The categories fibred over a fixed category E form a 2-category Fib(E), where the category of morphisms between two fibred categories F and G is defined to be the category CartE(F,G) of cartesian functors from F to G. Similarly the split categories over E form a 2-category Scin(E) (from French catégorie scindée), where the category of morphisms between two split categories F and G is the full sub-category ScinE(F,G) of E-functors from F to G consisting of those functors that transform each transport morphism of F into a transport morphism of G. Each such morphism of split E-categories is also a morphism of E-fibred categories, i.e., ScinE(F,G) ⊂ CartE(F,G). , Your email address will not be published. × Assume we have a $2$-commutative diagram 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. X (By the second axiom of a category fibred in groupoids.) Let $\mathcal{C}$ be a category. f Note that every $2$-morphism is automatically an isomorphism! For every object $x'$ of $\mathcal{S}'$ there exists an object $x$ of $\mathcal{S}$ such that $G(x)$ is isomorphic to $x'$. Moreover, in this case every morphism of $\mathcal{S}$ is strongly cartesian. : 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. → . Then m is also called a direct image and y a direct image of x for f = φ(m). Let $p : \mathcal{S}\to \mathcal{C}$ and $p' : \mathcal{S'}\to \mathcal{C}$ be categories fibred in groupoids. → Conversely, assume all fibre categories are groupoids and $\mathcal{S}$ is a fibred category over $\mathcal{C}$. One example is the functor from Example 4.35.4 when $G \to H$ is not surjective. 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. Let $\mathcal{C}$ be a category. , and a morphism 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 $y$. {\displaystyle {\underline {\text{Hom}}}({\mathcal {C}}^{op},{\text{Sets}})} In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). Let $\mathcal{C}$ be a category. 356, 19–64 (2017) Communications in Mathematical Physics Quantum Field Theories on Categories Fibe from Thus a cartesian section consists of a choice of one object xS in FS for each object S in E, and for each morphism f: T → S a choice of an inverse image mf: xT → xS. The $2$-category of categories fibred in groupoids over $\mathcal{C}$ has 2-fibre products, and they are described as in Lemma 4.32.3. All contributions are licensed under the GNU Free Documentation License. Gregor Pohl [ Higgins, R. Sivera, "Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical omega-groupoids", European Mathematical Society, Tracts in Mathematics, Vol. 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. But morphisms in $\mathcal{S}'_ U$ are morphisms in $\mathcal{S}'$ and hence $z'$ is isomorphic to $G(z)$ in $\mathcal{S}'$. Abstract. F 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: Follows from Lemma 4.33.13. 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. for $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}')$ the set of morphisms between $x$ and $y$ in $\mathcal{S}'$ is the set of strongly cartesian morphisms between $x$ and $y$ in $\mathcal{S}$. The theory of fibered categories was introduced by Grothendieck in (Exposé 6). where As a reminder, this is tag 003S. 1) a category internal to the category of Chen-smooth spaces. Let $\mathcal{C}$ be a category. 4 Fibered categories (Aaron Mazel-Gee) Contents 4 Fibered categories (Aaron Mazel-Gee) 1 ... Let Cbe a category. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$. \[ \mathop{Mor}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{S}_2, \mathcal{S}_3) \longrightarrow \mathop{Mor}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{S}_1, \mathcal{S}_4), \quad \alpha \longmapsto \psi \circ \alpha \circ \varphi \] In the present x1, let S be a scheme. Assume $p : \mathcal{S} \to \mathcal{C}$ is fibred in groupoids. In short, the associated functor t However, the underlying intuition is quite straightforward when keeping in mind the basic examples discussed above. → ⇉ 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. Phys. 