Stacks

Here I will add notes of the seminar that I am planning to conduct in School of Mathematics, IISER Thiruvananthapuram, India. First lecture is expected to happen on 14 August 2019. --- Some terms which I want to convey the meaning of in this Seminar. Manifold. Differential forms on Manifolds; pullbacks and differential of a Differential form. Lie group. Lie algebra of Lie group. Cohomology of Manifolds / Cohomology of Lie groups. Principal/Vector bundle. Connection (on principal/vector bundle). Curvature (of Connection on principal/vector bundle). Holonomy group. Ambrose-Singer theorem. Characteristic classes (Euler/Chern classes). Lecture notes/Articles : The Topology of Fiber Bundles --- Lecture Notes --- Ralph L. Cohen WHAT IS A CONNECTION? --- TIMOTHY E. GOLDBERG Books: The Topology of Fibre Bundles by Steenrod Foundations of Differentiable Manifolds and Lie Groups by Frank Warner Foundations of Differential Geometry by Kobayashi and Nomizu Introduction to Smooth Manifolds by John Lee Geometry of Differential forms by Shigeyuki Morita Topics in Differential Geometry by Peter W. Michor Differential Geometry - Connections, Curvature, and Characteristic Classes by Loring Tu An Introduction to Manifolds by Loring Tu Differential Geometry, Lie Groups, and Symmetric Spaces by Sigurdur Helgason Differential Forms in Algebraic Topology by Bott and Tu A Geometric Approach to Differential Forms by David Bachman Modern Differential Geometry for Physicists 2nd Edition by Chris J Isham Differential Forms and Connections by R. W. R. Darling Differential Forms - A Heuristic Introduction by M. Schreiber From Calculus to Cohomology by Madsen and Tornehave Manifolds, Sheaves, and Cohomology by Torsten Wedhorn Principal Bundles : The Classical Case by Stephen Bruce Sontz Introduction to the Theory of Lie Groups by Roger Godement Differential Geometry: Bundles, Connections, Metrics and Curvature by Clifford Henry Taubes YouTube videos : Fredric Schuller's YouTube channel MathOverflow/MathStackExchange questions/user pages: John M. Lee 's MathStackExchange page  

Here I add notes about notation I use in this blog. I might use the notion of fibered category and category fibered in groupoids as if there is no difference. I am mostly interested in fibered category \(\mathcal{F}\rightarrow \mathcal{C}\) where the fibre \(\mathcal{F}(U)\) is a groupoid for every object \(U\) of \(\mathcal{C}\). So, most of the times when I say fibered category, it is most likely that I mean fibred categroy whose fibres are groupoids i.e., category fibered in groupoids. Please let me know if there is some real confusion.

I understood most of this from Introduction to the language of stacks and gerbes (section 2) by Ieke Moerdijk and from stacks project Stackification of fibred categories. It is necessary to know what is the sheafification of a presheaf to understand what is the stackification. I studied sheafification from Hartshorne's Algebraic geometry book. You can choose what you are comfortable with. I will mention the result first as in Lemma \(8.8.1\). Lemma : Let \(\mathcal{C}\) be a site. Let \(p:\mathcal{S}\rightarrow \mathcal{C}\) be a fibred category over \(\mathcal{C}\). There exists a stack \(p':\mathcal{S}'\rightarrow \mathcal{C}\) and a morphisms \(G:\mathcal{S}\rightarrow \mathcal{S}'\) of fibred categories over \(\mathcal{C}\) such that for every \(U\in \text{Ob}(\mathcal{C})\) and \(x,y\in \mathcal{S}(U)\), the map \(\text{Mor}(x,y)\rightarrow \text{Mor}(G(x),G(y))\) induced by \(G\) identifies the right hand side with the sheafification of the left hand side. For \(U\in \mathcal{C}_0\) and \(x'\in \mathcal{S}'(U)\) there exists a covering \(\{U_i\rightarrow U\}\) such that each \(x'|_{U_i}\) is in the essential image of the functor \(G:\mathcal{S}(U)\rightarrow \mathcal{S}'(U)\). We recall what is \(\text{Mor}(a,b)\). This is a presheaf on \(U\) defined as follows. Given an inclusion \(i : V\hookrightarrow U\) we have \(i^*(a),i^*(b)\in \mathcal{S}(V)\).

In this page, I will give links of Category theory posts that I have made here. I learned some category theory from Hilton and Stammbach's book A Course in Homological Algebra. Angelo Vistoli's Descent theory notes.

