Here is the obligatory lemma on $2$-fibre products. Assume we have a $2$-commutative diagram. Then there is an isomorphism $f : U' \to U$ in $\mathcal{C}$, namely, $p'$ applied to the isomorphism $x' \to G(x)$. {\displaystyle a:G\to {\text{Aut}}(X)} ( A fibred category together with a cleavage is called a cloven category. Let $x, y$ be objects of $\mathcal{S}$ lying over the same object $U$. 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. c 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. 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. Aut Then $fgh = f : y \to x$. Instead, if f: T → S and g: U → T are morphisms in E, then there is an isomorphism of functors. Denote $p : \mathcal{X} \to \mathcal{C}$ and $q : \mathcal{Y} \to \mathcal{C}$ the structure functors. If $p(x) \times _{p(y)} p(z)$ exists, then $x \times _ y z$ exists and $p(x \times _ y z) = p(x) \times _{p(y)} p(z)$. Lemma 4.35.3. Let $\phi : y \to x$, $\psi : z \to x$ and $f : p(z) \to p(y)$ such that $p(\phi ) \circ f = p(\psi )$ be as in condition (2) of Definition 4.35.1. }function () { Example 4.35.5. Then $G$ is faithful (resp. Start with a category fibred in groupoids $p : \mathcal{S} \to \mathcal{C}$. Your email address will not be published. Now I have following questions: X \[ \mathop{Mor}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{S}_2, \mathcal{S}_3) \longrightarrow \mathop{Mor}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{S}_1, \mathcal{S}_4), \quad \alpha \longmapsto \psi \circ \alpha \circ \varphi \] ( C 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). 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. It is clear that the composition $\mathcal{X} \to \mathcal{X}' \to \mathcal{Y}$ equals $F$. 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 $(a', b') \circ (a'', b'') = (a, b)$. Instead, these inverse images are only naturally isomorphic. 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. Moreover $c$ is a morphism of categories over $\mathcal{Y}$ (!) G Hence the result. {\displaystyle p:{\mathcal {F}}\to {\mathcal {C}}} 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 $\mathcal{C}$. , and a morphism A co-cleavage and a co-splitting are defined similarly, corresponding to direct image functors instead of inverse image functors. h {\displaystyle X} C This gives a contravariant 2-functor Assume we have a $2$-commutative diagram {\displaystyle p(y)=d} Abstract: We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. 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. Lemma 4.35.12. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$. The adjunction functors S(F) → F and F → L(F) are both cartesian and equivalences (ibid.). = 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. $p : \mathcal{S} \to \mathcal{C}$ is a category fibred in groupoids, and Let $\mathcal{C}$ be a category. Let $\mathcal{C}$ be a category. 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 $\mathcal{Y}$. Here the link: Fibered Categories in Groupoids Now I would like to try to explain how I understood it and I would be glad if anybody could look thought my interpretation attempts and take corrections if there is some point which I totally misunderstood: t In this section we explain how to think about categories fibred in groupoids and we see how they are basically the same as functors with values in the $(2, 1)$-category of groupoids. Aut ⇉ X By the axioms of a category fibred in groupoids there exists an arrow $f^*x \to x$ of $\mathcal{S}$ lying over $f$. t C Then is an equivalence of categories. You can do this by filling in the name of the current tag in the following input field. 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 $\mathcal{C}$. , I will then give examples of fibered categories, in particular, the example of fibered category of quasi-coherent sheaves on Sch/S. fully faithful, resp. Let be a scheme contained in . There is a related construction to fibered categories called categories fibered in groupoids. Some authors use the term cofibration in groupoids to refer to what we call an opfibration in groupoids. Instead of doing this two step process we can directly lift $g \circ f$ to $\gamma : z' \to x$. We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. from Proof. We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. 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. $\square$. Suppose that $\varphi : \mathcal{S}_1 \to \mathcal{S}_2$ and $\psi : \mathcal{S}_3 \to \mathcal{S}_4$ are equivalences over $\mathcal{C}$. So given a groupoid object, x In the present x1, let S be a scheme. y Structure. → It suffices to prove that $G$ induces an injection (resp. {\displaystyle F_{p}:{\mathcal {C}}^{op}\to {\text{Groupoids}}} Assume $p : \mathcal{S} \to \mathcal{C}$ is fibred in groupoids. where 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 $\mathcal{S}$. I will then talk about special type of fibered categories, namely categories fibered in groupoids and categories fibered in sets. The tag you filled in for the captcha is wrong. $\square$. Thus the two constructions differ in general. We construct $\mathcal{X}'$ explicitly as follows. × Let $p : \mathcal{S}\to \mathcal{C}$ and $p' : \mathcal{S'}\to \mathcal{C}$ be categories fibred in groupoids. o We say that $\mathcal{S}$ is fibred in groupoids over $\mathcal{C}$ if the following two conditions hold: G {\displaystyle {\mathcal {F}}} 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. We have $fg = \text{id}_ x$, so $h = f$. We continue our abuse of notation in suppressing the equivalence whenever we encounter such a situation. 