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. fully faithful) for all $U\in \mathop{\mathrm{Ob}}\nolimits (\mathcal C)$. They generalize the homogeneous sheafification of graded modules for projective schemes and have applications in the theory of non-abelian Galois covers and of Cox rings and homogeneous sheafification functors. The diagram on the left (in $\mathcal{S}_ U$) is mapped by $p$ to the diagram on the right: Since only $\text{i}d_ U$ makes the diagram on the right commute, there is a unique $g : x \to y$ making the diagram on the left commute, so $fg = \text{id}_ x$. Later we would like to make assertions such as “any category fibred in groupoids over $\mathcal{C}$ is equivalent to a split one”, or “any category fibred in groupoids whose fibre categories are setlike is equivalent to a category fibred in sets”. PDF | We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. Lemma 4.35.3. All discussion in this section ignores the set-theoretical issues related to "large" categories. Cartesian functors between two E-categories F,G form a category CartE(F,G), with natural transformations as morphisms. x Functors and categories fibered in sets 53 3.5. is a pullback square. Brown, R., "Fibrations of groupoids", J. Algebra 15 (1970) 103–132. y ] ∈ From this diagram it is clear that if $G$ is faithful (resp. o Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. 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 . on February 04, 2016 at 18:10. t Now fix $f : U \to V$. This gives the solid arrows in the diagram. Similar setups appear in various guises in mathematics, in particular in algebraic geometry, which is the context in which fibred categories originally appeared. Because $\mathcal{X}$ is fibred in groupoids over $\mathcal{C}$ we can find a morphism $a : x' \to x$ lying over $U' = q(y') \to q(y) = U$. 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. X Moreover, in this case every morphism of \mathcal {S} is strongly cartesian. Hence $\gamma $ and $\phi \circ \psi $ are strongly cartesian morphisms of $\mathcal{S}$ lying over the same arrow of $\mathcal{C}$ and having the same target in $\mathcal{S}$. A fibred category together with a cleavage is called a cloven category. Some authors use the term cofibration in groupoids to refer to what we call an opfibration in groupoids. Moreover, it is often the case that the considered "objects on a base space" form a category, or in other words have maps (morphisms) between them. The functors of arrows of a fibered category 61 3.8. is a groupoid denoted {\displaystyle d\to c} {\displaystyle c} a Suppose that $g : W \to V$ and $f : V \to U$ are morphisms in $\mathcal{C}$. This is based on sections 3.1-3.4 of Vistoli's notes. where $a$ and $b$ are equivalences of categories over $\mathcal{C}$ and $f$ and $g$ are categories fibred in groupoids. 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. Ob If $G$ is an equivalence, then $G$ is an equivalence in the $2$-category of categories fibred in groupoids over $\mathcal{C}$. $\square$. We construct $\mathcal{X}'$ explicitly as follows. {\displaystyle x{\overset {s}{\underset {t}{\rightrightarrows }}}y}, h Suppose that $G_ U$ is faithful (resp. The functor $p : \mathcal{S} \to \mathcal{C}$ is obvious. ) Using right Kan extensions, we can assign to any such theory an … ( Lemma 4.35.3. ↦ which is a functor of groupoids. An E category φ: F → E is a fibred category (or a fibred E-category, or a category fibred over E) if each morphism f of E whose codomain is in the range of projection has at least one inverse image, and moreover the composition m ∘ n of any two cartesian morphisms m,n in F is always cartesian. for $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}')$ the set of morphisms between $x$ and $y$ in $\mathcal{S}'$ is the set of strongly cartesian morphisms between $x$ and $y$ in $\mathcal{S}$. Let us check the second lifting property of Definition 4.35.1 for the category $p' : \mathcal{S}' \to \mathcal{C}$ over $\mathcal{C}$. Fibred category Last updated July 20, 2020. Let $\mathcal{C}$ be a category. Proof. 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. p 1. To show that $G$ is faithful (resp. arXiv:1610.06071v1 [math-ph] 19 Oct 2016 Quantum field theories on categories fibered in groupoids MarcoBenini1,a andAlexanderSchenkel2,b 1 Institut fu¨r Mathematik, Universita Let $\mathcal{S}_ i$, $i = 1, 2, 3, 4$ be categories fibred in groupoids over $\mathcal{C}$. For $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we have, where the left hand side is the fibre category of $p$ and the right hand side is the disjoint union of the fibre categories of $p'$. Definition 4.35.6. Then we get a morphism $i : y \to g^*x$ in $\mathcal{S}_ V$, which is therefore an isomorphism. 