Let \(G\) be a Lie group and \(\pi_P:P\rightarrow M, \pi_Q:Q\rightarrow M\) be principal \(G\) bundles. Then, any \(G\)-equivariant map \(f:P\rightarrow Q\) inducing identity on \(M\) is a diffeomorphism. The same holds when we have Lie groupoids instead of Lie groups. Let \(\mathcal{G}\) be a Lie groupoid and \(P\rightarrow M, Q\rightarrow M\) be principal \(\mathcal{G}\) bundles. Then, any \(\mathcal{G}\)-equivariant map \(f:P\rightarrow Q\) inducing identity on \(M\) is a diffeomorphism. Above result is very basic thing when defining a stack associated for a Lie groupoid \(\mathcal{G}\). Given a Lie groupoid \(\mathcal{G}\), we define a category fibered in groupoids \(B\mathcal{G}\rightarrow \text{Man}\) by associating for each manifold \(U\) a category \(B\mathcal{G}(U)\) whose objects are principal \(\mathcal{G}\) bundles whose base space is \(U\) i.e., of the form \(P\rightarrow U\) and morphism from an object \(P\rightarrow U\) to another object \(Q\rightarrow U\) is a \(\mathcal{G}\)-equivariant map \(P\rightarrow Q\) that induces \(Id:U\rightarrow U\) on base space of those principal bundles. Thus, to say \(B\mathcal{G}(U)\) is a Lie groupoid, we need to prove that every arrow \((P\rightarrow U)\rightarrow (Q\rightarrow U)\) is an isomorphism which is what we are trying to prove. Let us see the proof for the case of Lie groups. See the set up as following diagram. Let \(p,p'\in P\) are such that \(f(p)=f(p')\), thus, \(\pi_Q(f(p))=\pi_Q(f(p'))\). As \(\pi_Q\circ f=\pi_P\), we have \(\pi_P(p)=\pi_P(p')\) i.e., there exists \(g\in G\) such that \(p'=p.g\). Thus, \(f(p')=f(pg)\). As \(f\) is \(G\)-equivariant, we have \(f(pg)=f(p)g\). Thus, we have \(f(p')=f(p)g\). As the action of \(G\) on \(Q\) is free, \(f(p')=f(p),f(p')=f(p)g\) implies \(g=1\). Thus, \(p'=p\). So, \(f\) is one to one mapping. Let \(q\in Q\). We have \(\pi_Q(q)\in M\). As \(\pi_P\) is surjective, there exists \(p\in P\) such that \(\pi_P(p)=\pi_Q(q)\). As \(\pi_Q\circ f=\pi_P\), we have \(\pi_Q(f(p))=\pi_P(p)=\pi_Q(q)\). As \(\pi_Q(f(p))=\pi_Q(q)\), there exists \(g\in G\) such that \(f(p)g=q\). As \(f\) is \(G\)-equivariant, we have \(f(p)g=f(pg)\). Thus, we have \(q=f(pg)\) which implies that \(f\) is an onto mapping. Suppose that \(\pi_P:P\rightarrow M\) is trivial \(G\) bundle, not for simplicity but because every principal \(G\) bundle is locally trivial and diffeomorphism is something that needs to be checked locally. As \(\pi_P:P\rightarrow M\) is trivial, it has a global section for \(\pi_P\) i.e., a smooth map \(\sigma:M\rightarrow P\) such that \(\pi_P\circ \sigma=1\). This gives a trivialization \(M\times G\xrightarrow{\Phi} P\) i.e., an isomorphism. Consider the cimposition \(f\circ \pi:M\rightarrow Q\). This is again a smooth map such that \(\pi_Q\circ (f\circ \sigma)=(\pi_Q\circ f)\circ \sigma=\pi_P\circ \sigma=1\) ...

A morphism of stacks \(F:\mathcal{D}\rightarrow \mathcal{C}\) is said to be a gerbe over stack if following two conditions hold : Given a manifold \(U\) and an object \(\xi\in \mathcal{C}(U)\), there exists a covering \(\{U_i\rightarrow U\}\) (depending on the Grothendieck topology that we have fixed on the category \(Man\) of manifolds) and objects \(x_i\in \mathcal{D}(U_i)\) with an isomorphism \(F(x_i)\rightarrow \xi|_{U_i}\) for each \(i\). Given a manifold \(U\) and an arrow \(\xi\rightarrow \eta\) in \(\mathcal{C}(U)\), there exists a covering \(\{U_i\rightarrow U\}\) (depending on the Grothendieck topology that we have fixed on the category \(Man\) of manifolds) and arrows \(x_i\rightarrow y_i\) in \(\mathcal{D}(U_i)\) such that

