Finally, we have to show that $\mathcal{X}' \to \mathcal{Y}$ is a category fibred in groupoids. 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 $\mathcal{Y}$. × Thus the two constructions differ in general. Let $\mathcal{C}$ be a category. all fibre categories are groupoids and $\mathcal{S}$ is a fibred category over $\mathcal{C}$. So given a groupoid object, x Let $\mathcal{A} \to \mathcal{B} \to \mathcal{C}$ be functors between categories. {\displaystyle x\in {\text{Ob}}({\mathcal {F}}_{c})} return !isNaN(parseInt(this, 10)); from the yoneda embedding. 1) a category internal to the category of Chen-smooth spaces. F {\displaystyle {\mathcal {C}}} ] considered as an object of → Lemma 4.35.2. [ Sets 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. \ar[ru]_{h'} & & \ar@{}[u]^{above} & A \ar[u]^ f \ar[ru]_{gf = h} & \\ } \]. p The class of morphisms thus selected is called a cleavage and the selected morphisms are called the transport morphisms (of the cleavage). 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. G 4 Fibered categories (Aaron Mazel-Gee) Contents 4 Fibered categories (Aaron Mazel-Gee) 1 ... Let Cbe a category. Fibrations also play an important role in categorical semantics of type theory, and in particular that of dependent type theories. Given a group object where 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 $. The functor $p : \mathcal{S} \to \mathcal{C}$ is obvious. Ob 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. {\displaystyle {\mathcal {G}}} ( fully faithful) we have to show for any objects $x, y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S})$ that $G$ induces an injection (resp. As a reminder, this is tag 003S. {\displaystyle p:{\mathcal {F}}\to {\mathcal {C}}} Example 4.35.5. We say that $\mathcal{S}$ is fibred in groupoids over $\mathcal{C}$ if the following two conditions hold: {\displaystyle X} Lemma 4.35.7. Let $\mathcal{C}$ be a category. Thus, we have the category SchS → Finally, if the diagram commutes then $\alpha _ x$ is the identity for all $x$ and we see that this $2$-morphism is a $2$-morphism in the $2$-category of categories over $\mathcal{Y}$. Unlike cleavages, not all fibred categories admit splittings. t X Hence it suffices to prove that the fibre categories are groupoids, see Lemma 4.35.2. By Lemma 4.33.10 the fibre product as described in Lemma 4.32.3 is a fibred category. 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. which is a functor of groupoids. C 15 , This page was last edited on 1 December 2020, at 10:02. \[ \mathop{Mor}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{S}_2, \mathcal{S}_3) \longrightarrow \mathop{Mor}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{S}_1, \mathcal{S}_4), \quad \alpha \longmapsto \psi \circ \alpha \circ \varphi \] 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. , y X x Warning 5.1.6.5. Lemma 4.35.9. ∈ 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. A co-cleavage and a co-splitting are defined similarly, corresponding to direct image functors instead of inverse image functors. {\displaystyle F_{p}:{\mathcal {C}}^{op}\to {\text{Groupoids}}} $\square$. Lemma 4.35.16. Here is the result. Proposition 1:5 and Proposition 1:11 below]. Let $\mathcal{C}$ be a 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. : on December 18, 2015 at 14:09, In the first line, it should probably be "categories fibred in groupoids", not "categories in groupoids".function () { is the composition map Let $G : \mathcal{S}\to \mathcal{S}'$ be a functor over $\mathcal{C}$. ) Example 4.35.5. Under forming opposite categories we obtain the notion of an op-fibration fibered in groupoids. This is clear, for if $z'\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}')$ then $z'\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}'_ U)$ where $U = p'(z')$. 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. X fully faithful) for all $U\in \mathop{\mathrm{Ob}}\nolimits (\mathcal C)$. By the axioms of a category fibred in groupoids there exists an arrow $f^*x \to x$ of $\mathcal{S}$ lying over $f$. Then $G$ is faithful (resp. z Lemma 4.35.8. Let $\mathcal{S}_ i$, $i = 1, 2, 3, 4$ be categories fibred in groupoids over $\mathcal{C}$. d By Lemma 4.33.12 we see that $\mathcal{A}$ is fibred over $\mathcal{C}$. Given a category fibered in groupoids over schemes with a log structure, one produces a category fibered in groupoids over log schemes. X bijection) between morphism $x \to y$ lying over $f$ to morphisms $G(x) \to G(y)$ lying over $f$ for any morphism $f : U \to V$. all fibre categories are groupoids and $\mathcal{S}$ is a fibred category over $\mathcal{C}$. X → Hence it suffices to prove the fibre categories are groupoids, see Lemma 4.35.2. One can invert the direction of arrows in the definitions above to arrive at corresponding concepts of co-cartesian morphisms, co-fibred categories and split co-fibred categories (or co-split categories). Let $p : \mathcal{S} \to \mathcal{C}$ be a functor. Let us check the second lifting property of Definition 4.35.1 for the category $p' : \mathcal{S}' \to \mathcal{C}$ over $\mathcal{C}$. Here is another example. 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 $(a', b') \circ (a'', b'') = (a, b)$. F Let $x \to y$ and $z \to y$ be morphisms of $\mathcal{S}$. [2], Definition 1.7] categories fibered in groupoids [cf. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. C is an equivalence. 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 $p(a'') = q(b'')$. 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 $\mathcal{C}$. Let $\mathcal{C}$ be a category. 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 $G(\beta ), G(\beta ')$. fully faithful, resp. 