Toposes are adhesive Stephen Lack1 and Pawe lSoboci´nski2? 1 School of Computing and Mathematics, University of Western Sydney, Australia 2 Computer Laboratory, University of Cambridge, United Kingdom Abstract. Adhesive categories have recently been proposed as a cate- gorical foundation for facets of the theory of graph transformation, and have also been used to study techniques from process algebra for reason- ing about concurrency. Here we continue our study of adhesive categories by showing that toposes are adhesive. The proof relies on exploiting the relationship between adhesive categories, Brown and Janelidze’s work on generalised van Kampen theorems as well as Grothendieck’s theory of descent. Introduction Adhesive categories [11,12] and their generalisations, quasiadhesive categories [11] and adhesive hlr categories [6], have recently begun to be used as a natural and relatively simple general foundation for aspects of the theory of graph transfor- mation, following on from previous work in this direction [5]. By covering several “graph-like” categories, they serve as a useful framework in which to prove struc- tural properties. They have also served as a bridge allowing the introduction of techniques from process algebra to the field of graph transformation [7, 13]. From a categorical point of view, the work follows in the footsteps of dis- tributive and extensive categories [4] in the sense that they study a particular relationship between certain finite limits and finite colimits. Indeed, whereas distributive categories are concerned with the distributivity of products over co- products and extensive categories with the relationship between coproducts and pullbacks, the various flavours of adhesive categories consider the relationship between certain pushouts and pullbacks. Adhesive categories are defined to be the categories with pullbacks where pushouts along monomorphisms are van Kampen [11] and as a consequence such pushouts can be considered as being well-behaved with respect to pullbacks. As we shall explain, related work includes Grothendieck’s theory of descent (see [9] for an overview) and generalised approaches to the van Kampen theorem [1]. As shown in [11], adhesive categories are closed under several categorical constructions, thus if C and D are adhesive then so is their product C × D, choosing any object C ∈ C, the slice category C/C and coslice category C/C are adhesive and, for any category X, the functor category [X, C] is adhesive. It is ? Research partially supported by epsrc grant EP/D066565/1. The first author grate- fully acknowledges the support of the Australian Research Council. also known that the category of sets and functions Set is adhesive, in particular this means that any presheaf topos [X, Set] is adhesive. These constructions are useful because adhesive categories satisfy many of the so-called hlr-axioms, and as a consequence, several important theorems about the rewriting theory of double-pushout transformations can be proved at the level of adhesive categories. Indeed, it is perhaps surprising that so many of the axioms, which were not known previously to be related, hold automatically in any adhesive category. One of the original contributions of this paper is a proof that adhesive categories satisfy the special pullback-pushout property, one of the aforementioned axioms. The central part of the paper is devoted to studying the relationship be- tween toposes and adhesive categories. Topos theory has many different facets and applications within mathematics and computer science. One of the interest- ing properties of toposes is that they have finite limits and colimits, and these behave somewhat as they do in Set. Indeed, while toposes enjoy much more structure than adhesive categories, adhesive categories themselves have certain finite colimits (pushouts along monomorphisms) and limits (pullbacks) which are well-behaved with respect to each other. The question of whether toposes are adhesive is thus a very natural one. As we have shown in [11, 12], there are adhesive categories which are not toposes. Here we prove that the converse is not true – indeed, the main contri- bution of the paper is Theorem 26, the conclusion of which states that toposes are adhesive. From a computer science perspective, this means that the theory developed for adhesive categories can be applied to any topos, not just a presheaf. This is a significant development, since topos theory is a well-established math- ematical discipline with many important results and wide-reaching applications. One interesting example of a category which is a topos and not a presheaf is the Schanuel topos. It has been used (see [8], for example) to model languages with name binding, such as the Pi-calculus. Our theorem thus allows us to apply the rewriting theory