This post is based on (wanted to write after reading) Lie Groupoids and Differentiable stacks by Matias L. del Hoyo. I would suggest this for any one who wants to know about Lie groupoids and Differentiable stacks. This is well written. By a manifold, we always mean a smooth manifold. A Groupoid is a category where every arrow is invertible. A Lie groupoid is a groupoid with additional smooth structures on object set/morphism set and maps between them. Definition : A Lie groupoid consists of a manifold \(\mathcal{G}_0\) of objects, a manifold \(\mathcal{G}_1\) of arrows and following maps : \(s:\mathcal{G}_1\rightarrow \mathcal{G}_0\), a submersion, called the source map. \(t:\mathcal{G}_1\rightarrow \mathcal{G}_0\), a submersion, called the target map. \(m:\mathcal{G}_1\times_{s,\mathcal{G}_0,t}\mathcal{G}_1\rightarrow \mathcal{G}_1\), a smooth map, called the multiplication map. \(u:\mathcal{G}_0\rightarrow \mathcal{G}_1\), a smooth map, called the unit map. \(i:\mathcal{G}_1\rightarrow \mathcal{G}_1\), a smooth map, called the inverse map. with some compatibility conditions. We denote this Lie groupoid by \(\mathcal{G}_1\rightrightarrows \mathcal{G}_0\). Definition : Let \(\mathcal{G}_1\rightrightarrows \mathcal{G}_0\) be a Lie groupoid and \(x\in \mathcal{G}_0\). The set \(s^{-1}(x)=:G(x,-)\) is called the \(s\)-fibre of \(x\) . The set \(t^{-1}(x)=:G(x,-)\) is called the \(s\)-fibre of \(x\) The set \(s^{-1}(x)\cap t^{-1}(x)=:G_x\) is called the Isotropy group of \(x\). The set \(t(s^{-1}(x))=\{y:x\rightarrow y\in \mathcal{G}_1\}=:O_x\) is called the orbit of \(x\). Proposition : Given a Lie groupoid \(\mathcal{G}_1\rightrightarrows \mathcal{G}_0\) and \(x,y\in \mathcal{G}_0\), the subset \(G(y,x)\subseteq G\) is an embedded submanifold. In particular, \(G_x\) is a Lie group. the subset \(O_x\) is a (may not be embedded) submanifold in a canonical way. By a morphism of Lie groupoids \(\phi: (\mathcal{G}_1\rightrightarrows \mathcal{G}_0)\rightarrow (\mathcal{H}_1\rightrightarrows \mathcal{H}_0)\) we mean a pair of smooth maps \(\phi^{ar}:\mathcal{G}_1\rightarrow \mathcal{H}_1\) and \(\phi^{ob}:\mathcal{G}_0\rightarrow \mathcal{H}_0\) compatible with structure maps \(s,t,m,u,i\). We write \(\phi\) for both \(\phi^{ar}\) and \(\phi^{ob}\).