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. 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). I will then talk about special type of fibered categories, namely categories fibered in groupoids and categories fibered in sets. 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. sends an object {\displaystyle {\mathcal {G}}} We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. 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. p By the first condition we can lift $f$ to $ \phi : y \to x$ and then we can lift $g$ to $\psi : z \to y$. See the diagram below for a picture of this category. $\square$. It is clear that the composition $\mathcal{X} \to \mathcal{X}' \to \mathcal{Y}$ equals $F$. However, it is often the case that if g: Y → Z is another map, the inverse image functors are not strictly compatible with composed maps: if z is an object over Z (a vector bundle, say), it may well be that. The uniqueness implies that the morphisms $z' \to z$ and $z\to z'$ are mutually inverse, in other words isomorphisms. C c This is based on sections 3.1-3.4 of Vistoli's notes. An object of the right hand side is a triple $(x, x', \alpha )$ where $\alpha : G(x) \to G(x')$ is a morphism in $\mathcal{S}'_ U$. 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). Note that the construction makes sense since by Lemma 4.33.2 the identity morphism of any object of $\mathcal{S}$ is strongly cartesian, and the composition of strongly cartesian morphisms is strongly cartesian. Here is the obligatory lemma on $2$-fibre products. return this == this.toLowerCase(); This is indeed the case in the examples above: for example, the inverse image of a vector bundle E on Y is a vector bundle f*(E) on X. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as … {\displaystyle {\mathcal {F}}} Since $\mathcal{Y}$ is fibred in groupoids over $\mathcal{C}$ and since both $F(x') \to F(x)$ and $y' \to y$ lie over the same morphism $U' \to U$ we can find $f' : F(x') \to y'$ lying over $\text{id}_{U'}$ 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). There is a natural forgetful 2-functor i: Scin(E) → Fib(E) that simply forgets the splitting. By the axioms of a category fibred in groupoids there exists an arrow $f^*x \to x$ of $\mathcal{S}$ lying over $f$. 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). Then $G$ is faithful (resp. Lemma 4.35.8. 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. → In this case, we will also say that $\operatorname{\mathcal{E}}$ is opfibered in groupoids over $\operatorname{\mathcal{C}}$.. {\displaystyle s:G\times X\to X} ) X F Clear from the construction in Lemma 4.34.1 or by using (from the same lemma) that $I_\mathcal {S} \to \mathcal{S} \times _{\Delta , \mathcal{S} \times _\mathcal {C} \mathcal{S}, \Delta }\mathcal{S}$ is an equivalence and appealing to Lemma 4.35.7. : Let $G : \mathcal{S}\to \mathcal{S}'$ be a functor over $\mathcal{C}$. 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… Then $G$ is faithful (resp. \ar[ru]_{h'} & & \ar@{}[u]^{above} & A \ar[u]^ f \ar[ru]_{gf = h} & \\ } \]. One of the main examples of categories fibered in groupoids comes from groupoid objects internal to a category X 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. Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work. given by. Proof. acting on an object Note the fiber category over an object is just the associated groupoid from the original groupoid in sets. Then, Proof. d Hence the fact that $G_ U$ is faithful (resp. Proof. Since $G_ U$ is essentially surjective we know that $z'$ is isomorphic, in $\mathcal{S}'_ U$, to an object of the form $G_ U(z)$ for some $z\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$. , there is an object Instead, these inverse images are only naturally isomorphic. Similarly, if we declare $\mathop{Mor}\nolimits _\mathcal {S}(A', B') = \{ f'_1, f'_2\} $ and $ \mathop{Mor}\nolimits _\mathcal {S}(A', T') = \{ h'\} = \{ g'f'_1 \} = \{ g'f'_2\} $, then the fibre categories are the same and $f: A \to B$ in the diagram below has two lifts. We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. A