Fundamental Groupoids As Generalised Pushouts of Codiscrete Groupoids

FUNDAMENTAL GROUPOIDS AS GENERALISED PUSHOUTS OF CODISCRETE GROUPOIDS

ETTORE CARLETTI AND MARCO GRANDIS

Abstract. Every differentiable manifold X has a ‘good cover’, where all open sets and their finite intersections are 1-connected (and even contractible). Using a generalised van Kampen theorem for open covers we deduce that the fundamental groupoid of X (or any space admitting such an open cover) is a ‘generalised pushout’ of codiscrete groupoids and inclusions. This fact motivates the present brief study of generalised pushouts. In particular, we show that every groupoid is, up to equivalence, a generalised pushout of codiscrete subgroupoids; moreover, in any category, generalised pushouts amount to connected colimits, and - in the finite case - to ordinary pushouts and coequalisers.

1. Introduction This article starts from two facts. (a) A well-known groupoid-version of the van Kampen theorem, by R. Brown [Bw1, Bw2], determines the fundamental groupoid of a space X as a pushout of fundamental groupoids of (convenient) subspaces X1,X2; the latter (and their intersection!) are not supposed to be path-connected - a crucial advantage with respect to the classical formulation for the fundamental group. This groupoid version can be easily extended to an open cover (Xi)i∈I of the space X: then the fundamental groupoid π1X is the ‘gluing’ of the fundamental groupoids π1Xi ‘over’ the groupoids π1(Xi ∩ Xj) - a colimit of a particular form, that is called here a ‘generalised pushout’ (see Sections 2 to 4). We readily note that generalised pushouts are connected, non simply connected colimits and therefore cannot be reduced to ordinary pushouts (see R. Paré[P1]). (Let us recall that in this terminology a connected category is assumed to be non-empty; we do not follow this convention for spaces.) (b) Then, in Section 5, we recall from the text of R. Bott and L.W. Tu [BT] that every differentiable n-manifold X has a good cover. This means an open cover (Xi)i∈I such that n all finite intersections Xi1 ∩ ... ∩ Xik are either diffeomorphic to R or empty.

Putting both facts together, we deduce that the fundamental groupoid π1X of a differentiable manifold X is the generalised pushout of a diagram of codiscrete groupoids and inclusions; if X is compact, we can use a finite cover and a finite set of ‘representative points’, getting a finite generalised pushout of finite codiscrete groupoids. More generally, this holds for any space X having a ‘suitably good’ open cover (Xi), see Theorem 5.1.

Work supported by a PRIN Research Project and a Research Contract of Universitàdi Genova. 2000 Mathematics Subject Classification: 55A5, 18A30, 53C22. Key words and phrases: fundamental groupoid, generalised pushout, connected colimit, differentiable manifold. 1 2 With this motivation we go on studying generalised pushouts, in a line well represented in category theory: exploring particular classes of (co)limits, like: - filtered colimits, - splitting of idempotents, - flexible (co)limits, - connected (co)limits, - connected, simply connected (co)limits. First we show, in Proposition 6.1, that - up to categorical equivalence - every groupoid is a generalised pushout of codiscrete groupoids and inclusions (in a form that is weaker than that of the previous theorem, besides being of no help for any computation of fundamental groupoids). Then, in the last three sections, we consider generalised pushouts in a general category. Applying results of a second paper of R. Paréon connected limits [P2], we show that generalised pushouts give all connected colimits, while finite generalised pushouts amount to finite connected colimits or, equivalently, to pushouts and coequalisers. As an extension, suggested by F.W. Lawvere, it would be interesting to investigate ‘codiscretely generated’ toposes, like symmetric simplicial sets [Gr], i.e. presheaves on finite non-empty sets. Another possible extension can be based on a paper by R. Brown and G. Janelidze [BwJ] on categorical descent theory, that studies wide extensions of the Van Kampen theorem to ‘lextensive categories’, comprising various settings of a topological or algebraic nature. The present results of Sections 2 to 5 can likely be extended to this framework, replacing - loosely speaking - the open cover of a topological space with a family of subobjects (Xi)i∈I of an object X such that the canonical map Σi Xi → X (defined on the sum of the given subobjects) is an effective descent morphism. One should be aware that, for the standard topological case, this version applies to spaces where the fundamental groupoid classifies all coverings, as indicated in [BwJ], p. 256. (See also [BoJ].) Finally, let us remark that the concrete computation of fundamental groupoids is already well covered in the literature, see [Bw2, BwS] and references therein. In particular the present Theorem 3.2 is the Main Theorem of [BwS], in a version that makes no use of disconnected colimits (namely categorical sums) but says nothing more for computation. Our goal is rather to isolate a kind of connected colimit motivated by the previous arguments and study its categorical aspects. We gratefully acknowledge helpful information and suggestions from R. Brown, G. Janelidze, F.W. Lawvere and R. Paré.