Given a manifold \(M\) we have the concept of open cover of \(M\). We usually write an open cover of a manifold \(M\) as a collection of open subsets \(\{U_i\}\) (such that \(\bigcup U_i=M\)). In this note we see an open cover of \(M\) as a collection of maps (inclusions) \(\{U_i\rightarrow M\}\). Some properties of "open cover" are. (Pull back exists and gives an open cover) Suppose \(\{U_i\rightarrow M\}\) is an open cover for \(M\) and \(\pi:V\rightarrow M\) is a smooth map. Then, \(\{\pi^{-1}(U_i) \rightarrow V\}\) is a cover for \(V\). (Diffeomorphisms gives open cover) For any manifold \(M\), \(M\) itself is considered as an open cover \(\{M\rightarrow M\}\). More generally, for any diffeomorphism \(M'\rightarrow M\), \(\{M'\rightarrow M\}\) is considered as an open cover. (Open cover of open cover is an open cover) Let \(\{U_\alpha\rightarrow U\}\) be an open cover for \(U\) i.e., \(\bigcup_{\alpha} U_\alpha=U\). Suppose \(\{V_{\alpha\beta}\rightarrow U_\alpha\}\) is an open cover for \(U_\alpha\) for each \(\alpha\) i.e., \(\bigcup_{\beta}V_{\alpha\beta}=U_\alpha\). Then, \(\bigcup_{\alpha\beta}V_{\alpha\beta}=U\) i.e., \(\{V_{\alpha\beta}\rightarrow U\}\) is an open cover for \(U\). For a category \(\mathcal{C}\) and an object \(U\) of \(\mathcal{C}\), a collection of arrows \(\{U_i\rightarrow U\}\) is said to be a cover for \(U\). Definition : Let \(\mathcal{C}\) be a category. A Grothendieck topology on \(\mathcal{C}\) is given by a collection of covers \(\mathcal{W}=\{\{U_i\rightarrow U\}: U\in \mathcal{C}_0\}\) satisfying following conditions. (Pullbacks exists and gives a cover) Suppose \(\{U_i\rightarrow U\}\in \mathcal{W}\) and \(\pi:V\rightarrow U\) be an arrow. Then, the pull back \(U_i\times_UV\) exists (as an object in \(\mathcal{C}\)) and \(\{U_i\times_UV \rightarrow V\}\) is a cover for \(V\). (Isomorphisms gives an open cover) Suppose \(V\in \mathcal{C}_0\) and \(V\rightarrow U\) is an isomorphism in \(\mathcal{C}\) then, \(\{V\rightarrow U\}\in \mathcal{W}\). (cover of a cover is a cover) Suppose \(\{U_\alpha\rightarrow U\}\in \mathcal{W}\) and \(\{U_{\alpha\beta}\rightarrow U_\alpha\}\in \mathcal{W}\) for each \(\alpha\). Then, the collection of compositions \(\{U_{\alpha\beta}\rightarrow U_\alpha\rightarrow U\}\in \mathcal{W}\). To talk about a stack over category \(\mathcal{C}\) we fix a Grothendieck topology \(\mathcal{W}\) on \(\mathcal{C}\). When we say cover, we mean it belongs to \(\mathcal{W}\). Let \(\mathcal{D}\) be a category fibered in groupoids over \(\mathcal{C}\) i.e., we have a functor \(F:\mathcal{D}\rightarrow \mathcal{C}\) satisfying some conditions. Given an object \(U\) of \(\mathcal{C}\) we have what is called fibre of \(U\) in \(\mathcal{D}\) usually denoted by \(\mathcal{D}(U)\). Given an object \(U\) of \(\mathcal{C}\) and a cover \(\{U_i\rightarrow U\}\) (i.e., it belongs to \(\mathcal{W}\)) we have what is called descent category associated to the cover \(\{U_i\rightarrow U\}\), usually denoted by \(\mathcal{D}(\{U_i\rightarrow U\})\). There is an obvious functor \(\mathcal{D}(U)\rightarrow \mathcal{D}(\{U_i\rightarrow U\})\). Definition : Let \(\mathcal{C}\) be a category with Grothendieck topology \(\mathcal{W}\). A category fibered in groupoids \(\mathcal{D}\rightarrow \mathcal{C}\) is said to be a stack over \(\mathcal{C}\) if, for every object \(U\) of \(\mathcal{C}\) and every cover \(\{U_i\rightarrow U\}\), the functor \(\mathcal{D}(U)\rightarrow \mathcal{D}(\{U_i\rightarrow U\})\) is an equivalence of categories. The fibre categroy \(\mathcal{D}(U)\) is a category whose objects are that of \(\mathcal{D}\) which map to \(U\) under \(F\) i.e., \(\mathcal{D}(U)_0=\{V\in \mathcal{D}_0:F(V)=U\}\). ...

Here, I will add links to WordPress pages where I have written something about Stacks. Papers I am reading are Differentiable Stacks and Gerbes by Kai Behrend and Ping Xu. Orbifolds as Stacks by Eugene Lerman. Non abelian Differentiable Gerbes by Camille, Stienon and Ping Xu. -- --  

