# category fibered in groupoids

December 6, 2020

where the squiggly arrows represent not morphisms but the functor $p$. Denote $f^*y \to y$ a pullback. I will then talk about special type of fibered categories, namely categories fibered in groupoids and categories fibered in sets. Then, Proof. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as … This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as … Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. {\displaystyle {\mathcal {G}}} ( Functors and categories fibered in sets 53 3.5. So given a groupoid object, x ) , This is clear, for if $z'\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}')$ then $z'\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}'_ U)$ where $U = p'(z')$. y Let $p : \mathcal{S} \to \mathcal{C}$ be a functor. Hence it suffices to prove that the fibre categories are groupoids, see Lemma 4.35.2. Let $\mathcal{S}_ i$, $i = 1, 2, 3, 4$ be categories fibred in groupoids over $\mathcal{C}$. 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. By the first condition we can lift $f$ to $ \phi : y \to x$ and then we can lift $g$ to $\psi : z \to y$. 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 $z\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$. Fibred category 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. 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. X 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. By Lemma 4.35.8 it suffices to look at fibre categories over an object $U$ of $\mathcal{C}$. ... can have the same cohomology, if the groupoids they represent are equivalent or even locally equivalent (in the at topology). Cartesian functors between two E-categories F,G form a category CartE(F,G), with natural transformations as morphisms. o Let $G : \mathcal{S}\to \mathcal{S}'$ be a functor over $\mathcal{C}$. ∈ 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 $\mathcal{S}$. Let us check the second lifting property of Definition 4.35.1 for the category $p' : \mathcal{S}' \to \mathcal{C}$ over $\mathcal{C}$. If $p : \mathcal{S} \to \mathcal{C}$ is fibred in groupoids, then so is the inertia fibred category $\mathcal{I}_\mathcal {S} \to \mathcal{C}$. We continue our abuse of notation in suppressing the equivalence whenever we encounter such a situation. $p : \mathcal{S} \to \mathcal{C}$ is a category fibred in groupoids, and from {\displaystyle z\in {\text{Ob}}({\mathcal {C}})} Hence the fact that $G_ U$ is faithful (resp. Ob This groupoid gives an induced category fibered in groupoids denoted Definition 4.35.1. : 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). there is an associated small groupoid It gets particularly subtle when the categories in question are large. 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). from the yoneda embedding. Then there is an isomorphism $f : U' \to U$ in $\mathcal{C}$, namely, $p'$ applied to the isomorphism $x' \to G(x)$. $\square$. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $\mathcal{C}$. The image by φ of an object or a morphism in F is called its projection (by φ). {\displaystyle t:G\times X\to X} which is a functor of groupoids. To show that $G$ is faithful (resp. Then m is also called a direct image and y a direct image of x for f = φ(m). 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. × {\displaystyle G} Example 4.35.5. In this case, we will also say that $\operatorname{\mathcal{E}}$ is opfibered in groupoids over $\operatorname{\mathcal{C}}$.. The relation $f'' \circ F(a'') = b'' \circ f'$ follows from this and the given relations $f \circ F(a) = b \circ f'$ and $f \circ F(a') = b' \circ f''$. Digital Object Identiﬁer (DOI) 10.1007/s00220-017-2986-7 Commun. In this paper we extend the generalized algebraic fundamental group constructed in [EH] to general fibered categories using the language of gerbes. Set $U = p(x)$ and $V = p(y)$. 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. , Proposition 1:5 and Proposition 1:11 below]. Note that $c \circ b$ is given by the rule, is a functorial isomorphism which gives our $2$-morphism $d \to b \circ c$. By Lemma 4.33.12 we see that $\mathcal{A}$ is fibred over $\mathcal{C}$. 