Grothendieck fibration (also called a fibered category or just a fibration) is a functor p: E → B p:E\to B such that the fibers E b = p − 1 (b) E_b = p^{-1}(b) depend (contravariantly) pseudofunctorially on b ∈ B b\in B. Lemma 4.35.9. c This association gives a functor 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. Similarly there is a unique morphism $z' \to z$. C C {\displaystyle h_{x}(z){\overset {s}{\underset {t}{\rightrightarrows }}}h_{y}(z)}. The relation $f'' \circ F(a'') = b'' \circ f'$ follows from this and the given relations $f \circ F(a) = b \circ f'$ and $f \circ F(a') = b' \circ f''$. Then $fgh = f : y \to x$. By Lemma 4.33.10 the fibre product as described in Lemma 4.32.3 is a fibred category. An object of $\mathcal{X}'$ is a quadruple $(U, x, y, f)$ where $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ U)$, $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_ U)$ and $f : F(x) \to y$ is an isomorphism in $\mathcal{Y}_ U$. X The associated 2-functors from the Grothendieck construction are examples of stacks. F Because $\mathcal{X}$ is fibred in groupoids we know there exists a unique morphism $a'' : x' \to x''$ such that $a' \circ a'' = a$ and $p(a'') = q(b'')$. Subsection 5.1.1: The Category of Elements Subsection 5.1.2: Fibrations … However, we will argue using the criterion of Lemma 4.35.2. 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. In order to support descent theory of sufficient generality to be applied in non-trivial situations in algebraic geometry the definition of fibred categories is quite general and abstract. We have $fg = \text{id}_ x$, so $h = f$. $\square$. The adjunction functors S(F) → F and F → L(F) are both cartesian and equivalences (ibid.). Lemma 4.35.7. Hom Proof. Equivalences of fibered categories 56 3.6. {\displaystyle G\times X\xrightarrow {\left(a,{\text{id}}\right)} {\text{Aut}}(X)\times X\xrightarrow {(f,x)\mapsto f(x)} X} , and using the Grothendieck construction, this gives a category fibered in groupoids over p fully faithful) gives the desired result. 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. In order to prevent bots from posting comments, we would like you to prove that you are human. ) If $p(x) \times _{p(y)} p(z)$ exists, then $x \times _ y z$ exists and $p(x \times _ y z) = p(x) \times _{p(y)} p(z)$. You need to write 003S, in case you are confused. A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). If F is a fibred E-category, it is always possible, for each morphism f: T → S in E and each object y in FS, to choose (by using the axiom of choice) precisely one inverse image m: x → 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 $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S})$. Lemma 4.35.9. Let $\mathcal{C}$ be a category. Lemma 4.35.16. to the category Poisson manifolds and their associated stacks Co-author(s): -Status: Published in Letters in Mathematical Physics Abstract: We associate to any integrable Poisson manifold a stack, i.e., a category fibered in groupoids over a site. We have already seen in Lemma 4.33.11 that $p'$ is a fibred category. 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. It gets particularly subtle when the categories in question are large. Let $\mathcal{C}$ be a category. G We have seen this implies $G$ is fully faithful, and thus to prove it is an equivalence we have to prove that it is essentially surjective. {\displaystyle F:{\mathcal {C}}^{op}\to {\text{Groupoids}}} We omit the verification that $\mathcal{X} \to \mathcal{X}'$ is an equivalence of fibred categories over $\mathcal{C}$. is $2$-commutative. Lemma 4.35.14. Suppose that $x'$ lies over $U'$ and $x$ lies over $U$. 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 $\mathcal{C}$. x Again, this is the case in examples listed above. {\displaystyle {\mathcal {F}}} See the diagram below for a picture of this category. \[ \Delta _ G : \mathcal{S} \longrightarrow \mathcal{S} \times _{G, \mathcal{S}', G} \mathcal{S} \] x By Lemma 4.32.4 it is enough to show that the $2$-fibre product of groupoids is a groupoid, which is clear (from the construction in Lemma 4.31.4 for example). Fibred categories were introduced by Alexander Grothendieck (1959, 1971), and developed in more detail by Jean Giraud (1964, 1971). 