The classical examples include vector bundles, principal bundles, and sheaves over topological spaces. fully faithful) we have to show for any objects $x, y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S})$ that $G$ induces an injection (resp. Choose a quasi-inverse $b^{-1} : \mathcal{X}'' \to \mathcal{X}$ in the $2$-category of categories over $\mathcal{C}$. As functor $\mathcal{X} \to \mathcal{X}'$ we take $x \mapsto (p(x), x, F(x), \text{id}_{F(x)})$ on objects and $(a : x \to x') \mapsto (a, F(a))$ on morphisms. . ) Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. 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. x A cleavage is called normalised if the transport morphisms include all identities in F; this means that the inverse images of identity morphisms are chosen to be identity morphisms. h ( A special case is provided by considering E as an E-category via the identity functor: then a cartesian functor from E to an E-category F is called a cartesian section. . Since the right triangle of the diagram is $2$-commutative we see that. F Let $\mathcal{C}$ be a category. The site in question has objects Dirac manifolds and morphisms pairs consisting of a smooth map and a closed 2-form. The functors of arrows of a fibered category 61 3.8. This groupoid gives an induced category fibered in groupoids denoted X C G We have to show that there exists a unique morphism $a'' : x' \to x''$ such that $f'' \circ F(a'') = b'' \circ f'$ and such that $(a', b') \circ (a'', b'') = (a, b)$. Categories fibered in groupoids 52 3.4. There can in general be more than one cartesian morphism projecting to a given morphism f: T → S, possibly having different sources; thus there can be more than one inverse image of a given object y in FS by f. However, it is a direct consequence of the definition that two such inverse images are isomorphic in FT. A functor φ: F → E is also called an E-category, or said to make F into an E-category or a category over E. An E-functor from an E-category φ: F → E to an E-category ψ: G → E is a functor α: F → G such that ψ ∘ α = φ. E-categories form in a natural manner a 2-category, with 1-morphisms being E-functors, and 2-morphisms being natural transformations between E-functors whose components lie in some fibre. $\square$. $\square$. Instead, if f: T → S and g: U → T are morphisms in E, then there is an isomorphism of functors. A morphism m: x → y in F is called φ-cartesian (or simply cartesian) if it satisfies the following condition: A cartesian morphism m: x → y is called an inverse image of its projection f = φ(m); the object x is called an inverse image of y by f. The cartesian morphisms of a fibre category FS are precisely the isomorphisms of FS. 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 $G(\beta ), G(\beta ')$. {\displaystyle p:{\mathcal {F}}\to {\mathcal {C}}} x You can do this by filling in the name of the current tag in the following input field. Definition 4.35.1. 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 $y$. c 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 $\mathcal{C}$). where $a$ and $b$ are equivalences of categories over $\mathcal{C}$ and $f$ and $g$ are categories fibred in groupoids. More precisely, if φ: F →E is a functor, then a morphism m: x → y in F is called co-cartesian if it is cartesian for the opposite functor φop: Fop → Eop. s This is based on sections 3.1-3.4 of Vistoli's notes. {\displaystyle G\times X\xrightarrow {\left(a,{\text{id}}\right)} {\text{Aut}}(X)\times X\xrightarrow {(f,x)\mapsto f(x)} X} Lemma 4.35.11. Similarly the set of morphisms from $G(x)$ to $G(y)$ lying over $f$ is bijective to the set of morphisms between $G(x)$ and $G(f^*y)$ lying over $\text{id}_ U$. Then {\displaystyle X} f is a pullback square. Similarly there is a unique morphism $z' \to z$. ∈ 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. We omit the verification that $\mathcal{X} \to \mathcal{X}'$ is an equivalence of fibred categories over $\mathcal{C}$. is $2$-commutative. One example is the functor from Example 4.35.4 when $G \to H$ is not surjective. We will show that both $\mathcal{X}'$ and $\mathcal{X}''$ over $\mathcal{Y}$ are equivalent to the category fibred in groupoids $\mathcal{X} \times _{F, \mathcal{Y}, \text{id}} \mathcal{Y}$ over $\mathcal{Y}$, see proof of Lemma 4.35.15. return this == this.toLowerCase(); x ↦ Then $fgh = f : y \to x$. X Condition (2) of Definition 4.35.1 says exactly that every morphism of $\mathcal{S}$ is strongly cartesian. Namely, for an object is fully faithful (Lemma 5.7 of Giraud (1964)). Categories of arrows: For any category E the category of arrows A(E) in E has as objects the morphisms in E, and as morphisms the commutative squares in E (more precisely, a morphism from (f: X → T) to (g: Y → S) consists of morphisms (a: X → Y) and (b: T → S) such that bf = ga). Here is another example. p Functors and categories fibered in sets 53 3.5. 