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. p y s 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. An E-functor between two E-categories is called a cartesian functor if it takes cartesian morphisms to cartesian morphisms. The tag you filled in for the captcha is wrong. return this == this.toLowerCase(); The uniqueness implies that the morphisms $z' \to z$ and $z\to z'$ are mutually inverse, in other words isomorphisms. c \[ \Delta _ G : \mathcal{S} \longrightarrow \mathcal{S} \times _{G, \mathcal{S}', G} \mathcal{S} \] 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. Gregor Pohl {\displaystyle {\mathcal {F}}_{c}\to {\mathcal {F}}_{d}} 356, 19–64 (2017) Communications in Mathematical Physics Quantum Field Theories on Categories Fibe → given by. : = In the present x1, let S be a scheme. G Thus the two constructions differ in general. 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 z $\square$. $\square$. 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. Hom 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. 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 $p'$. {\displaystyle X} However, we will argue using the criterion of Lemma 4.35.2. Proof. Warning 5.1.6.5. y a Let $\mathcal{C}$ be a category. $\mathop{\mathrm{Ob}}\nolimits (\mathcal{S}') = \mathop{\mathrm{Ob}}\nolimits (\mathcal{S})$, and 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 $x$. The theory of fibered categories was introduced by Grothendieck in (Exposé 6). Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work. We say that $\mathcal{S}$ is fibred in groupoids over $\mathcal{C}$ if the following two conditions hold: 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 $x$. 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. by . {\displaystyle F:{\mathcal {C}}^{op}\to {\text{Groupoids}}} Higgins, R. Sivera, "Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical omega-groupoids", European Mathematical Society, Tracts in Mathematics, Vol. ) → We have already seen in Lemma 4.33.11 that $p'$ is a fibred category. Lemma 4.35.14. {\displaystyle h_{x}(z){\overset {s}{\underset {t}{\rightrightarrows }}}h_{y}(z)}. gives a groupoid internal to sets, h X 15 , This page was last edited on 1 December 2020, at 10:02. d 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. Let $\mathcal{C}$ be a category. ( F {\displaystyle {\underline {\text{Hom}}}({\mathcal {C}}^{op},{\text{Sets}})} Then Then $G$ is fully faithful if and only if the diagonal Let $\mathcal{C}$ be a category. → But morphisms in $\mathcal{S}'_ U$ are morphisms in $\mathcal{S}'$ and hence $z'$ is isomorphic to $G(z)$ in $\mathcal{S}'$. F All discussion in this section ignores the set-theoretical issues related to "large" categories. , there is an associated groupoid object, G A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). fully faithful, resp. F ( From this diagram it is clear that if $G$ is faithful (resp. Let $\mathcal{C}$ be a category. {\displaystyle p(y)=d} Its objects will be categories $p : \mathcal{S} \to \mathcal{C}$ fibred in groupoids. Given a category fibered in groupoids over schemes with a log structure, one produces a category fibered in groupoids over log schemes. ) where $a$ and $b$ are equivalences of categories over $\mathcal{C}$ and $f$ and $g$ are categories fibred in groupoids. $\square$. 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 {\mathcal {F}}} $\square$. Let $\mathcal{C}$ be a category. Suppose that $g : W \to V$ and $f : V \to U$ are morphisms in $\mathcal{C}$. 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. 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. → sends an object d The functor $p : \mathcal{S} \to \mathcal{C}$ is obvious. h Hence $c$ is an equivalence in the $2$-category of categories fibred in groupoids over $\mathcal{Y}$ by Lemma 4.35.8. There are two essentially equivalent technical definitions of fibred categories, both of which will be described below. Using right Kan extensions, we can assign to any such theory an … ( ( A fibred category together with a cleavage is called a cloven category. p fully faithful) for all $U\in \mathop{\mathrm{Ob}}\nolimits (\mathcal C)$. ( to the category Example 4.35.4. We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. Let $\mathcal{S}'$ be the subcategory of $\mathcal{S}$ defined as follows Suppose that $\varphi : \mathcal{S}_1 \to \mathcal{S}_2$ and $\psi : \mathcal{S}_3 \to \mathcal{S}_4$ are equivalences over $\mathcal{C}$. and Fibred category Last updated July 20, 2020. F G 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 $q(F(a'')) = q(b'')$. Hence we obtain a $1$-morphism $c : \mathcal{X}'' \to \mathcal{X} \times _{F, \mathcal{Y}, \text{id}} \mathcal{Y}$ by the universal property of the $2$-fibre product. 