Let us check for limit/limsup/liminf of the sequence \(\frac{n}{10^{\lceil \log_{10}n \rceil}}\), where the notation \(\lceil x \rceil\) means the smallest integer greater than or equal to \(x\). For example, \(\lceil 0.1 \rceil=1, \lceil 0.9 \rceil=1, \lceil -1.2 \rceil=-1, \lceil -2.5 \rceil=-2\) To compute limit (to have a hope of computing), we need to know it converge (which we can check by checking it is Cauchy sequence). ...
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 \(\mathcal{D}\).
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 \(F:\mathcal{I}\rightarrow \mathcal{C}\) is a functor. This is also called as diagram indexed by \(\mathcal{I}\). By the Limit of this diagram, we mean an object (universal) \(L\) of \(\mathcal{C}\) and a collection of arrows (universal again) \(\pi_i:L\rightarrow F(i)\) such that, for each arrow \(m:i\rightarrow j\) in \(\mathcal{I}\) the following diagram is commutative. This is usually denoted by \(\varprojlim_{\mathcal{I}}F(i)\) or simply by \(\varprojlim_{\mathcal{I}}F\). Fixing an object \(X\) in \(\mathcal{C}\), I want to prove that \(\varprojlim_{\mathcal{I}}(\text{Hom}_{\mathcal{C}}(X,F(i))) =\text{Hom}_{\mathcal{C}}(X,\varprojlim_{\mathcal{I}}F(i))\) ...
Given a morphism of Lie groups \(\theta:G\rightarrow H\) and a principal \(G\) bundle \(\pi:P\rightarrow M\) there are (at least) two ways to assign a principal \(H\) bundle. See that the morphism of Lie groups \(\theta:G\rightarrow H\) gives an action of \(G\) on \(H\) by \(g.h=\theta(g).h\). Given an action of \(G\) on manifold (Lie group in this case) \(H\) there is an associated fibre bundle \(P\times_G H\rightarrow M\) with fibre \(H\). This gives a principal \(H\) bundle. For principal bundle \(\pi:P\rightarrow M\), we can find an open cover \(\{U_\alpha\}\) of \(M\) and (transition) maps \(g_\alpha g_\beta:U_{\alpha\beta}\rightarrow G\) satifsying the cocycle condition \(g_{\alpha\beta}g_{\beta\gamma}=g_{\alpha\gamma}\) on \(U_\alpha\cap U_\beta\cap U_\gamma\). Then the compositions \(\tau_{\alpha\beta}=\theta\circ g_{\alpha\beta}:U_{\alpha\beta}\rightarrow G\rightarrow H\) also satifies the cocycle condition \(\tau_{\alpha\beta}\tau_{\beta\gamma}=\tau_{\alpha\gamma}\) on \(U_\alpha\cap U_\beta\cap U_\gamma\). One can then produce a principal \(H\) bundle over \(M\) given this open cover \(\{U_\alpha\}\) of \(M\) and smooth maps \(\tau_{\alpha\beta}:U_\alpha\cap U_\beta\rightarrow H\) satisfying the cocycle condition. This gives a principal \(H\) bundle. It is a good exercise (that I have not tried) to check that principal \(H\) bundles obtained from above two methods are (naturally) isomorphic i.e., one and the same. Given a Lie group \(G\), let \(BG\) denote the category of principal \(G\) bundles. Objects are principal \(G\) bundles and morphisms are \(G\)-equivariant morphisms. Given a morphism of Lie groups \(\theta:G\rightarrow H\), above construction gives a functor (at the level of objects) \(B\theta:BG\rightarrow BH\). It is not difficult to see that, a \(G\)-equivarint map induce a \(H\)-equivariant map. This gives a functor \(BG\rightarrow BH\).
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., \(\mathcal{D}(U)_0=\{V\in \mathcal{D}_0:F(V)=U\}\). ...
Definition : Let \(\mathcal{C}\) be a category and \(f:A\rightarrow C\) and \(g:B\rightarrow C\) be morphisms. We define pull back of \(f,g\) to be a triple \((P,f',g')\) where \(P\) is an object of \(\mathcal{C}\) and \(f':P\rightarrow B, g':P\rightarrow A\) with \(g\circ f'=f\circ g'\) such that given any object \(P'\) of \(\mathcal{C}\) and morphisms \(g'':P'\rightarrow A\) and \(f'':P'\rightarrow B\) with \(g\circ f''=f\circ g''\), there exists a unique morphism \(\eta:P'\rightarrow P\) such that \(g''=g'\circ \eta\) and \(f''=f'\circ \eta\) as in the following commutative diagram. Definition : Let \(\mathcal{C}\) be a category and \(f:A\rightarrow C\) and \(g:B\rightarrow C\) be morphisms. We define pull back of \(f,g\) to be a triple \((P,f',g')\) where \(P\) is an object of \(\mathcal{C}\) and \(f':P\rightarrow B, g':P\rightarrow A\) with \(g\circ f'=f\circ g'\) such that given any object \(P'\) of \(\mathcal{C}\) and morphisms \(g'':P'\rightarrow A\) and \(f'':P'\rightarrow B\) with \(g\circ f''=f\circ g''\), there exists a unique morphism \(\eta:P'\rightarrow P\) such that \(g''=g'\circ \eta\) and \(f''=f'\circ \eta\) as in the following commutative diagram.