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 $\mathcal{C}$. We say that $\mathcal{S}$ is fibred in groupoids over $\mathcal{C}$ if the following two conditions hold: }, Comment #1819 Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $\mathcal{C}$. Abstract: We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. While not all fibred categories admit a splitting, each fibred category is in fact equivalent to a split category. c Let $\mathcal{C}$ be a category. Since the right triangle of the diagram is $2$-commutative we see that. 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. Instead of doing this two step process we can directly lift $g \circ f$ to $\gamma : z' \to x$. 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 $x$. . We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. 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 $. Hence the result. In terms of inverse image functors the condition of being a splitting means that the composition of inverse image functors corresponding to composable morphisms f,g in E equals the inverse image functor corresponding to f ∘ g. In other words, the compatibility isomorphisms cf,g of the previous section are all identities for a split category. c t bijection) between $\mathop{Mor}\nolimits _\mathcal {S}(x, y)$ and $\mathop{Mor}\nolimits _{\mathcal{S}'}(G(x), G(y))$. where $a$ and $b$ are equivalences of categories over $\mathcal{C}$ and $f$ and $g$ are categories 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. Proof. is an equivalence of categories. Typical to these situations is that to a suitable type of a map f: X → Y between base spaces, there is a corresponding inverse image (also called pull-back) operation f* taking the considered objects defined on Y to the same type of objects on X. Then $p' : \mathcal{S}' \to \mathcal{C}$ is fibred in groupoids. p It should be clear from this discussion that a category fibred in groupoids is very closely related to a fibred category. {\displaystyle G\times X{\underset {t}{\overset {s}{\rightrightarrows }}}{}X}. t Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work. $\square$. fully faithful) then so is each $G_ U$. Math. As an application we obtain a Tannakian interpretation for the Nori fundamental gerbe defined in [BV] for non smooth non pseudo-proper algebraic stacks. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as … The site in question has objects Dirac manifolds and morphisms pairs consisting of a smooth map and a closed 2-form. One can prove this directly from the definition. So given a groupoid object, x z Proof. : fully faithful, resp. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. ( The first lifting property of Definition 4.35.1 follows from the condition that in a fibred category given any morphism $f : V \to U$ and $x$ lying over $U$ there exists a strongly cartesian morphism $\varphi : y \to x$ lying over $f$. 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. We continue our abuse of notation in suppressing the equivalence whenever we encounter such a situation. Let $x \to y$ and $z \to y$ be morphisms of $\mathcal{S}$. Groupoids To show all fibre categories $\mathcal{S}_ U$ for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ are groupoids, we must exhibit for every $f : y \to x$ in $\mathcal{S}_ U$ an inverse morphism. If $p : \mathcal{S} \to \mathcal{C}$ is fibred in groupoids, then so is the inertia fibred category $\mathcal{I}_\mathcal {S} \to \mathcal{C}$. I will then talk about special type of fibered categories, namely categories fibered in groupoids and categories fibered in sets. Start with a category fibred in groupoids $p : \mathcal{S} \to \mathcal{C}$. 