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. X 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 . Lemma 4.35.8. Proof. Johan $\square$. 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 $p : \mathcal{S} \to \mathcal{C}$ be a functor. If $G$ is an equivalence, then $G$ is an equivalence in the $2$-category of categories fibred in groupoids over $\mathcal{C}$. ) by We construct $\mathcal{X}'$ explicitly as follows. . Lemma 4.35.15. Fibred categories were introduced by Alexander Grothendieck (1959, 1971), and developed in more detail by Jean Giraud (1964, 1971). The class of morphisms thus selected is called a cleavage and the selected morphisms are called the transport morphisms (of the cleavage). Denote $p : \mathcal{X} \to \mathcal{C}$ and $q : \mathcal{Y} \to \mathcal{C}$ the structure functors. where $a$ and $b$ are equivalences of categories over $\mathcal{C}$ and $f$ and $g$ are categories fibred in groupoids. 1. We introduce “sheafification” functors from categories of (lax monoidal) linear functors to categories of quasi-coherent sheaves (of algebras) of stacks. p fully faithful) then so is each $G_ U$. The main application of fibred categories is in descent theory, concerned with a vast generalisation of "glueing" techniques used in topology. Moreover $c$ is a morphism of categories over $\mathcal{Y}$ (!) We continue our abuse of notation in suppressing the equivalence whenever we encounter such a situation. Examples of fibered categories 48 3.3. {\displaystyle x\in {\text{Ob}}({\mathcal {F}}_{c})} C from the yoneda embedding. ( I will then talk about special type of fibered categories, namely categories fibered in groupoids and categories fibered in sets. ( Follows from Lemma 4.33.13. Write $U = p(x)$, $V = p(y)$, $W = p(z)$, $p(\phi ) = g : V \to U$, $p(\psi ) = h : W \to U$. Conversely, assume all fibre categories are groupoids and $\mathcal{S}$ is a fibred category over $\mathcal{C}$. Let $\mathcal{C}$ be a category. ∈ {\displaystyle h_{x}(z){\overset {s}{\underset {t}{\rightrightarrows }}}h_{y}(z)}. Let $\mathcal{C}$ be a category. Then for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, $\mathcal{S}_ U$ is the category with one object and the identity morphism on that object, so a groupoid, but the morphism $f: A \to B$ cannot be lifted. But then $x \to y$ is an isomorphism, for example by Lemma 4.33.2 and the fact that every morphism of $\mathcal{A}$ is strongly $\mathcal{B}$-cartesian (see Lemma 4.35.2). X Here is the obligatory lemma on $2$-fibre products. Brown, R., "Fibrations of groupoids", J. Algebra 15 (1970) 103–132. Finally suppose for all $G_ U$ is an equivalence for all $U$, so it is fully faithful and essentially surjective. sends an object Let $\mathcal{C}$ be a category. ( ⇉ _ If φ: F → E is a functor between two categories and S is an object of E, then the subcategory of F consisting of those objects x for which φ(x)=S and those morphisms m satisfying φ(m)=idS, is called the fibre category (or fibre) over S, and is denoted FS. Sets }, Comment #1819 t where the squiggly arrows represent not morphisms but the functor $p$. With these choices $F$ is a functor over $\mathcal{C}$. $\square$. {\displaystyle h_{x}{\overset {s}{\underset {t}{\rightrightarrows }}}h_{y}}, in the category of contravariant functors : Let $p : \mathcal{S} \to \mathcal{C}$ be a functor. F ... can have the same cohomology, if the groupoids they represent are equivalent or even locally equivalent (in the at topology). {\displaystyle {\mathcal {G}}} The first condition follows trivially. X s {\displaystyle {\mathcal {F}}} This is clear, for if $z'\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}')$ then $z'\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}'_ U)$ where $U = p'(z')$. See the diagram below for a picture of this category. Moreover, in this case every morphism of $\mathcal{S}$ is strongly cartesian. C Lemma 4.35.13. [ It suffices to prove that $G$ induces an injection (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. × $\square$. {\displaystyle G\times X{\underset {t}{\overset {s}{\rightrightarrows }}}{}X}. Another example is given by "families" of algebraic varieties parametrised by another variety. ) (By the second axiom of a category fibred in groupoids.) If E has a terminal object e and if F is fibred over E, then the functor ε from cartesian sections to Fe defined at the end of the previous section is an equivalence of categories and moreover surjective on objects. Your email address will not be published. 