi- cal one, coming precisely from mathematical practice: Championing either set theory or category theory often implies that one extrapolates a tradition within mathematics (analysis and logical semantics in the case of set theory, algebraic geometry and mod- ern algebraic topology in the case of category theory) to the whole of mathematics. Nevertheless, the rivalry between set theory and category theory does certainly not boil down to a mere historical or sociological issue. It has a deep mathematical core. This is precisely the reason why several mathematical theories feed upon the combination of 1See [Freyd, 2004]. 2The Yoneda lemma applies only to a locally small category, and the notion of local smallness presup- poses the notion of set, so sets should be given beforehand anyway. But, once the category of sets is assumed, then most mathematical constructions can be understood as pertaining to presheaves of sets, and thus as be- ing pointwise set-theoretic in nature. And this is a result of category theory itself. I thank an anonymous referee for pointing this out to me. 3See [MacLane and Moerdijk, 1992] (p. 163) for a detailed exposition. 4[Leinster, 2012], p. 2. 2 set theory and category theory, which is much more fruitful than the endeavor to grant either theory the status of true foundations. This paper is devoted to a central, although too neglected, example of such a combination of both theories, namely Algebraic Set Theory (AST), a theory founded and built up almost from scratch in the nineties by two preeminent mathematicians, André Joyal and Ieke Moerdijk. Set theory may have a peculiar, foundational status, but it constitutes also a specific branch of mathematics, and as such interacts with the other branches of mathemat- ics. On that score, one can look at set theory from a non-set-theoretic point of view. This sounds very common (think of topos theory), but the purpose of AST, rather than replacing ZFC with something else, is to deepen ZFC by mixing it with a genuinely geometric perspective, where the shift from local data to a global item comes to the fore (this will become clearer in a while). Actually, AST can be described as a re- construction of ZFC in a category-theoretic framework inspired by modern algebraic geometry —and more specifically, as will presently appear, by descent theory, which calls for the theory of fibered categories. This is what this paper will argue and explain. Focusing on the first three axioms of the original axiomatization of AST by Joyal and Moerdijk, it aims to show in detail that these axioms make sense essentially in refer- ence to descent theory. One remarkable feature of AST is thus that it grafts the theory of fibered categories onto ZFC and thereby crossbreeds a new concept of set, or rather a new conception of what the very sets of ZFC, not the ones of some alternative topos, are. In doing so, AST goes beyond the unfruitful opposition often established between set theory and category theory. Furthermore, it challenges the linear order that would lead from foundational theories to specialized ones, and also holds out hopes of further applications of fibered categories to set theory and logic. 1 The first axioms of Algebraic Set Theory Up to now, AST has been explored along two main different approaches: a geometric one, embodied by the seminal work of Joyal and Moerdijk;5 and a logical one, rep- resented by Steve Awodey and Alex Simpson, among many others.6 To put it briefly, whereas Joyal and Moerdijk seek to generalize models of ZF with the notion of “free ZF-algebra,” in order to apply algebraic tools to models of set theory, the research program pursued by Awodey and his coworkers is to establish a detailed range of com- pleteness results between different axiomatizations of set theory (intuitionistic set the- ory, predicative set theory, and so on), on the one hand, and, on the other, different collections of “categories of classes” (i.e., different collections of models of variants of AST). In what follows, I will be interested only in the former approach, and leave aside the latter. 5[Joyal and Moerdijk, 1995]. 6For an overview, see [Awodey et al., 2014].
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