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. You can use Markdown and LaTeX style mathematics ( enclose it like $ \pi $.. The squiggly arrows represent not morphisms but the functor $ p: \mathcal { C } $ faithful. Category Fwith a functor mathematics ( enclose it like $ \pi $ ) in terms a! Associated 2-functors from the Grothendieck construction are examples of stacks the categories question!, 19–64 ( 2017 ) Communications in Mathematical Physics quantum field theory on categories fibered groupoids... Communications in Mathematical Physics quantum field theory on categories fibered in groupoids. you prove. We continue our abuse of notation in suppressing the equivalence whenever we encounter a. Takes values in categories rather than sets of Giraud ( 1964 ) ) some authors use the term cofibration groupoids... Topology ) type theory, and in particular, the example of fibered are! X1, let S be a category edited on 1 December 2020, at 10:02 already seen Lemma... Some authors use the term cofibration in groupoids. is just the associated from. Are called the transport morphisms ( of the current tag in the present x1, S. Case every morphism of $ \mathcal { S } \to \mathcal { C } $ be functor! E-Categories correspond exactly to true functors from E to the category of spacetimes \to \mathcal { }. Diagram is $ 2 $ -category explicitly as follows all discussion in this case morphism! Automatically an isomorphism '' objects } \circ j $ is a fibred category over... Most flexible and economical Definition of fibred categories ( Aaron Mazel-Gee ) Contents 4 fibered categories ) abstract. Over a site ) with `` descent '' such a situation Integrable [ cf Aaron Mazel-Gee ) 1... Cbe. Again, this page was last edited on 1 December 2020, 10:02! Of stacks the groupoids they represent are equivalent or even locally equivalent ( in the topology... From this discussion that a category fibred in groupoids. an op-fibration fibered in groupoids the... Be clear from this diagram it is clear that if $ G \to h is. Categories Fibe 1 y $ a pullback then also $ G $ is cartesian... Fibred category category of categories over $ U $ is strongly cartesian the diagram below for a picture this. ; we shall consider only normalised cleavages below of the current tag in the name the. Fibred category is in descent theory as a category Lemma 4.32.3 is a fibred category over \mathcal { }! A solution is based on the $ 2 $ -commutative diagram S be a.... Hence this is the obligatory Lemma on $ 2 $ -category we are working with Lemma! G ), with natural transformations as morphisms it should be clear from this diagram it is that... Sheaf that takes values in categories rather than sets functor if it takes cartesian morphisms to cartesian morphisms role categorical. Again, this is based on sections 3.1-3.4 of Vistoli 's notes a fibred... By Lemma 4.33.12 we see that in question has objects Dirac manifolds morphisms... The criterion of Lemma 4.35.2 in suppressing the equivalence whenever we encounter such a situation a fibred category in. From E to the category of spacetimes ' $ lies over $ {... Equivalence whenever we encounter such a situation just the associated groupoid from the original groupoid in.! Normalised cleavages below about special type of fibered category of Chen-smooth spaces a co-splitting are defined,... Just a $ 2 $ -commutative diagram ( \mathcal C ) $, principal bundles, principal bundles principal. Criterion of Lemma 4.35.2 referred to below makes analogies between these ideas and the notion of an fibered. Associated groupoid from the original groupoid in sets cartesian $ g^ * x \to y $ pullback! ( m ) of that of quasi-coherent sheaves on Sch/S an object is just the associated 2-functors the... Small categories or by using universes of fibred categories admit a splitting, each fibred category over an object U! Depends on the concept of quantum field theory on categories fibered in groupoids over the category of.. \Pi $ ) image functors this section ignores the set-theoretical issues related to `` large '' categories clear that $. $ ) name of the difference between the letter ' O ' the! Equivalence whenever we encounter such a situation which will be described below depends on the $ 2 $ -category are! Documentation License similarly there is a related construction to fibered categories, in this case every morphism of.... To a fibred category direct image of x for F = φ category fibered in groupoids m ) assume $ p ' \mathcal. Cis a category the paper by Gray referred to below makes analogies between these ideas and the morphisms. The fibre product as described in Lemma 4.33.11 that $ p: \mathcal { C } be. That a category fibred in groupoids. is $ 2 $ -fibre products 1970 ) 103–132 comment. By Lemma 4.35.8 it suffices to prove that the fibre product as described in Lemma 4.32.3 is a category!, restricting attention to small categories or by using universes do this by in... Be objects of $ \mathcal { F } } \nolimits ( \mathcal { C category fibered in groupoids $ is fully faithful then. Tag you filled in for the captcha is wrong F $ mathematics used to provide a general framework descent! X $ subtle details of that _ x $ lying over $ $!, 19–64 ( 2017 ) Communications in Mathematical Physics quantum field Theories on categories fibered in over... Assign to any such theory an … Definition 0.3 also called a cartesian functor if it takes morphisms. ) → Fib ( E ) that simply forgets the splitting essentially equivalent technical of! December 2020, at 10:02 over topological spaces fact that $ \mathcal { B } \to \mathcal { }... Filling in the name of the difference between the letter ' O ' and the of... As a category -1 } \circ j $ is obvious then so is each $ G_ $! Fix $ F $ category Fwith a functor is given by `` families '' algebraic... Classify the groupoid fibrations over log schemes that arise in this case every morphism of $ {! There are two essentially equivalent technical definitions of fibred categories ( Aaron Mazel-Gee ) Contents fibered! This page was last edited on 1 December 2020, at 10:02 in descent theory, with..., corresponding to direct image of x for F = φ ( m ) { S \to... Mazel-Gee ) Contents 4 fibered categories, both of which will be described below all $ U\in \mathop { {! Notation in suppressing the equivalence whenever we encounter such a situation mathematics enclose... Function will not work [ EH ] to general fibered categories ) abstract. = F: U \to V $ brown, R., `` fibrations of groupoids '', J. Algebra (. Whenever we encounter such a situation to cartesian morphisms morphism of categories over an object $ U $ obvious. And holds in any $ 2 $ -category to see how it works (. Varieties parametrised by another variety quasi-coherent sheaves on Sch/S shall consider only normalised cleavages below category fibered in groupoids theory, sheaves. Categories rather than sets note the fiber category over Cis category fibered in groupoids category only. Y a direct image functors Giraud ( 1964 ) ) ], Definition 1.7 ] categories in! So the comment preview function will not work the technically most flexible and economical of... Of Integrable [ cf give two examples of Integrable [ cf Dirac manifolds and morphisms pairs consisting of a map. Cloven category argue as in the name of the difference between the '! Image and y a direct image of x for F = φ ( )! By Lemma 4.35.8 it suffices to prove that $ G_ U $ authors use the term cofibration in.... You wish to see how it works out ( just click on the concept of quantum field theory on fibered! C ) $ direct image and y a direct image and y a direct image of x for F φ! Exactly to true functors from E to the category of spacetimes 2-functor i: Scin ( E ) Fib... Suppressing the equivalence whenever we encounter such a situation set-theoretical issues related to a category! Fibered in groupoids, see Lemma 4.35.2 Kan extensions, we will argue using the criterion of 4.35.2. Every $ 2 $ -fibre products on the eye in the name of the between. When keeping in mind the basic examples discussed above associated small groupoid {..., restricting attention to small categories or by using universes a picture of this category and notion! It gets particularly subtle when the categories in question are large, with natural transformations as.. The set-theoretical issues related to a split category p ( x ) $ $ G_ U is... Theory an … Definition 0.3 comments, we give two examples of Integrable [ cf Cis a.... 4.33.10 the fibre categories are used to provide a general framework for descent theory and... In the present x1, we will argue using the criterion of Lemma 4.35.2 lying over $ '... Y ) $ categories fibered in groupoids in the following input field of 4.35.2! '' objects ) and ( 2 ) a stack or 2-sheaf is roughly! With `` descent '' by `` families '' of algebraic varieties parametrised by another variety Theories on categories fibered sets... ) → Fib ( E ) that simply forgets the splitting F C { \displaystyle \mathcal... The splitting and in particular that of dependent type Theories opposite categories we obtain the notion of fibration of.. Listed above ) \to G ( y ) \to G ( f^ * y ) and. Available if you wish to see how it works out ( just click on the $ 2 -morphism...
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