2. A van Kampen theorem for groupoids and open covers

We begin by recalling R. Brown’s formulation of the van Kampen theorem for fundamental groupoids; a stronger version will be recalled and used in the next section. 3

2.1. Theorem. (R. Brown) Let X be a topological space, with two subspaces X1,X2 such that X = int(X1) ∪ int(X2). Then the following diagram of functors induced by inclusions is a pushout of groupoids

π1(X1 ∩ X2) / π1(X1) (1) π1(X2) / π1(X) Proof. See [Bw2], Section 6.7.2, ﬁrst part: ‘the case A = X’. The crucial advantage with respect to the classical formulation for the fundamental group is that, here, we need not assume that the spaces X1,X2 and X1 ∩ X2 be path- connected; for instance, this formulation allows one to compute the fundamental groupoid of the circle (and deduce its fundamental group). Now, the previous statement can be generalised to a space X equipped with a family of subspaces (Xi)i∈I such that X is covered by their interior parts. Then the fundamental groupoid π1X is the ‘gluing’ of the fundamental groupoids π1Xi ‘over’ the groupoids π1(Xi ∩ Xj). (A similar result, based on ﬁnite instead of binary intersections, is suggested in Exercise 6 of [Bw2], Section 6.7.) More precisely, we have the following result. We start from a non-empty set I and form the set Iˆ of its subsets {i, j} having one or two elements, ordered by the relation {i, j} ≤ {i}. Plainly, Iˆ is a connected category (a term which - as usual - also means non-empty), with maps indexed on the pairs (i, j) (i, j): {i, j} → i, (i, j) ∈ I2, (2) (together with other identities, of no relevance for limits or colimits).

2.2. Theorem. In the hypotheses above, π1X is the colimit of a composed functor π1F : Iˆ → Gpd with values in the category of small groupoids, where the functor F : Iˆ → Top is deﬁned as follows

F ({i, j}) = Xij = Xi ∩ Xj,F ({i, j} → {i}) = uij : Xij ⊆ Xi, (3) Proof. The proof is essentially the same as in [Bw2] (for I = {1, 2}), and we only sketch it. −1 A (Moore) path a: [0, p] → X gives an open cover Vi = a (int(Xi)) of the compact metric space [0, p] (independently of how big the set of indices I is). Fixing a Lebesgue number for this cover, we can decompose a into a ﬁnite concatenation of paths a = a1 ∗...∗an so that the image of each of them is contained in some Xi. Similarly, a homotopy with ﬁxed endpoints h: [0, p]×[0, q] → X can be decomposed on ‘small rectangles’, so that the image of h on each of them is contained in some Xi. Then we prove the universal property for the cocone of the functor π1F consisting of the canonical morphisms π1Xi → π1X (i ∈ I). Given a cocone of the functor π1F with vertex in a groupoid G, namely a family (fi)i∈I such that

fi : π1Xi → G, fiuij = fjuji (i, j ∈ I), (4) 4 one can (and must) define f : π1X → G on the class of the previous path a = a1 ∗ ... ∗ an letting f[ar] = fi[ar], if Im(ar) ⊆ Xi, f[a] = f[a1] ∗ ... ∗ f[an]. The result does not depend on the various choices we have made because of: - the cocone condition (4), - the usual argument on refining concatenations of the same path, - some computations on the decomposition of a path-homotopy on ‘small rectangles’ (that one can find in the cited book).

3. Representative sets of points Actually, R. Brown states from the beginning (in [Bw1, Bw2]) a stronger version of The- orem 2.1, more useful for computation. For a subset A of the space X, we shall write π1X|A the restriction of π1X to A, i.e. the full subgroupoid of π1X with vertices in A (the notation of [Bw2] is π1XA); more 0 0 0 generally, for a subspace X of X, we write π1X |A for the restriction of π1X to the subset X0 ∩ A. The subset A is said to be representative in X0 if A meets all its path-connected 0 0 components; it is then easy to show that the groupoid π1X |A is a retract of π1X .

