This is clear, for if $z'\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}')$ then $z'\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}'_ U)$ where $U = p'(z')$. 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. Objects as fibered categories and the 2-Yoneda Lemma 59 3.7. × Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work. If $G$ is an equivalence, then $G$ is an equivalence in the $2$-category of categories fibred in groupoids over $\mathcal{C}$. y return this == this.toLowerCase(); Let $p : \mathcal{S}\to \mathcal{C}$ and $p' : \mathcal{S'}\to \mathcal{C}$ be categories fibred in groupoids. X ) By the discussion following Definition 4.33.1 this means these two arrows are isomorphic as desired (here we use also that any isomorphism in $\mathcal{S}$ is strongly cartesian, by Lemma 4.33.2 again). The functor $p : \mathcal{S} \to \mathcal{C}$ is obvious. , Then $p' : \mathcal{S}' \to \mathcal{C}$ is fibred in groupoids. X fully faithful) we have to show for any objects $x, y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S})$ that $G$ induces an injection (resp. 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 $(a', b') \circ (a'', b'') = (a, b)$. d and an equivalence (by the assumption that $b$ is an equivalence, see Lemma 4.31.7). Then $G$ is fully faithful if and only if the diagonal, Proof. 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 $\mathcal{Y}$. Digital Object Identiﬁer (DOI) 10.1007/s00220-017-2986-7 Commun. Start with a category fibred in groupoids $p : \mathcal{S} \to \mathcal{C}$. There are many examples in topology and geometry where some types of objects are considered to exist on or above or over some underlying base space. Let $p : \mathcal{S} \to \mathcal{C}$ be 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. Lemma 4.35.3. Hence we obtain $(a, b) : (U', x', y', f') \to (U, x, y, f)$. X The functors of arrows of a fibered category 61 3.8. {\displaystyle G\times X\xrightarrow {\left(a,{\text{id}}\right)} {\text{Aut}}(X)\times X\xrightarrow {(f,x)\mapsto f(x)} X} We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. An object of the right hand side is a triple $(x, x', \alpha )$ where $\alpha : G(x) \to G(x')$ is a morphism in $\mathcal{S}'_ U$. 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 G} 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 We continue our abuse of notation in suppressing the equivalence whenever we encounter such a situation. t Since this diagram applied to an object 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. {\displaystyle p(y)=d} = The site in question has objects Dirac manifolds and morphisms pairs consisting of a smooth map and a closed 2-form. $\square$. Let $\mathcal{C}$ be a category. The final statement follows directly from Lemma 4.33.7. 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 $\mathcal{S}$. G Hence it suffices to prove that the fibre categories are groupoids, see Lemma 4.35.2. In short, the associated functor X . Namely, if $x \to y$ is a morphism of $\mathcal{A}_ U$, then its image in $\mathcal{B}$ is an isomorphism as $\mathcal{B}_ U$ is a groupoid. PDF | We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. Using right Kan extensions, we can assign to any such theory an … A cartesian section is thus a (strictly) compatible system of inverse images over objects of E. The category of cartesian sections of F is denoted by, In the important case where E has a terminal object e (thus in particular when E is a topos or the category E/S of arrows with target S in E) the functor. 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. See the diagram below for a picture of this category. It is clear that the composition $\mathcal{X} \to \mathcal{X}' \to \mathcal{Y}$ equals $F$. We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. Condition (2) phrased differently says that applying the functor $p$ gives a bijection between the sets of dotted arrows in the following commutative diagram below: Another way to think about the second condition is the following. X Some authors use the term cofibration in groupoids to refer to what we call an opfibration in groupoids. Denote $p : \mathcal{X} \to \mathcal{C}$ and $q : \mathcal{Y} \to \mathcal{C}$ the structure functors. $\square$, Comment #1768 : 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). The paper by Gray referred to below makes analogies between these ideas and the notion of fibration of spaces. Then there exists an equivalence $h : \mathcal{X}'' \to \mathcal{X}'$ of categories over $\mathcal{Y}$ such that $h \circ b$ is $2$-isomorphic to $a$ as $1$-morphisms of categories over $\mathcal{C}$. to the category A fibred category together with a cleavage is called a cloven category. Categories fibered in groupoids 52 3.4. . This is based on sections 3.1-3.4 of Vistoli's notes. We omit the verification that $\mathcal{X} \to \mathcal{X}'$ is an equivalence of fibred categories over $\mathcal{C}$. Lemma 4.35.2. 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 . By Lemma 4.35.8 it suffices to look at fibre categories over an object $U$ of $\mathcal{C}$. The following are equivalent all fibre categories are groupoids and $\mathcal{S}$ is a fibred category over $\mathcal{C}$. 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. . Two Examples of Integrable Category Fibered in Groupoids In the present x1, we give two examples of integrable [cf. an equivalence). 1. ( All contributions are licensed under the GNU Free Documentation License. s p Indeed, there are two canonical ways to construct an equivalent split category for a given fibred category F over E. More precisely, the forgetful 2-functor i: Scin(E) → Fib(E) admits a right 2-adjoint S and a left 2-adjoint L (Theorems 2.4.2 and 2.4.4 of Giraud 1971), and S(F) and L(F) are the two associated split categories. 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). The uniqueness implies that the morphisms $z' \to z$ and $z\to z'$ are mutually inverse, in other words isomorphisms. As a reminder, this is tag 003S. 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. x Definition 4.35.6. where $a$ and $b$ are equivalences of categories over $\mathcal{C}$ and $f$ and $g$ are categories fibred in groupoids. all fibre categories are groupoids and $\mathcal{S}$ is a fibred category over $\mathcal{C}$. p : \mathcal {S} \to \mathcal {C} is a category fibred in groupoids, and. 2) a stack over manifolds – which is the same as a category fibered over manifolds. 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. h from G A generality and holds in any $ 2 $ -category { id } U. 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