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Fundamental Groupoids As Generalised Pushouts of Codiscrete Groupoids

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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)i2I 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. Par´e[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)i2I 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 dif- ferentiable 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 Universit`adi 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. Par´eon 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)i2I 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 al- ready 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 computa- tion. 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. Par´e. 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)i2I 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 I^ of its subsets fi; jg having one or two elements, ordered by the relation fi; jg ≤ fig. Plainly, I^ is a connected category (a term which - as usual - also means non-empty), with maps indexed on the pairs (i; j) (i; j): fi; jg ! i; (i; j) 2 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 : I^ ! Gpd with values in the category of small groupoids, where the functor F : I^ ! Top is defined as follows F (fi; jg) = Xij = Xi \ Xj;F (fi; jg ! fig) = uij : Xij ⊆ Xi; (3) Proof. The proof is essentially the same as in [Bw2] (for I = f1; 2g), 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 2 I). Given a cocone of the functor π1F with vertex in a groupoid G, namely a family (fi)i2I such that fi : π1Xi ! G; fiuij = fjuji (i; j 2 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 π1XjA 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 jA 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 jA 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.
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