Definition : A stack \(\mathcal{D}\rightarrow \text{Man}\) is differentiable if there exists a manifold \(X\) with an atlas \(p:\underline{X}\rightarrow \mathcal{D}\) i.e., \(p\) is representable surjective submersion. We see a criterion for a map \(p:\underline{X}\rightarrow \mathcal{D}\) to be an atlas. By \(p:\underline{X}\rightarrow \mathcal{D}\) to be representable surjective submersion, we mean given a map of stacks \(\underline{Y}\rightarrow \mathcal{D}\) the fibered product \(\underline{X}\times_{\mathcal{D}}\underline{Y}\) is representable by a manifold and that the map of manifolds \(\underline{X}\times_{\mathcal{D}}\underline{Y}\rightarrow \underline{Y}\) is a surjective submersion. As \(\underline{X}\times_{\mathcal{D}}\underline{Y}\) is representable by a manifold for any map of stacks \(\underline{Y}\rightarrow \mathcal{D}\), in particular, taking \(\underline{Y}\rightarrow \mathcal{D}\) to be the same map \(\underline{X}\rightarrow \mathcal{D}\) we see that, in particular \(\underline{X}\times_{\mathcal{D}}\underline{X}\) is representable by a manifold. Remark : If \(p:\underline{X}\rightarrow \mathcal{D}\) is an atlas for \(\mathcal{D}\) then \(\underline{X}\times_{\mathcal{D}}\underline{X}\) is representable by a manifold. As \(\underline{X}\times_{\mathcal{D}}\underline{Y}\rightarrow \underline{Y}\) is a submersion for any map of stacks \(\underline{Y}\rightarrow \mathcal{D}\), in particular, taking \(\underline{Y}\rightarrow \mathcal{D}\) to be the same map \(\underline{X}\rightarrow \mathcal{D}\) we see that, projecion map \(\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) is a submersion. It is not relevant which projection is it as both maps are same. So, both projection maps \(pr_1:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) and \(pr_2:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) are submersions. Remark : If \(p:\underline{X}\rightarrow \mathcal{D}\) is an atlas for stack \(\mathcal{D}\) then projection maps \(pr_1:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) and \(pr_2:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) are submersions. As any representable surjective submersion is an epimorphism we have following remark. Remark : If \(p:\underline{X}\rightarrow \mathcal{D}\) is an atlas for \(\mathcal{D}\) then \(p:\underline{X}\rightarrow \mathcal{D}\) is an epimorphism. Combining all these remarks we have following remark. If \(p:\underline{X}\rightarrow \mathcal{D}\) is an atlas for \(\mathcal{D}\) then, \(\underline{X}\times_{\mathcal{D}}\underline{X}\) is representable by a manifold and projection maps \(pr_1:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) and \(pr_2:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) are submersions and \(p:\underline{X}\rightarrow \mathcal{D}\) is an epimorphism. It turns out that converse of above remark is true. Proposition : Let \(p:\underline{X}\rightarrow \mathcal{D}\) is a morphism of stacks such that \(\underline{X}\times_{\mathcal{D}}\underline{X}\) is representable by a manifold and projection maps \(pr_1:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) and \(pr_2:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) are submersions and that \(p:\underline{X}\rightarrow \mathcal{D}\) is an epimorphism. Then, Then, \(p:\underline{X}\rightarrow \mathcal{D}\) is a representable surjective submersion i.e., an atlas for \(\mathcal{D}\). Before we give proof of this, we recall a result. Lemma : Let \(\mathcal{D}\rightarrow\mathcal{C}\) be a morphism of stacks. Suppose \(U\) be a manifold and \(\underline{U}\rightarrow \mathcal{C}\) is an epimorphism of stacks such that fiber product \(\mathcal{D}\times_{\mathcal{C}}\underline{U}\) is represented by a manifold and the map of manifolds \(\mathcal{D}\times_{\mathcal{C}}\underline{U}\rightarrow U\) is a submersion. Then, \(\mathcal{D}\rightarrow \mathcal{C}\) is a representable submersion. To prove \(\underline{X}\rightarrow \mathcal{D}\) is a representable submersion, consider an epimorphism of stacks, namely \(\underline{X}\rightarrow \mathcal{D}\) (it is given to be an epimorphism, condition \(2\) above). See that the fibre product \(\underline{X}\times_{\mathcal{D}}\underline{X}\) is representable by a manifold (it is given in condition \(1\) above) and that the projection map \(\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow X\) is a submersion (it is in condition \(1\) above). Thus, by above lemma, \(p:\underline{X}\rightarrow \mathcal{D}\) is a representable submersion. Note that, a representable submersion that is an epimorphism is a representable surjective submersion. Thus, \(p:\underline{X}\rightarrow \mathcal{D}\) is an atlas for \(\mathcal{D}\). So, we have the following result. Proposition : Let \(p:\underline{X}\rightarrow \mathcal{D}\) is a morphism of stacks such that \(\underline{X}\times_{\mathcal{D}}\underline{X}\) is representable by a manifold and projection maps \(pr_1:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) and \(pr_2:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) are submersions and that \(p:\underline{X}\rightarrow \mathcal{D}\) is an epimorphism. Then, Then, \(p:\underline{X}\rightarrow \mathcal{D}\) is a representable surjective submersion i.e., an atlas for \(\mathcal{D}\).