In the theory of Lie algebras, we have the notion of representation of a Lie algebra \(\mathfrak{g}\), which consists of a vector space \(V\), and a morphism of Lie algebras \(\mathfrak{g}\rightarrow \mathfrak{gl}(V)\). We are so used to thinking of \(\mathfrak{gl}(V)\) as a Lie algebra, that, we might not remember that the underlying set \(End(V)\) has a structure of an associative algebra, and that we made the underlying set into a Lie algebra by considering the binary operation \([f,g]=fg-gf\) for \(f,g\in End(V)\). This is where the notion of enveloping algebra comes into picture. ...

Once we have a reasonably good notion of an object, we would look at a notion of morphisms. Let \((L,A,\rho,\tau)\) to \((L',A',\rho',\tau')\) be Lie-Rinehart algebras. Our experience suggests that the data of a morphism of Lie-Rinehart algebras from \((L,A,\rho,\tau)\) to \((L',A',\rho',\tau')\) should at least have two morphisms, one a morphism of Lie algebras \(\Phi:L\rightarrow L'\) and a morphism of associative algebras \(\Psi:A\rightarrow A'\) such that the following diagram commute, ...

Any notion of an "algebra" comes with two binary operations: \(A\times A\rightarrow A\), called the addition map, \(A\times A\rightarrow A\), called the multiplication map. Two properties that are assumed for addition map are that of commutativity and associativity. By the very definition, we would have \(a+b=b+a\) and \(a+(b+c)=(a+b)+c\) for all \(a,b,c\in A\). ...

These are “notes” I have written for myself when reading the book Model Categories by Mark Hovey. This book has some typos, there is an errata by its Author. There might be some more typos. I am assuming some notation and results about topological spaces (fibrations, cofibrations, etc) and homological algebra (chain complexes, etc). Other references for Model categories are : An Introduction to Homotopical categories by Julie Bergner.  

Before we move to the notion of "a model structure on a category", we need to recall (or introduce) some definitions. Definition : Let \(\mathcal{C}\) be a category. An object \(X\) of \(\mathcal{C}\) is said to be a retract of an object \(\mathcal{C}\) if there exists arrows \(X\xrightarrow{f} Y\xrightarrow{g} X\) such that, the composition \(g\circ f:X\rightarrow X\) is equal to the identity arrow \(1_X:X\rightarrow X\). Definition : Let \(\mathcal{C}\) be a category. We define the morphism category of \(\mathcal{C}\), denoted by \(\text{Map}(\mathcal{C})\) whose objects are the arrows of \(\mathcal{C}\), morphisms are commutative diagrams in \(\mathcal{C}\). Definition : Let \(\mathcal{C}\) be a category. A morphism \(f\) in \(\mathcal{C}\) is said to be a retract of a morphism \(g\) in \(\mathcal{C}\), if, \(f\) is a retract of \(g\), when both \(f\) and \(g\) are seen as objects of \(\text{Map}(\mathcal{C})\). Definition : Let \(\mathcal{C}\) be a category. Let \(i:A\rightarrow B\) and \(p:X\rightarrow Y\) be morphisms in \(\mathcal{C}\). We say that \(i\) has the left lifting property with respect to \(p\) or \(p\) has the right lifting property with respect to \(i\) if, for every commutative diagram there exists an arrow \(h:B\rightarrow X\) such that \(h\circ i=f\) and \(p\circ h=g\). Definition : Let \(\mathcal{C}\) be a category. A model structure on \(\mathcal{C}\) consists of the following data : a subcategory of \(\mathcal{C}\) called “weak equivalences”, a subcategory of \(\mathcal{C}\) called “fibrations”, a subcategory of \(\mathcal{C}\) called “cofibrations”, satisfying certain conditions: If \(f,g\) are morphisms of \(\mathcal{C}\) such that \(gf\) is defined and two of \(f,g,gf\) are "weak equivalences" then so is the third. A retract of a "weak equivalece" is a "weak equivalence". A retract of a "fibration" is a "fibration". A retract of a "cofibration" is a "cofibration". factorizing property (part \(1\)) of arrows of \(\mathcal{C}\) : for each \(f\) in \(\text{Mor}(\mathcal{C})\), there exists a cofibration \(i\) and a traivial fibration \(q\) such that \(f=qi\). factorizing property (part \(2\)) of arrows of \(\mathcal{C}\) : for each \(f\) in \(\text{Mor}(\mathcal{C})\), there exists a fibration \(p\) and a trivial cofibration \(j\) such that \(f=pj\). Any commutative diagram of the type has lifting property if either \(i\) or \(p\) is a "weak equivalence". Definition : A model category is defined to be a category that has all small limits, all small colimits, a model structure in \(\mathcal{C}\). Construction of new model categories from old model categories: Let \(\mathcal{C}\) and \(\mathcal{D}\) be model categories. Defining the collection of fibrations (cofibrations, weak equivalences) as pairs \((f,g)\) where both \(f\) and \(g\) are fibrations (cofibrations, weak equivalences) defined a model structure on the product category \(\mathcal{C}\times \mathcal{D}\). This model category is called the product model category produced from model categories \(\mathcal{C}\) and \(\mathcal{D}\).

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 \(i^*(a),i^*(b)\in \mathcal{S}(V)\).

Let \(\pi:P\rightarrow M\) be a principal \(G\) bundle. We choose an open covering \(\{U_\alpha\}\) of \(M\) and trivializations \(\psi_\alpha:\pi^{-1}(U_\alpha)\rightarrow U_\alpha\times G\) defined as \(\psi_\alpha(u)= (\pi(u),\varphi_\alpha(u))\) such that \(\varphi_\alpha(ua)=\varphi_\alpha(u)a\) for all \(u\in \pi^{-1}(U_\alpha)\) and \(a\in G\). Let \(x\in U_\alpha\cap U_\beta\). Given \(v\in \pi^{-1}(x)\subseteq \pi^{-1}(U_\alpha)\cap \pi^{-1}(U_\beta)\), we have \(\varphi_\alpha(v)\in G\) and \(\varphi_\beta(v)\in G\). For \(v'\in \pi^{-1}(x)\) there exists \(g\in G\) such that \(v'=vg\). Then, we have \(\varphi_\alpha(v')\varphi_\beta(v')^{-1}= \varphi_\alpha(vg)\varphi_\beta(ua)^{-1} =\varphi_\alpha(u)aa^{-1}\varphi_\beta(u)^{-1} =\varphi_\alpha(u)\varphi_\beta(u)^{-1}\) Thus, for any \(v,v'\in \pi^{-1}(x)\), we have \(\varphi_\alpha(v)\varphi_\beta(v)^{-1}=\varphi_\alpha(v')\varphi_\beta(v')^{-1}.\) ...

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.