In order to prevent bots from posting comments, we would like you to prove that you are human. I will then talk about special type of fibered categories, namely categories fibered in groupoids and categories fibered in sets. Let $\mathcal{S}'$ be the subcategory of $\mathcal{S}$ defined as follows {\displaystyle d\to c} G Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory.They formalise the various situations in geometry and algebra in which inverse images (or pull-backs) of objects such as vector bundles can be defined. Proof. Given $p : \mathcal{S} \to \mathcal{C}$, we can ask: if the fibre category $\mathcal{S}_ U$ is a groupoid for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, must $\mathcal{S}$ be fibred in groupoids over $\mathcal{C}$? : $\square$. is an equivalence of categories. Instead, these inverse images are only naturally isomorphic. Some authors use the term cofibration in groupoids to refer to what we call an opfibration in groupoids. and an equivalence (by the assumption that $b$ is an equivalence, see Lemma 4.31.7). Fibrations also play an important role in categorical semantics of type theory, and in particular that of dependent type theories. This groupoid gives an induced category fibered in groupoids denoted Suppose that $g : W \to V$ and $f : V \to U$ are morphisms in $\mathcal{C}$. x The two preceding constructions of split categories are used in a critical way in the construction of the stack associated to a fibred category (and in particular stack associated to a pre-stack). {\displaystyle a:G\to {\text{Aut}}(X)} {\displaystyle p:{\mathcal {F}}\to {\mathcal {C}}} Proof. G However, the underlying intuition is quite straightforward when keeping in mind the basic examples discussed above. Because $\mathcal{X}$ is fibred in groupoids over $\mathcal{C}$ we can find a morphism $a : x' \to x$ lying over $U' = q(y') \to q(y) = U$. Let $\mathcal{C}$ be a category. Let $\mathcal{C}$ be a category. 4 Fibered categories (Aaron Mazel-Gee) Contents 4 Fibered categories (Aaron Mazel-Gee) 1 ... Let Cbe a category. Two Examples of Integrable Category Fibered in Groupoids In the present x1, we give two examples of integrable [cf. id Let $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) = \{ A, B, T\}$ and $\mathop{Mor}\nolimits _\mathcal {C}(A, B) = \{ f\}$, $\mathop{Mor}\nolimits _\mathcal {C}(B, T) = \{ g\}$, $\mathop{Mor}\nolimits _\mathcal {C}(A, T) = \{ h\} = \{ gf\} ,$ plus the identity morphism for each object. The $2$-category of categories fibred in groupoids over $\mathcal{C}$ is the sub $2$-category of the $2$-category of fibred categories over $\mathcal{C}$ (see Definition 4.33.9) defined as follows: Its objects will be categories $p : \mathcal{S} \to \mathcal{C}$ fibred in groupoids. o Let $p : \mathcal{S}\to \mathcal{C}$ and $p' : \mathcal{S'}\to \mathcal{C}$ be categories fibred in groupoids, and suppose that $G : \mathcal{S}\to \mathcal{S}'$ is a functor over $\mathcal{C}$. Thus, we have the category SchS In other words, an E-category is a fibred category if inverse images always exist (for morphisms whose codomains are in the range of projection) and are transitive. Then $G$ is faithful (resp. We classify the groupoid fibrations over log schemes that arise in this manner in terms of a categorical notion of "minimal" objects. G By Lemma 4.33.10 the fibre product as described in Lemma 4.32.3 is a fibred category. As an example, for each topological space there is the category of vector bundles on the space, and for every continuous map from a topological space X to another topological space Y is associated the pullback functor taking bundles on Y to bundles on X. Fibred categories formalise the system consisting of these categories and inverse image functors. The functors of arrows of a fibered category 61 3.8. In such cases the inverse image operation is often compatible with composition of these maps between objects, or in more technical terms is a functor. ( Let $p : \mathcal{S} \to \mathcal{C}$ be a functor. → $\square$. An object of the right hand side is a triple $(x, x', \alpha )$ where $\alpha : G(x) \to G(x')$ is a morphism in $\mathcal{S}'_ U$. To see (2) let $(a, b) : (U', x', y', f') \to (U, x, y, f)$ and $(a', b') : (U'', x'', y'', f'') \to (U, x, y, f)$ be morphisms of $\mathcal{X}'$ and let $b'' : y' \to y''$ be a morphism of $\mathcal{Y}$ such that $b' \circ b'' = b$. Lemma 4.35.12. It suffices to prove that $G$ induces an injection (resp. The morphisms of FS are called S-morphisms, and for x,y objects of FS, the set of S-morphisms is denoted by HomS(x,y). C Because $\mathcal{Y}$ is fibred in groupoids we see that $F(a'')$ is the unique morphism $F(x') \to F(x'')$ such that $F(a') \circ F(a'') = F(a)$ and $q(F(a'')) = q(b'')$. Assume we have a $2$-commutative diagram {\displaystyle {\mathcal {F}}_{c}\to {\mathcal {F}}_{d}} Then one checks that $\chi = i^{-1} \circ j$ is a solution. {\displaystyle s:G\times X\to X} Then m is also called a direct image and y a direct image of x for f = φ(m). → Fibred categories were introduced by Alexander Grothendieck (1959, 1971), and developed in more detail by Jean Giraud (1964, 1971). Example 4.35.5. d Under forming opposite categories we obtain the notion of an op-fibration fibered in groupoids. Given a category fibered in groupoids over schemes with a log structure, one produces a category fibered in groupoids over log schemes. A fibration fibered in groupoids is a functor p: E → B such that the corresponding (strict) functor Bop → Cat classifying p under the Grothendieck construction factors through the inclusion Grpd ↪ Cat. {\displaystyle h_{x}(z){\overset {s}{\underset {t}{\rightrightarrows }}}h_{y}(z)}. Instead, if f: T → S and g: U → T are morphisms in E, then there is an isomorphism of functors. $\xymatrix{ \mathcal{X}' \ar[rd]_ f & \mathcal{X} \ar[l]^ a \ar[d]^ F \ar[r]_ b & \mathcal{X}'' \ar[ld]^ g \\ & \mathcal{Y} }$ We omit the verification that $G \circ F$ and $F \circ G$ are $2$-isomorphic to the respective identity functors (in the $2$-category of categories fibred in groupoids over $\mathcal{C}$). Higgins, R. Sivera, "Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical omega-groupoids", European Mathematical Society, Tracts in Mathematics, Vol. A fibred category together with a cleavage is called a cloven category. For an example, see below. Fibered categories are used to define stacks, which are fibered categories (over a site) with "descent". Let $x \to y$ and $z \to y$ be morphisms of $\mathcal{S}$. c One can prove this directly from the definition. Proof. to the category Then also $G(f^*y) \to G(y)$ is a pullback. from : From this diagram it is clear that if $G$ is faithful (resp. Beware of the difference between the letter 'O' and the digit '0'. × fully faithful, resp. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as bundles, connections and spin structures. Suppose that $\varphi : \mathcal{S}_1 \to \mathcal{S}_2$ and $\psi : \mathcal{S}_3 \to \mathcal{S}_4$ are equivalences over $\mathcal{C}$. A category fibred in groupoids is called representable by an algebraic space over if there exists an algebraic space over and an equivalence of categories over . C Let $G : \mathcal{S}\to \mathcal{S}'$ be a functor over $\mathcal{C}$. A homomorphism of groups $p : G \to H$ gives rise to a functor $p : \mathcal{S}\to \mathcal{C}$ as in Example 4.2.12. fully faithful, resp. , and using the Grothendieck construction, this gives a category fibered in groupoids over Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work. is a pullback square. It is clear that the composition $\mathcal{X} \to \mathcal{X}' \to \mathcal{Y}$ equals $F$. In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A morphism $(a, b) : (U, x, y, f) \to (U', x', y', f')$ is given by $a : x \to x'$ and $b : y \to y'$ with $p(a) = q(b)$ and such that $f' \circ F(a) = b \circ f$. Its $1$-morphisms $(\mathcal{S}, p) \to (\mathcal{S}', p')$ will be functors $G : \mathcal{S} \to \mathcal{S}'$ such that $p' \circ G = p$ (since every morphism is strongly cartesian $G$ automatically preserves them). Let $b : y' \to y$ be a morphism in $\mathcal{Y}$ and let $(U, x, y, f)$ be an object of $\mathcal{X}'$ lying over $y$. Thus in Diagram 4.35.1.1 the morphisms $\phi$, $\psi$ and $\gamma$ are strongly cartesian morphisms of $\mathcal{S}$. × \ar[ru]_{h'} & & \ar@{}[u]^{above} & A \ar[u]^ f \ar[ru]_{gf = h} & \\ } \]. Let $p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids. t We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. If $\mathcal{A}$ is fibred in groupoids over $\mathcal{B}$ and $\mathcal{B}$ is fibred in groupoids over $\mathcal{C}$, then $\mathcal{A}$ is fibred in groupoids over $\mathcal{C}$. This is based on sections 3.1-3.4 of Vistoli's notes. G Condition (2) of Definition 4.35.1 says exactly that every morphism of $\mathcal{S}$ is strongly cartesian. → Ob acting on an object . Since the right triangle of the diagram is $2$-commutative we see that. Then, Proof. on February 04, 2016 at 18:10. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$. To show that $G$ is faithful (resp. Suppose that $\varphi : \mathcal{S}_1 \to \mathcal{S}_2$ and $\psi : \mathcal{S}_3 \to \mathcal{S}_4$ are equivalences over $\mathcal{C}$. $\mathop{Mor}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{S}_2, \mathcal{S}_3) \longrightarrow \mathop{Mor}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{S}_1, \mathcal{S}_4), \quad \alpha \longmapsto \psi \circ \alpha \circ \varphi$ where the squiggly arrows represent not morphisms but the functor $p$. C See the diagram below for a picture of this category. Let $G : \mathcal{S}\to \mathcal{S}'$ be a functor over $\mathcal{C}$. Let $x, y$ be objects of $\mathcal{S}$ lying over the same object $U$. We still have to construct a $2$-isomorphism between $c \circ b$ and the functor $d : \mathcal{X} \to \mathcal{X} \times _{F, \mathcal{Y}, \text{id}} \mathcal{Y}$, $x \mapsto (p(x), x, F(x), \text{id}_{F(x)})$ constructed in the proof of Lemma 4.35.15. {\displaystyle t:G\times X\to X} $\square$. → This gives a contravariant 2-functor Then we get a morphism $i : y \to g^*x$ in $\mathcal{S}_ V$, which is therefore an isomorphism. The associated 2-functors from the Grothendieck construction are examples of stacks. . C We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. It should be clear from this discussion that a category fibred in groupoids is very closely related to a fibred category. → ↦ , there is an associated groupoid object, G ) The $2$-category of categories fibred in groupoids over $\mathcal{C}$ is the sub $2$-category of the $2$-category of fibred categories over $\mathcal{C}$ (see Definition 4.33.9) defined as follows: there is an associated small groupoid s The choice of a (normalised) cleavage for a fibred E-category F specifies, for each morphism f: T → S in E, a functor f*: FS → FT: on objects f* is simply the inverse image by the corresponding transport morphism, and on morphisms it is defined in a natural manner by the defining universal property of cartesian morphisms. c Hence in order for $\Delta _ G$ to be an equivalence, every $\alpha$ has to be the image of a morphism $\beta : x \to x'$, and also every two distinct morphisms $\beta , \beta ' : x \to x'$ have to give distinct morphisms $G(\beta ), G(\beta ')$. Then $G$ is fully faithful if and only if the diagonal To do this we argue as in the discussion following Definition 4.35.1. For every pair of morphisms $\phi : y \to x$ and $\psi : z \to x$ and any morphism $f : p(z) \to p(y)$ such that $p(\phi ) \circ f = p(\psi )$ there exists a unique lift $\chi : z \to y$ of $f$ such that $\phi \circ \chi = \psi$. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. Since this diagram applied to an object ∈ We have to check conditions (1) and (2) of Definition 4.35.1. $\Delta _ G : \mathcal{S} \longrightarrow \mathcal{S} \times _{G, \mathcal{S}', G} \mathcal{S}$ is an equivalence. {\displaystyle p(y)=d} from the yoneda embedding. Lemma 4.35.15. If f is a morphism of E, then those morphisms of F that project to f are called f-morphisms, and the set of f-morphisms between objects x and y in F is denoted by Homf(x,y). The uniqueness implies that the morphisms $z' \to z$ and $z\to z'$ are mutually inverse, in other words isomorphisms. → A co-fibred E-category is anE-category such that direct image exists for each morphism in E and that the composition of direct images is a direct image. C Let $p' : \mathcal{S}' \to \mathcal{C}$ be the restriction of $p$ to $\mathcal{S}'$. such that any subcategory of The operation which associates to an object S of E the fibre category FS and to a morphism f the inverse image functor f* is almost a contravariant functor from E to the category of categories. Then $G$ is faithful (resp. Now let $\mathop{\mathrm{Ob}}\nolimits (\mathcal{S}) = \{ A', B', T'\}$ and $\mathop{Mor}\nolimits _\mathcal {S}(A', B') = \emptyset$, $\mathop{Mor}\nolimits _\mathcal {S}(B', T') = \{ g'\}$, $\mathop{Mor}\nolimits _\mathcal {S}(A', T') = \{ h'\} ,$ plus the identity morphisms. After some final words on Grothendieck topology, we will take the last step towards defining stacks over categories and discuss categories fibered in groupoids (CFGs for short). p Lemma 4.35.11. Thus split E-categories correspond exactly to true functors from E to the category of categories. It is equivalent to a definition in terms of cleavages, the latter definition being actually the original one presented in Grothendieck (1959); the definition in terms of cartesian morphisms was introduced in Grothendieck (1971) in 1960–1961. Math. Let $\mathcal{C}$ be a category. Proof. If $p : \mathcal{S} \to \mathcal{C}$ is fibred in groupoids, then so is the inertia fibred category $\mathcal{I}_\mathcal {S} \to \mathcal{C}$. F Its objects will be categories $p : \mathcal{S} \to \mathcal{C}$ fibred in groupoids. In the present x1, let S be a scheme. 1) a category internal to the category of Chen-smooth spaces. ) This introduction of some "slack" in the system of inverse images causes some delicate issues to appear, and it is this set-up that fibred categories formalise. x $\xymatrix{ y \ar[r] & x & p(y) \ar[r] & p(x) \\ z \ar@{-->}[u] \ar[ru] & & p(z) \ar@{-->}[u]\ar[ru] & \\ }$, \begin{equation} \label{categories-equation-fibred-groupoids} \vcenter { \xymatrix{ z' \ar@{-->}[d]\ar[rrd]^\gamma & & \\ z \ar@{-->}[u] \ar[r]^\psi \ar@{~>}[d]^ p & y \ar[r]^\phi \ar@{~>}[d]^ p & x \ar@{~>}[d]^ p \\ W \ar[r]^ g & V \ar[r]^ f & U \\ } } \end{equation}, $\xymatrix{ y \ar[r]^ f & x & U \ar[r]^{\text{id}_ U} & U \\ x \ar@{-->}[u] \ar[ru]_{\text{id}_ x} & & U \ar@{-->}[u]\ar[ru]_{\text{id}_ U} & \\ }$, \[ \xymatrix{ B' \ar[r]^{g'} & T' & & B \ar[r]^ g & T & \\ A' \ar@{-->}[u]^{??} . Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $\mathcal{C}$. Lemma 4.35.10. gives a groupoid internal to sets, h Let $\mathcal{C}$ be a category. Aut X Here is the result. Here is the obligatory lemma on $2$-fibre products. If the diagram above actually commutes, then we can arrange it so that $h \circ b$ is $2$-isomorphic to $a$ as $1$-morphisms of categories over $\mathcal{Y}$. bijection) between $\mathop{Mor}\nolimits _\mathcal {S}(x, y)$ and $\mathop{Mor}\nolimits _{\mathcal{S}'}(G(x), G(y))$. Lemma 4.35.13. on December 18, 2015 at 14:09, In the first line, it should probably be "categories fibred in groupoids", not "categories in groupoids".function () { an equivalence) if and only if for each $U\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ the induced functor $G_ U : \mathcal{S}_ U\to \mathcal{S}'_ U$ is faithful (resp. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as … Johan ) There exists a factorization $\mathcal{X} \to \mathcal{X}' \to \mathcal{Y}$ by $1$-morphisms of categories fibred in groupoids over $\mathcal{C}$ such that $\mathcal{X} \to \mathcal{X}'$ is an equivalence over $\mathcal{C}$ and such that $\mathcal{X}'$ is a category fibred in groupoids over $\mathcal{Y}$. From this point on we proceed as usual (see proof of Lemma 4.2.19) to produce an inverse functor $F : \mathcal{S}' \to \mathcal{S}$, by taking $x' \mapsto o_{x'}$ and $\varphi ' : x' \to y'$ to the unique arrow $\varphi _{\varphi '} : o_{x'} \to o_{y'}$ with $\alpha _{y'}^{-1} \circ G(\varphi _{\varphi '}) \circ \alpha _{x'} = \varphi '$. These ideas simplify in the case of groupoids, as shown in the paper of Brown referred to below, which obtains a useful family of exact sequences from a fibration of groupoids. fully faithful) gives the desired result. ) F {\displaystyle G} A (normalised) cleavage such that the composition of two transport morphisms is always a transport morphism is called a splitting, and a fibred category with a splitting is called a split (fibred) category. PDF | We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. {\displaystyle G\times X{\underset {t}{\overset {s}{\rightrightarrows }}}{}X}. Lemma 4.35.3. Choose a quasi-inverse $b^{-1} : \mathcal{X}'' \to \mathcal{X}$ in the $2$-category of categories over $\mathcal{C}$. F by Now fix $f : U \to V$. X z We construct $\mathcal{X}'$ explicitly as follows. 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