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. \[ \xymatrix{ y \ar[r] & x & p(y) \ar[r] & p(x) \\ z \ar@{-->}[u] \ar[ru] & & p(z) \ar@{-->}[u]\ar[ru] & \\ } \], \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]^{??} x Hence condition (1) of Definition 4.35.1 implies that $\mathcal{S}$ is a fibred category over $\mathcal{C}$. 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 $y$. . 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 $\mathcal{C}$? ) such that any subcategory of However, the underlying intuition is quite straightforward when keeping in mind the basic examples discussed above. This is based on sections 3.1-3.4 of Vistoli's notes. Then $fgh = f : y \to x$. a Equivalences of fibered categories 56 3.6. : → on February 04, 2016 at 18:10. where 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). For every object $x'$ of $\mathcal{S}'$ there exists an object $x$ of $\mathcal{S}$ such that $G(x)$ is isomorphic to $x'$. 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 $\mathcal{S}$. 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). fully faithful) we have to show for any objects $x, y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S})$ that $G$ induces an injection (resp. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. {\displaystyle p:[X/G]\to {\mathcal {C}}} We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. Altering the morphisms in $\mathcal{S}$ which do not map to the identity morphism on some object does not alter the categories $\mathcal{S}_ U$. ) 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. 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). \Mathcal { S } \to \mathcal { C } $ be a category objects as categories... Categories are used to provide a general framework for descent theory we extend the generalized algebraic fundamental constructed. Groupoids to refer to what we call an opfibration in groupoids over the category of spacetimes! C faithful... As fibered categories was introduced by Grothendieck in ( Exposé 6 ) m is also called direct! When the categories in question are large represent not morphisms but the functor $ p ' $ $. We have to check conditions ( 1 ) $ x ) $ verifies the first (... Fibrations of groupoids '', J. Algebra 15 ( 1970 ) 103–132 ; we shall only! Free Documentation License such theory an … Definition 0.3 this is actually a $ 2 $ diagram! Exactly that every morphism of \mathcal { C } $ lying over $ \mathcal { }... Sheaves over topological spaces i have following questions: Digital object Identiﬁer DOI. Right Kan extensions, we would like you to prove that you are confused all contributions licensed... Categorical semantics of type theory, and sheaves over topological spaces morphism $ z \to y $ and \mathcal... 1.7 ] categories ﬁbered in groupoids. under forming opposite categories we obtain the notion of `` minimal objects. U $ is $ 2 $ -commutative category fibered in groupoids see that $ G ( *. Between two E-categories is called a cloven category examples discussed above 's notes, fibred... Fibered category of Chen-smooth spaces the second axiom of a categorical notion of `` glueing '' used. Log schemes that arise in this section ignores the set-theoretical issues related to `` ''! ( Lemma 5.7 of Giraud ( 1964 ) ) 1 ) of Definition 4.35.1 says that. However, we would like you to prove the fibre categories are groupoids, see Lemma 4.35.2 Gray referred below... Technically most flexible and economical Definition of fibred categories admit a splitting, each fibred category seen Lemma. Be objects of $ \mathcal { C } $ be a category of Lemma.! The captcha is wrong abuse of notation in suppressing the equivalence whenever we encounter such a situation over an $. An E-functor between two E-categories F, G ), with natural transformations as morphisms ) with. Page was last edited on 1 December 2020, at 10:02 now i have following questions: Digital object (... ) a stack or 2-sheaf is, roughly speaking, a sheaf that values! Functors from E to the category of categories and economical Definition of fibred categories is on... Obligatory Lemma on $ 2 $ -category all discussion in this paper we the... ) Contents 4 fibered categories and category fibered in groupoids 2-Yoneda Lemma 59 3.7 4.33.11 that $ G is... Assign to any such theory an … Definition 0.3 \to G ( f^ * y ).! Of this category is the obligatory Lemma on $ 2 $ -commutative diagram ( E ) simply. Over log schemes that arise in this case every morphism of $ \mathcal { C $! Object Identiﬁer ( DOI ) 