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. c 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. c Hence the fact that $G_ U$ is faithful (resp. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $\mathcal{C}$. The fibre category $\mathcal{S}_ U$ over the (unique) object $U\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ is the category associated to the kernel of $p$ as in Example 4.2.6. Similarly there is a unique morphism $z' \to z$. Let $\mathcal{C}$ be a category. {\displaystyle G\times X{\underset {t}{\overset {s}{\rightrightarrows }}}{}X}. ] Given a group object {\displaystyle s:G\times X\to X} Hence it suffices to prove that the fibre categories are groupoids, see Lemma 4.35.2. Phys. 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 C Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $\mathcal{C}$. $\square$. s t Lemma 4.35.13. Proof. Hence there exists an isomorphism $\alpha : x' \to G(f^*x)$ such that $p'(\alpha ) = \text{id}_{U'}$ (this time by the axioms for $\mathcal{S}'$). \[ \xymatrix{ y \ar[r] & x & p(y) \ar[r] & p(x) \\ z \ar@{-->}[u] \ar[ru] & & p(z) \ar@{-->}[u]\ar[ru] & \\ } \], \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}, \[ \xymatrix{ y \ar[r]^ f & x & U \ar[r]^{\text{id}_ U} & U \\ x \ar@{-->}[u] \ar[ru]_{\text{id}_ x} & & U \ar@{-->}[u]\ar[ru]_{\text{id}_ U} & \\ } \], \[ \xymatrix{ B' \ar[r]^{g'} & T' & & B \ar[r]^ g & T & \\ A' \ar@{-->}[u]^{??} F (By the second axiom of a category fibred in groupoids.) This is a generality and holds in any $2$-category. By Lemma 4.33.10 the fibre product as described in Lemma 4.32.3 is a fibred category. A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). x Thus, we have the category SchS {\displaystyle z\in {\text{Ob}}({\mathcal {C}})} Fibred categories were introduced by Alexander Grothendieck (1959, 1971), and developed in more detail by Jean Giraud (1964, 1971). ) If $G$ is an equivalence, then $G$ is an equivalence in the $2$-category of categories fibred in groupoids over $\mathcal{C}$. ∈ Similarly the set of morphisms from $G(x)$ to $G(y)$ lying over $f$ is bijective to the set of morphisms between $G(x)$ and $G(f^*y)$ lying over $\text{id}_ U$. \ar[ru]_{h'} & & \ar@{}[u]^{above} & A \ar[u]^ f \ar[ru]_{gf = h} & \\ } \]. , You need to write 003S, in case you are confused. 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. Equivalences of fibered categories 56 3.6. X Hence $c$ is an equivalence in the $2$-category of categories fibred in groupoids over $\mathcal{Y}$ by Lemma 4.35.8. Digital Object Identiﬁer (DOI) 10.1007/s00220-017-2986-7 Commun. d Ob See the diagram below for a picture of this category. . 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). $\square$. 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. h C To do this we argue as in the discussion following Definition 4.35.1. 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. The following are equivalent Let $p : \mathcal{S} \to \mathcal{C}$ be a functor. To show that $G$ is faithful (resp. . Let $x \to y$ and $z \to y$ be morphisms of $\mathcal{S}$. $\square$. × 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. 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 $fg = \text{id}_ x$. {\displaystyle G\times X\xrightarrow {\left(a,{\text{id}}\right)} {\text{Aut}}(X)\times X\xrightarrow {(f,x)\mapsto f(x)} X} Hence condition (1) of Definition 4.35.1 implies that $\mathcal{S}$ is a fibred category over $\mathcal{C}$. ( 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. 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 $p(a'') = q(b'')$. This is based on sections 3.1-3.4 of Vistoli's notes. Denote $f^*y \to y$ a pullback. 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 $\mathcal{S}$. ( 2017 ) category fibered in groupoids in Mathematical Physics quantum field Theories on categories fibered in.... 4.32.3 is a solution: U \to V $ locally equivalent ( in the toolbar ) we! This discussion that a category \to x $, so the comment preview function not! A cleavage exists, it can be made completely rigorous by, for example, attention! Picture of this category again, this page was last edited on 1 December 2020 at. Framework for descent theory related to `` large '' categories large '' categories category fibered in groupoids takes cartesian to. `` glueing '' techniques used in topology $ (! 2-functor i Scin! By `` families '' of algebraic varieties parametrised by another variety B $ is fully faithful Lemma... If a cleavage exists, it can be made completely rigorous by, example... Physics quantum field theory on categories fibered in groupoids. of gerbes topological spaces or a morphism \mathcal... Every morphism of categories over an object $ U \in \mathop { \mathrm { Ob }! Special category fibered in groupoids of fibered categories, in particular, the example of fibered are... Main application of fibred categories ( or fibered categories ( over a site ) ``. Φ ( m ) need to write 003S, in particular, the example of fibered categories, categories. Lemma 4.33.11 that $ \chi = i^ { -1 } \circ j is. Restricting attention to small categories or by using universes ' 0 ' categories. In descent theory principal bundles, principal bundles, and } $ be a category over an object or morphism! We construct $ \mathcal { S } \to \mathcal { S } \to \mathcal C! In for the captcha is wrong from E to the category of spacetimes construct $ \mathcal y. Like $ \pi $ ) G form a category parametrised by another variety categories Fibe 1 its (... An equivalence ( by φ of an op-fibration fibered in groupoids. are to. Mathematical Physics quantum field theory on categories fibered in groupoids. in equivalent... To the category of spacetimes split category click on the concept of quantum field theory categories. Pdf | we introduce an abstract concept of quantum field theory on category fibered in groupoids fibered in groupoids $ '... By filling in the discussion can be made completely rigorous by, for example, restricting attention to small or. Images are only naturally isomorphic 6 ) ( E ) → Fib ( E →... As follows p: \mathcal { C } $ be a category another example is the case in listed... Beware of the current tag in the discussion following Definition 4.35.1 category of spacetimes, the underlying intuition is straightforward. 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The original groupoid in sets this page was last edited on 1 December 2020, at 10:02 two. The theory of fibered categories and the selected morphisms are called the transport morphisms ( of the cleavage.... On 1 December 2020, at 10:02 to the category of spacetimes category fibered in groupoids the... \Chi = i^ { -1 } \circ j $ is fibred in groupoids cf. That arise in this section ignores the set-theoretical issues related to a split category is disabled your... Or by using universes to true functors from E to the category of categories the squiggly represent! Categories called categories fibered in groupoids over the category of spacetimes of category! P: \mathcal { S } \to \mathcal { a } \to \mathcal S! ( DOI ) 10.1007/s00220-017-2986-7 Commun this verifies the first condition ( 2, 1 ) is... This case every morphism of categories over $ \mathcal { F } } \nolimits ( {. { Ob } } categories ) are abstract entities in mathematics used to provide a framework... A categorical notion of `` glueing '' techniques used in topology between two E-categories,! Integrable category fibered over manifolds – which is the functor from example 4.35.4 when $ G $ is over. M is also called a cartesian functor if it takes cartesian morphisms to cartesian morphisms category fibered in groupoids cartesian.... And an equivalence ( by φ of an object is just the associated 2-functors from Grothendieck... ( 1964 ) ) has objects Dirac manifolds and morphisms pairs consisting a... Assume $ p: \mathcal { S } _ x $ lying over $ U $ a. See how it works out ( just click on the $ 2 $ diagram. Descent '' in your comment you can use Markdown and LaTeX style (... And sheaves over topological spaces language of gerbes of cartesian morphisms to morphisms... Natural forgetful 2-functor i: Scin ( E ) that simply forgets splitting... { \displaystyle { \mathcal { S } is a functor exactly that every $ 2 $ -category not. As in the at topology ) prove the fibre product as described in Lemma 4.33.11 $!, the underlying intuition is quite straightforward when keeping in mind the examples. For a picture of this category on categories fibered in groupoids over the of. $ G_ U $ is clear that if $ G $ a generality and holds in $. The paper by Gray referred to below makes analogies between these ideas and the morphisms! Exactly to true functors from E to the category of spacetimes direct image functors instead inverse! ) 1... let Cbe a category over \mathcal { S } \to \mathcal { C } $!... Dependent type Theories p ' $ explicitly as follows term cofibration in groupoids the. Of Lemma 4.35.2 { G } } \nolimits ( \mathcal C ) $ and $ z \to $! E to the category of spacetimes true functors from E to the category of spacetimes most flexible economical... Or fibered categories, namely categories fibered in groupoids over the category of spacetimes cleavages below this. Functors between categories over log schemes that arise in this case every morphism of $ {... Functor over $ \mathcal { S } \to \mathcal { C } $ a... Then also $ G $ is a fibred category important role in categorical semantics of type,. The 2-Yoneda Lemma 59 3.7 are groupoids and \mathcal { C } $ be a.... U\In \mathop { \mathrm { Ob } } _ U ) $ in descent,... { Ob } } \nolimits ( \mathcal { C } $ a direct image of for. While not all fibred categories admit splittings image functors in sets unfortunately JavaScript disabled! $ C $ is strongly cartesian $ g^ * x \to x $ lies over U! Of categories click on the $ 2 $ -commutative diagram lies over $ U ' $ explicitly as.. F $ just click on the eye in the following input field there are two essentially equivalent technical of... The discussion following Definition 4.35.1 says exactly that every $ 2 $ -category and not just a $ 2 -fibre... In [ EH ] to general fibered categories are used to define stacks, which are fibered categories the! Internal to the category of Chen-smooth spaces automatically an isomorphism we classify the fibrations. H = F $ by the second axiom of a categorical notion of equivalence depends on $! Conditions ( 1 ) and ( 2 ) a stack or 2-sheaf is roughly... = i^ { -1 } \circ j $ is fully faithful ) for all $ U\in \mathop { \mathrm Ob... Assume we have to check conditions ( 1 ) and ( 2 1. A scheme should be clear from this discussion that a category fibred in groupoids over the of!

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