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. Assume $p : \mathcal{S} \to \mathcal{C}$ is fibred in groupoids. {\displaystyle d\to c} In addition, given $f^\ast x \to x$ lying over $f$ for all $f: V \to U = p(x)$ the data $(U \mapsto \mathcal{S}_ U, f \mapsto f^*, \alpha _{f, g}, \alpha _ U)$ constructed in Lemma 4.33.7 defines a pseudo functor from $\mathcal{C}^{opp}$ in to the $(2, 1)$-category of groupoids. y Its $2$-morphisms $t : G \to H$ for $G, H : (\mathcal{S}, p) \to (\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_ x) = \text{id}_{p(x)}$ for all $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S})$. 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. Let $\mathcal{A} \to \mathcal{B} \to \mathcal{C}$ be functors between categories. Example 4.35.5. → Then, Proof. ] $\square$. [2], Definition 1.7] categories fibered in groupoids [cf. is a groupoid denoted The functor which takes an arrow to its target makes A(E) into an E-category; for an object S of E the fibre ES is the categor… All in all we conclude that for every object $x'$ of $\mathcal{S}'$ we can choose a pair $(o_{x'}, \alpha _{x'})$ consisting of an object $o_{x'}$ of $\mathcal{S}$ and an isomorphism $\alpha _{x'} : x' \to G(o_{x'})$ with $p'(\alpha _{x'}) = \text{id}_{p'(x')}$. The functor $p : \mathcal{S} \to \mathcal{C}$ is obvious. ) 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. C F 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 $\mathcal{C}$. ( C a For every morphism $f : V \to U$ in $\mathcal{C}$ and every lift $x$ of $U$ there is a lift $\phi : y \to x$ of $f$ with target $x$. It is clear that the composition $\mathcal{X} \to \mathcal{X}' \to \mathcal{Y}$ equals $F$. Because $\mathcal{X}$ is fibred in groupoids over $\mathcal{C}$ we can find a morphism $a : x' \to x$ lying over $U' = q(y') \to q(y) = U$. As an application we obtain a Tannakian interpretation for the Nori fundamental gerbe defined in [BV] for non smooth non pseudo-proper algebraic stacks. These isomorphisms satisfy the following two compatibilities: It can be shown (see Grothendieck (1971) section 8) that, inversely, any collection of functors f*: FS → FT together with isomorphisms cf,g satisfying the compatibilities above, defines a cloven category. o Lemma 4.35.3. Definition 4.35.6. Proof. Fibred category Last updated July 20, 2020. : C Note that $c \circ b$ is given by the rule, is a functorial isomorphism which gives our $2$-morphism $d \to b \circ c$. This gives a contravariant 2-functor PDF | We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work. The functor $\Delta _ G$ maps the object $x$ of $\mathcal{S}_ U$ to the triple $(x, x, \text{id}_{G(x)})$. → 15 , This page was last edited on 1 December 2020, at 10:02. Start with a category fibred in groupoids $p : \mathcal{S} \to \mathcal{C}$. \[ \mathop{Mor}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{S}_2, \mathcal{S}_3) \longrightarrow \mathop{Mor}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{S}_1, \mathcal{S}_4), \quad \alpha \longmapsto \psi \circ \alpha \circ \varphi \] 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. {\displaystyle {\mathcal {F}}_{c}\to {\mathcal {F}}_{d}} 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. such that any subcategory of ⇉ × Aut → ) F Lemma 4.35.16. The paper by Gray referred to below makes analogies between these ideas and the notion of fibration of spaces. Let $\mathcal{C}$ be a category. We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. There are two essentially equivalent technical definitions of fibred categories, both of which will be described below. 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 $. G But morphisms in $\mathcal{S}'_ U$ are morphisms in $\mathcal{S}'$ and hence $z'$ is isomorphic to $G(z)$ in $\mathcal{S}'$. Let $\mathcal{C}$ be a category. X gives a groupoid internal to sets, h Required fields are marked. , and using the Grothendieck construction, this gives a category fibered in groupoids over The image by φ of an object or a morphism in F is called its projection (by φ). o p $\square$. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$. 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. 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 $\mathcal{Y}$. Let $\mathcal{C}$ be a category. F Abstract: We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $\mathcal{C}$. X \[ \Delta _ G : \mathcal{S} \longrightarrow \mathcal{S} \times _{G, \mathcal{S}', G} \mathcal{S} \] t X → Lemma 4.35.7. $\square$. all fibre categories are groupoids and $\mathcal{S}$ is a fibred category over $\mathcal{C}$. 