3.1. Theorem. (R. Brown) Let X be again a topological space with two subspaces X1,X2 whose interior parts cover X. Let A be a subset of X that is representative in X1,X2 and X0 = X1 ∩ X2. Then the following diagram of functors induced by inclusions is a pushout of groupoids

π1X0|A / π1X1|A (5) π1X2|A / π1X|A Proof. See [Bw2], Section 6.7.2, second part: ‘the general case’, where the present result is deduced from ‘the case A = X’ (here stated in Theorem 2.1) showing that the commutative square above is a retract of the commutative square (1), and applying a general result on a square diagram that is a retract of a pushout square ([Bw2], Section 6.6.7). Also this theorem can be extended to a cover indexed on an arbitrary set I. Here the previous argument based on retracts would fail (because of coherence problems in the diagram of retracts), but one can apply a result by R. Brown and A.R. Salleh [BwS]. 3.2. Theorem. (Brown-Salleh) Let us consider a space X equipped with a family of subspaces (Xi)i∈I such that X is covered by their interior parts; let A be a subset of X that is representative in each Xi ∩ Xj ∩ Xk, for i, j, k ∈ I. Then the groupoid π1X|A is the gluing of the groupoids π1Xi|A ‘over’ the groupoids π1(Xi ∩ Xj)|A (as made precise in Section 2). 5

Proof. The present statement is a rewriting of the Main Theorem of [BwS], where π1X|A is expressed as the coequaliser of two morphisms between sums of groupoids (i.e. disjoint unions) / a, b:Σ(i,j) π1(Xi ∩ Xj)|A / Σi π1Xi|A, (6) respectively determined by the inclusions Xi ∩ Xj ⊆ Xi and Xi ∩ Xj ⊆ Xj. In fact, when categorical sums exist, one can easily rewrite a generalised pushout in such a form, by a simpliﬁed version of the usual construction of (general) colimits. This is made explicit below, in Remark 4.1.

3.3. Remark. The article [BwS] shows, by a counterexample, that it is not suﬃcient to assume that A be representative in each binary intersection Xi ∩ Xj. In Hatcher’s text [Ha], p. 44, one can ﬁnd another counterexample, easier to de- scribe: the space X is the suspension of the discrete space on three (distinct) points P = {p1, p2, p3} and Xi = X \ {pi}; then the singleton A = {x0} at a vertex of the suspension is representative in each binary intersection Xi ∩Xj, but not in the ternary one X \ P which is disconnected. And indeed π1X|A = π1(X, x0) is the free group on two generators while the coequaliser of the maps above would give the free group on three generators. 0 On the other hand, the set A = {x0, x1} formed by both vertices of the suspension is also representative in X \ P , and gives the correct result. The interested reader can extend these considerations letting P be the discrete space on four points (or more). Now the vertex A = {x0} is representative in all intersections Xi ∩ Xj ∩ Xk, and the theorem holds true even though A is not representative in the ‘total’ intersection X \ P , which is disconnected. The authors are indebted to R. Brown for helpful comments on this point and for correcting an error in a previous version of this note.

4. Generalised pushouts We are thus interested in colimits (in an arbitrary category C) of a particular form, that will be called ‘generalised pushouts’. We start from a non-empty set I and form, as above, the set Iˆ of its subsets {i, j} having one or two elements, ordered by the relation {i, j} ≤ {i} and viewed as a connected category. A functor X : Iˆ → C will be represented as the left diagram below

u X X ij 5 i 5 i fi X (i, j ∈ I),X ( Y ij ij 6 (7) uji ) ) fj Xj Xj

A cocone (of vertex Y ) amounts to family fi : Xi → Y of morphisms of C such that all the right-hand squares above commute. The colimit will be called the generalised 6 pushout of the objects Xi over the objects Xij. We speak of a ﬁnite generalised pushout, or n-generalised pushout when the set I is ﬁnite, with n elements. Plainly, 1-generalised pushouts are trivial and 2-generalised pushouts are ordinary pushouts. A 3-generalised pushout is the colimit of a diagram X : Iˆ → C with I = {1, 2, 3}

3 X3 9 e < b ˆ {1, 3} {2, 3} IX13 X23 X (8)

} ! ~ 1 o {1, 2} / 2 X1 o X12 / X2

We prove below that they give all finite generalised pushouts and amount to pushouts and coequalisers (Section 7). The fact that pushouts are not sufficient to construct all finite generalised pushouts is already known from a paper of R. Paré[P1] (cf. Theorem 2) that characterises the limits that can be constructed with pullbacks. In fact, the categories Iˆ and Iôp associated to I = {1, 2, 3} (or any larger set) are not simply connected: the left figure above shows a non-trivial loop in Iˆ, whence a non-trivial endomorphism in the fundamental groupoid ˆ ôp π1(I) = π1(I ) (cf. [P1]).