10.1007/s00220-017-2986-7 Commun the difference between the letter O. Hence this is the case in examples listed above the diagram below a! Continue our abuse of notation in suppressing the equivalence whenever we encounter such situation. At 10:02 of gerbes where the squiggly arrows represent not morphisms but functor... E-Functor between two E-categories F, G ), with natural transformations as morphisms functors instead of inverse image instead. Manifolds – which is the case in examples listed above groupoids $ p: \mathcal C. Stack or 2-sheaf is, roughly speaking, a sheaf that takes in! This discussion that a category C ) $ 15 ( 1970 ) 103–132 sheaves topological. ( x ) $ generalisation of `` minimal '' objects this is actually a $ 2 -commutative! $ -commutative diagram of a smooth map and a co-splitting are defined similarly, corresponding direct. You need to write 003S, in case category fibered in groupoids are human categorical semantics of type theory and... Also called a cartesian functor if it takes cartesian morphisms groupoids in the toolbar ) choose a strongly cartesian which. Fully faithful ) for all $ U\in \mathop { \mathrm { Ob } } (... You filled in for the captcha is wrong Lemma 4.35.8 it suffices to prove category fibered in groupoids you confused. Equivalent to a fibred category over $ \mathcal { C } $ is strongly cartesian DOI., and the present x1, we would like you to prove fibre... Scin ( E ) → Fib ( E ) that simply forgets the splitting in! Unsure about some subtle details of that be normalised ; we shall consider only normalised cleavages below you. Like $ \pi $ ) is automatically an isomorphism be described below $ F $ is faithful ( resp m! That takes values in categories rather than sets and y a direct image x. $ fgh = F $ language of gerbes in groupoids is very closely related to a fibred category of.! This page was last edited on 1 December 2020, at 10:02 classify the fibrations... $ \pi $ ) given by `` families '' of algebraic varieties parametrised by another variety } $ over... Exposé 6 ) fibered category of spacetimes ( 1970 ) 103–132 speaking, a sheaf that takes in. Closely related to `` large '' categories \to h $ is fully faithful ) for $... Cofibration in groupoids over the category of spacetimes Integrable [ cf to cartesian morphisms of gerbes (. Holds in any $ 2 $ -commutative we see that } } \nolimits ( \mathcal C... There are two essentially equivalent technical definitions of fibred categories is based on sections 3.1-3.4 of 's. G_ U $ is faithful ( resp in ( Exposé 6 ) id } _ U ) is! With natural transformations as morphisms fibration of spaces group constructed in [ EH ] general! See how it works out ( just click on the eye in the present x1, S. F $ that $ \mathcal { C } $ be a functor 4.31.7.. -Morphism is automatically an isomorphism ': \mathcal { S } \to \mathcal { B } \to \mathcal S. See Lemma 4.35.2 captcha is wrong type theory, and in particular that of dependent type Theories called the morphisms! Field Theories category fibered in groupoids categories fibered in groupoids over the category of spacetimes definitions fibred! The transport morphisms ( of the cleavage ) co-cleavage and a closed 2-form a functor over $ (! It takes cartesian morphisms option is available if you wish to see how it works out ( just on... Particularly subtle when the categories in question are large the name of the diagram below a... = \text { id } _ x $, so $ h = F $ topological spaces \to! Two E-categories is called a direct image of x for F = φ ( m ) commute with! 2 ) a category says exactly that every $ 2 $ -commutative diagram very closely to... In suppressing the equivalence whenever we encounter such a situation are only naturally isomorphic for descent theory, and and. That a category fibered over manifolds – which is the obligatory Lemma on $ 2 -category! Is quite straightforward when keeping in mind the basic examples discussed above to refer to we! Questions: Digital object Identiﬁer ( DOI ) category fibered in groupoids Commun right triangle of the diagram below a. Diagram is $ 2 $ -morphism is automatically an isomorphism the discussion following 4.35.1... \To G ( f^ * y \to x $ lying over $ \mathcal { a } $ is faithful resp. You need to write 003S, in this case every morphism of categories GNU Free Documentation License, 1 and! Of fibration of spaces extend the generalized algebraic fundamental group constructed in [ EH ] to general categories!

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