developed for adhesive categories to such a setting. While we do not study this example in detail within the present paper, we plan to study such systems as part of future work. The Schanuel topos actually arises as a full subcategory of a presheaf category with objects the (atomic) sheaves. It is a general fact that any such category of sheaves is a topos. The proof of our theorem relies on exploiting the connections between adhe- sive categories (or more generally, van Kampen squares), Brown and Janelidze’s generalised van Kampen theorems and Grothedieck’s theory of descent. Indeed, in order to prove that toposes are adhesive, we must show that pushouts along monomorphisms are van Kampen. To do so, we split such a pushout into two: one pushout with all morphisms mono, and one with two monomorphisms and two epimorphisms. The former is also a pullback in any topos, and a theorem of Brown and Janelidze’s (here recalled as Theorem 23) guarantees that it satisfies the van Kampen theorem – which implies that the original pushout is a van Kampen square. Here we also prove that pushouts of the latter kind are van Kampen in a topos, it is the most difficult and technical result of the paper. 2 f C / B m n A g / D Fig. 1. Pushout diagram. This completes the proof because, as we show in Lemma 2, van Kampen squares compose in categories with pullbacks and pushouts. Structure of the paper. In Section 1 we recall two equivalent ways of defining van Kampen squares and show that van Kampen squares compose in categories with pushouts and pullbacks. We recall the definition of adhesive categories and prove that adhesive categories enjoy the special pullback-pushout property. Section 2 recalls the fragments of descent theory and topos theory necessary for our main result. In Section 3, we recall the theorem of Brown and Janelidze, which forms one of the ingredients of the proof of our main Theorem 26. Theorem 25 is the other main ingredient, and its proof relies on the background introduced in Section 2. We conclude in Section 4 with several directions for future work. The paper is relatively self-contained, although we omit the proofs of well-known results and instead provide references to standard sources. 1 Van Kampen squares and adhesive categories Adhesive and quasiadhesive categories are defined using certain pushout dia- grams which are called van Kampen squares. We refer the reader to [11, 12] for an introduction to, the enumeration of the basic properties of, and the appli- cations of adhesive and quasiadhesive categories. Here we shall concentrate on the definitions of van Kampen squares and derive the properties which will be needed for the proof of our main result. We shall also prove that adhesive cat- egories satisfy the so-called special pullback-pushout property, which is one of the many hlr axioms, the majority of which have already been shown in [11,12] to hold in any adhesive category. We shall need both the axiomatic and the “equivalence of categories” versions of the definitions of van Kampen squares [12]. In particular, using the former, we shall show that van Kampen squares compose in categories with pushouts and pullbacks, and that adhesive categories satisfy the special pullback-pushout property. The fact that van Kampen squares compose, together with the latter, equivalent, way of defining van Kampen squares will be used in the proof of our main Theorem 26. The latter definition of van Kampen squares also makes it possible to establish a relationship between van Kampen squares and Brown and Janelidze’s generalised van Kampen theorems, as we shall explain in §3. 3 Definition 1. A van Kampen square is a pushout di- 0 0 m0 ii C K f agram as in Fig 1 which satisfies the following condi- iiii KK 0 tiii c % 0 A i B tion: KKK iiii g0 % iii D0 ti n0 – for any commutative cube, as illustrated, of which a b Fig 1 forms the bottom face and the back faces are m C f iiii KKK pullbacks: the front faces are pullbacks iff the top iii K% A ti d B KK iiii face is a pushout. gKK iiin % D tii The following lemma shows that, in categories with pushouts and pullbacks, van Kampen squares paste together to give van Kampen squares. Lemma 2. Consider the illustrated commutative diagram · / · / · in a category with pushouts and pullbacks. If (1) and (2) (1) (2) are van Kampen then so is (1)+(2). · / · / · Proof. Straightforward; in order to show that the combined pushout is stable under pullback it suffices to break up a cube into two cubes, using the existence of pullbacks. Conversely, a cube with its top face a pushout, can be split into two using the existence of pullbacks and pushouts. ut We shall now recall an equivalent definition of van Kampen squares which will be useful for the purposes of this paper. The reader is referred to [12] for the proof that the definitions are equivalent.