4.1. Remark. Of course, categorical sums are disconnected colimits and are out of the main line of our approach. Let us recall however that, assuming that the category C has sums, a generalised pushout in C (or any colimit) amounts to a coequaliser of two maps between sums. This has already been used above to reduce Theorem 3.2 to the original form of [BwS]. To this eﬀect, we should replace the standard construction of a colimit with a simpliﬁed form - in the same way as one can construct a pushout as a quotient of a binary sum, instead of a ternary one. Concretely, let a functor X : Iˆ → C be given, with notation as above. Then its colimit is the same as the coequaliser of a pair of morphisms (where (i, j) ∈ I2 and i ∈ I)

/ a, b:Σ(i,j) Xij / Σi Xi. (9)

The latter are determined as follows, letting wij and wi denote the injections of the two sums:

awij = wiuij : Xij → Σi Xi, bwij = wjuji : Xij → Σi Xi.

In fact, a map f :Σi Xi → Y amounts to a family of maps fi : Xi → Y (i ∈ I), while 2 the condition fa = fb amounts to fawij = fbwij i.e. fiuij = fjuji (for all (i, j) ∈ I ), which is the cocone condition for X, in (7). 7 5. Good covers of manifolds and spaces As proved in the text of R. Bott and L.W. Tu on differential forms ([BT], Theorem 5.1), every differentiable n-manifold X has a good cover. This means an open cover (Xi)i∈I n such that all the non-empty finite intersections Xi1 ∩ ... ∩ Xik are diffeomorphic to R . The open subsets Xi are constructed as geodesically convex subsets, for a Riemannian metric on X. Since we only need to know that their non-empty intersections are 1- connected, it would be sufficient to apply a more elementary result: a non-empty geodesically convex (or just star shaped) subspace of a differentiable manifold is contractible ([Co], Lemma 10.5.5); this is indeed a weaker property [Wh]. (It would be interesting to know if the previous result on good covers can be extended to topological manifolds, replacing diffeomorphic to Rn with contractible, or just 1- connected. Let us recall that, while in dimension up to 3 every topological manifolds has a differentiable structure [Mo], this is no longer true in higher dimension [Sc].) More generally, we now consider a topological space X having an open cover (Xi) such that all subspaces Xi ∩ Xj (including the original Xi, of course) are 1-connected: in other words we assume that the following groupoids are codiscrete, or chaotic (i.e. have precisely one arrow x → y for any two vertices x, y)

π1(Xi) = C(Xi), π1(Xi ∩ Xj) = C(Xi ∩ Xj) = C(Xi) ∩ C(Xj), (10) where we write C(S) for the codiscrete groupoid on a set of objects S, possibly empty.

5.1. Theorem. Let us suppose that the topological space X has an open cover (Xi) such that all subspaces Xi ∩ Xj are 1-connected. ˆ (a) Then the fundamental groupoid π1X is the generalised pushout of a diagram C : I → Gpd of codiscrete groupoids and inclusions

uij 4 Ci

Cij Ci = C(Xi),Cij = C(Xi ∩ Xj)(i, j ∈ I). (11) uji * Cj

(b) All the objects of diagram (11) are subgroupoids of π1X (the colimit), and moreover

Cij = Ci ∩ Cj. (12)

(c) If X has a finite cover (Xi) of the previous type (which is a consequence of the previous assumption, for a compact X), then π1X is a finite generalised pushout of codiscrete groupoids and inclusions. Moreover, if each subspace Xijk = Xi ∩ Xj ∩ Xk has a finite number of path components, we can choose a finite subset A of X representative in each Xijk so that the restricted groupoid π1X|A is a finite generalised pushout of finite codiscrete subgroupoids and inclusions. Proof. It is a straightforward consequence of Theorems 2.2, 3.2 and the obvious fact that a codiscrete groupoid on finitely many vertices is finite. 8

5.2. Corollary. Such results are automatically true for every diﬀerentiable manifold X. If X is compact, also the additional hypotheses of 5.1(c) automatically hold.

Proof. Follows from the existence of good covers recalled above.

6. Generalised pushouts of groupoids

We prove now a general result on groupoids, similar to the particular case that we have examined in the previous section, yet less powerful. In fact: - it works up to categorical equivalence, - it does not cover the points (b), (c) of Theorem 5.1, - it says nothing about computing fundamental groupoids (obviously).