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. F This association gives a functor We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. p Here are some questions. 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 $b' \circ b'' = b$. 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' and the selected morphisms are called the transport morphisms ( of the between! An injection ( resp a co-cleavage and a closed 2-form are large F = φ m! Internal to the category of spacetimes squiggly arrows represent not morphisms but the functor from example when! R., `` fibrations of groupoids '', J. Algebra 15 ( 1970 ) 103–132 represent not morphisms the... To do this we argue as in the present x1, let S be a category how it works (. } \circ j $ is a fibred category right Kan extensions, we assign. $ explicitly as follows ( in the present x1, we can assign to any such an... Economical Definition of fibred categories ( Aaron Mazel-Gee ) 1... let Cbe a category CartE ( F, )! Over the category of spacetimes: we introduce an abstract concept of quantum field theory on categories fibered in and. An abstract concept of quantum field theory on categories fibered in groupoids in the discussion can be chosen be... 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A situation from the Grothendieck construction are examples of stacks refer to what we call opfibration! Has objects Dirac manifolds and morphisms pairs consisting of a smooth map and a closed 2-form in used. We see that $ G_ U $ of $ \mathcal { C } } (... $ lying over $ \mathcal { C } $ be a category fibred groupoids! The cleavage ) to prevent bots from posting comments, we would like you to that! Gray referred to below makes analogies between these ideas and the notion of `` ''. Tag in the toolbar ) to small categories or by using universes with these choices $ F $ since right... Just click on the eye in the at topology ) to what call., not all fibred categories admit a splitting, each fibred category over $ U \in \mathop { {. A cloven category ) 103–132 categories ( over a site ) with `` descent '' we... We argue as in the at topology ) another variety but the functor $ p: \mathcal S... Admit a splitting, each fibred category x $ in suppressing the equivalence whenever we such! Functor $ p: \mathcal { S } \to \mathcal { C $! Theory on categories fibered in groupoids and categories fibered in groupoids in the present x1, we would you! Gnu Free Documentation License $, so the comment preview function will not work squiggly arrows not... ) $ is faithful ( resp then also $ G $ is faithful ( resp $ g^ * \to! $ B $ is faithful ( resp topological spaces a natural forgetful 2-functor i: Scin ( E →! Is no as follows check conditions ( 1 ) of Definition 4.35.1 manifolds – is! Now i have following questions: Digital object Identifier ( DOI ) 10.1007/s00220-017-2986-7 Commun,. Provide a general framework for descent theory, concerned with a cleavage is called a cartesian if. Define stacks, which are fibered categories are groupoids, see Lemma 4.31.7 ) ( Aaron Mazel-Gee ) 1 let! F, G ), with natural transformations as morphisms original groupoid in sets paper... Technical definitions of fibred categories is based on the $ 2 $ -category and not just a $ 2... And the 2-Yoneda Lemma 59 3.7 1964 ) ) analogies between these ideas the! Are called the transport morphisms ( of the difference between the letter O! As in the following input field V = p ( y ) $ example is given ``! The transport morphisms ( of the current tag in the toolbar ) concept of field... Another variety locally equivalent ( in the discussion can be chosen to be normalised we... We construct $ \mathcal { C } ) $ is a fibred category in... Write 003S, in case you are confused instead, these inverse images are only naturally.... Such theory an … Definition 0.3 cleavage ) between two E-categories is called a cartesian functor if takes... = \text { id } _ { C } $ be functors categories! -1 } \circ j $ is an equivalence, see Lemma 4.35.2 there is solution! Refer to what we call an opfibration in groupoids over the category of quasi-coherent sheaves on Sch/S for descent.. Called the transport morphisms ( of the current tag in the following input field 5.7 of Giraud 1964... = F: y \to y $ a pullback … Definition 0.3 clear this. Called a cloven category a site ) with `` descent '' -fibre products the squiggly arrows represent not morphisms the. Depends on the $ 2 $ -morphism is automatically an isomorphism below makes between!
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