6.1. Proposition. Every groupoid is categorically equivalent to some groupoid G that is a generalised pushout of codiscrete groupoids and inclusions. These groupoids can be assumed to be subgroupoids of G, and non-empty if G is connected.

Proof. As a motivation of making appeal to categorical equivalence, note that a group has only one codiscrete subgroupoid, the trivial one, and cannot be a generalised pushout of codiscrete subgroupoids - unless it is trivial. (But one can prove, by an argument similar to the following one, that every connected groupoid on at least three vertices is a generalised pushout of codiscrete subgroupoids and inclusions.) We can suppose that our groupoid is connected (non-empty); then it is equivalent to its skeleton, which is a group G0, and we replace the latter with an equivalent groupoid G on three vertices, say 1, 2, 3. Let I be the set of commutative diagrams in G of the following form 2 x 7 y ' (13) 1 z / 3 z = yx. It will be convenient to denote this diagram by the triple (x, y, z), even though each pair of these arrows determines the third; we write F (x, y, z) the subgroupoid of G generated by these arrows: it is formed by all the objects, their identities, the three given maps and their inverses. I can be identified with the set of all the wide codiscrete subgroupoids of G. It is also easy to see that the set I is in bijective (non-canonical) correspondence with G0 × G0: after fixing a diagram (13) and identifying G(1, 1) with G0, each pair 0 (g, h) ∈ G0 ×G0 determines a triple (xg, y , zh). We now form a diagram F : Iˆ → Gpd of codiscrete groupoids and inclusions. F is already defined on the triples (x, y, z) ∈ I. For two distinct triples (x, y, z), (x0, y0, z0) we distinguish two cases: (i) if x = x0 or y = y0 or z = z0, we let

F (x, y, z; x0, y0, z0) = (F (x, y, z) ∩ F (x0, y0, z0))ˆ, 9 where (−)ˆ means taking out the isolated vertex (since the intersection itself is not codiscrete), (ii) otherwise we let F (x, y, z; x0, y0, z0) be the (co)discrete groupoid on the vertex 1 (since we do not want to use the empty groupoid).

It is now evident that every cocone fxyz : F (x, y, z) → H (for (x, y, z) ∈ I) of F has precisely one extension to a ‘mapping’ f : G → H, which necessarily preserves identities and composition. The only point that is not completely trivial is showing that two arbitrary morphisms fxyz and fx0y0z0 coincide on each vertex. Working for instance on the vertex 3, it is suﬃcient to consider the objects

F (x, y, z),F (x00, y0, z),F (x0, y0, z0)(x00 = z−1y0),

and remark that the cocone condition gives: fxyz(3) = fx00y0z(3) = fx0y0z0 (3).

7. Generalised pushouts and connected colimits We shall now study generalised pushouts in a general category C. We already noted that generalised pushouts are connected, non-simply connected colimits, and therefore cannot be reduced to ordinary pushouts [P1]. We shall now make use of a second paper of R. Paréon connected limits [P2], where it is proved that all of them can be constructed with fibred products and equalisers. A fibred product is the limit of a family hi : Xi → X of morphisms indexed on a non-empty set I. Plainly, the existence of finite fibred products is equivalent to the existence of pullbacks.

7.1. Theorem. (R. Paré)The category C has arbitrary (resp. finite) connected colimits if and only if it has arbitrary (resp. finite) fibred coproducts and coequalisers. Proof. See [P2] where the dual case is proved; the finite case (and also the general one) can be found in Section 4.

7.2. Lemma. If the category C has arbitrary (resp. finite) generalised pushouts then it has arbitrary (resp. finite) fibred coproducts.

Proof. Starting from a family hi : X → Xi of morphisms of C indexed on a non-empty ˆ set I, we transform it into a diagram X : I → C. After the given objects Xi, we let:

Xij = X, for i 6= j,

uij = hi : X → Xi, for i 6= j; uii = idXi.

The given diagram (hi : X → Xi)i∈I and the new one have the same cocones, namely the families of morphisms fi : Xi → Y (i ∈ I) such that fihi = fjhj for all pairs i, j. Therefore the existence of (finite) generalised pushouts implies the existence of (finite) fibred coproducts. 10

7.3. Lemma. If the category C has 3-generalised pushouts then it has ordinary pushouts and coequalisers.

Proof. Suppose that C has 3-generalised pushouts. To prove the existence of pushouts, starting from the left diagram below

X X X u12 5 1 u13 5 1 u23 5 2 X12 X13 X23 (14) u21 ) u31 ) u32 ) X2 X3 X3

we extend it to a diagram on I = {1, 2, 3}, where X3 = X2, the second span above coincides with the ﬁrst and the third consists of identities u23 = u32 = idX2. Plainly the extended diagram has the same cocones as the original one and its colimit is the pushout of the latter. To prove that C has coequalisers, we start from two arrows u, v : X → Y and construct a new diagram on I = {1, 2, 3}, where X1 = X2 = X3 = Y and the three spans are speciﬁed below

X X X u 5 1 id 5 1 id 5 2 X Y Y (15) v ) id ) id ) X2 X3 X3

For the new diagram, a cocone (fi : Xi → Z) is a map f = f1 = f2 = f3 : Y → Z such that fu = fv; the 3-generalised pushout is thus the coequaliser of u, v.

7.4. Corollary. The following conditions on a category C are equivalent: (a) C has generalised pushouts, (b) C has ﬁbred coproducts and coequalisers, (c) C has connected colimits.

Proof. (a) ⇒ (b) From the previous lemmas. (b) ⇒ (c) From Theorem 7.1. (c) ⇒ (a) Obvious.

7.5. Corollary. The following conditions on a category C are equivalent: (a) C has ﬁnite generalised pushouts, (b) C has 3-generalised pushouts, (c) C has pushouts and coequalisers. (d) C has ﬁnite connected colimits.

Proof. (a) ⇒ (b) Obvious. (b) ⇒ (c) From Lemma 7.3. (c) ⇒ (d) From Theorem 7.1. (d) ⇒ (a) Obvious. A direct proof of the implication (c) ⇒ (a), without using connected colimits, will be given in the last section. 11 8. Finite generalised pushouts

In a general category C, the ‘weight’ of generalised pushouts within finite colimits can be measured noting that each of the following conditions is strictly implied by the previous one (the equivalences in (b) have been proved in the last corollary above) (a) C has finite colimits, (b) C has finite generalised pushouts, or (equivalently) finite connected colimits, or pushouts and coequalisers, or 3-generalised pushouts, (c) C has pushouts. The implications (a) ⇒ (b) ⇒ (c) cannot be reversed. To wit, the category of non- empty sets has all colimits of non-empty diagrams but lacks an initial object. Secondly, the groupoid of sets and bijections has pushouts (trivially constructed) but lacks coequalisers. More generally, every groupoid has (trivial) pullbacks and pushouts; but it has coequalisers if and only if it is an equivalence relation (i.e. a subgroupoid of a codiscrete one), if and only if it has equalisers. One can also note that the category of sets and surjective mappings has pushouts and coequalisers but no initial object nor binary sums, while the category of sets and injective mappings has dual properties.

9. A direct proof

Finally, we give a direct, elementary prove of the implication (c) ⇒ (a) in Corollary 7.5, without going through connected colimits: if C has pushouts and coequalisers, then it also has ﬁnite generalised pushouts. Actually, we prove the dual fact, which is easier to follow (thinking of limits of sets, that have an easy construction). We suppose that C has pullbacks and equalisers and construct the limit of a diagram X : Iˆop → C, where I = {1, ..., n} and n ≥ 1

Xi uij * X (i, j ∈ I). 4 ij (16) uji Xj

One cannot work directly with iterated pullbacks, because the procedure ‘would never end’; indeed, we know that pullbacks are not suﬃcient to construct the limit.

First we restrict the diagram to the consecutive pairs (i, i + 1). The limit (P, v1, ..., vn) 12 of the restricted diagram can be constructed with a ﬁnite number of pullbacks

X1 u = 12 , 2 X12 v1 X u21 6 2 , X23 v2 . . P . . (17) vn−1 Xn−2,n−1 ' 2 vn Xn−1 un−1,n , 2 Xn−1,n

un,n−1 Xn

Now, for every pair i < j of diﬀerent indices we have two morphisms

fij, fji : P → Xij, fij = uijvi, fji = ujivj, (18)

(that already coincide when j = i + 1, but we can forget about that). Let

eij : Eij → P (i < j), (19) be the equaliser of this pair, and let w : E → P be the intersection of this ﬁnite family of subobjects of P (which again can be constructed with a ﬁnite number of pullbacks, starting from the family of all morphisms eij). Finally, the family wi = viw : E → Xi (i ∈ I) is a cone of the original diagram, because uijviw = ujivjw, and it is obviously the universal one.

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