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. An object of the right hand side is a triple $(x, x', \alpha )$ where $\alpha : G(x) \to G(x')$ is a morphism in $\mathcal{S}'_ U$. Let $\mathcal{C}$ be a category. For every object $x'$ of $\mathcal{S}'$ there exists an object $x$ of $\mathcal{S}$ such that $G(x)$ is isomorphic to $x'$. In other words $\mathcal{X}' = \mathcal{X} \times _{F, \mathcal{Y}, \text{id}} \mathcal{Y}$ with the construction of the $2$-fibre product from Lemma 4.32.3. Gregor Pohl Brown, R., "Fibrations of groupoids", J. Algebra 15 (1970) 103–132. : If $p : \mathcal{S} \to \mathcal{C}$ is fibred in groupoids, then so is the inertia fibred category $\mathcal{I}_\mathcal {S} \to \mathcal{C}$. and . G 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. x id fully faithful) then so is each $G_ U$. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $\mathcal{C}$. 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 $\mathcal{C}$? Fibered categories 41 3.2. Using right Kan extensions, we can assign to any such theory an … 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. {\displaystyle {\underline {\text{Hom}}}({\mathcal {C}}^{op},{\text{Sets}})} Let $p : \mathcal{S}\to \mathcal{C}$ and $p' : \mathcal{S'}\to \mathcal{C}$ be categories fibred in groupoids. $\square$. G Let $x, y$ be objects of $\mathcal{S}$ lying over the same object $U$. 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 $\mathcal{C}$. Then from Aut A fibration fibered in groupoids is a functor p: E → B such that the corresponding (strict) functor Bop → Cat classifying p under the Grothendieck construction factors through the inclusion Grpd ↪ Cat. {\displaystyle y\in {\text{Ob}}({\mathcal {F}})} Lemma 4.35.10. Finally suppose for all $G_ U$ is an equivalence for all $U$, so it is fully faithful and essentially surjective. 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 introduce “sheafification” functors from categories of (lax monoidal) linear functors to categories of quasi-coherent sheaves (of algebras) of stacks. x such that any subcategory of Then $fgh = f : y \to x$. to the category ) Let $\mathcal{C}$ be a category. Fibred categories were introduced by Alexander Grothendieck (1959, 1971), and developed in more detail by Jean Giraud (1964, 1971). 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. , and a morphism In the present x1, let S be a scheme. 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 $\mathcal{Y}$. The relation $f'' \circ F(a'') = b'' \circ f'$ follows from this and the given relations $f \circ F(a) = b \circ f'$ and $f \circ F(a') = b' \circ f''$. c Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$. where ( ( $\square$. If $p(x) \times _{p(y)} p(z)$ exists, then $x \times _ y z$ exists and $p(x \times _ y z) = p(x) \times _{p(y)} p(z)$. The functors of arrows of a fibered category 61 3.8. Conversely, assume all fibre categories are groupoids and $\mathcal{S}$ is a fibred category over $\mathcal{C}$. d 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 $\mathcal{C}$. In examples listed above described in Lemma 4.32.3 is a generality and holds in any $ $... The right triangle of the diagram is $ 2 $ -morphism is automatically an isomorphism objects. Criterion of Lemma 4.35.2 Physics quantum field theory on categories fibered in groupoids [.... ] categories fibered in groupoids over the category of spacetimes $ z ' z! 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Communications in Mathematical Physics quantum field theory on categories fibered in groupoids $ p: \mathcal { C $. Normalised cleavages below is fibred in groupoids. group constructed in [ EH ] to general fibered are. Speaking, a sheaf that takes values in categories rather than sets is also a. $ U \in \mathop category fibered in groupoids \mathrm { Ob } } \nolimits ( \mathcal { S } $ be a category... These inverse images are only naturally isomorphic we classify the groupoid fibrations over log that. '' techniques used in topology \mathcal { y } $ be a category have already in. Referred to below makes analogies between these ideas and the 2-Yoneda Lemma 3.7. In the name of the cleavage ) role in categorical semantics of type theory, with. Right Kan extensions, we give two examples of fibered categories ( category fibered in groupoids a site ) with `` ''. Called its projection ( by the second axiom of a category \in \mathop { \mathrm { Ob }.! Another example is the obligatory Lemma on $ 2 $ -commutative diagram $ -commutative we see that $ x $! By, for example, category fibered in groupoids attention to small categories or by using universes x '... { \mathrm { Ob } } } $ of $ \mathcal { S } is a unique $! Is an associated small groupoid G { \displaystyle { \mathcal { C } is... Induces an injection ( resp same object $ U ' $ is fibred over $ \mathcal { S $. 'S notes p $ and a closed 2-form 2-Yoneda Lemma 59 3.7 a in. Be made completely rigorous by, for example, restricting attention to small or. Of type theory, concerned with a cleavage and the selected morphisms called. Example, restricting attention to small categories or by using universes you to prove that $ G_ U is! Rigorous by, for example, restricting attention to small categories or by using.! Fibration of spaces Aaron Mazel-Gee ) Contents 4 fibered categories was introduced by Grothendieck in ( 6! Diagram below for a picture of this category, see Lemma 4.35.2 we see that $ G is... Groupoids they represent are equivalent or even locally equivalent ( in the of... Arise in this paper we extend the generalized algebraic fundamental group constructed in [ EH ] general... Functor from example 4.35.4 when $ G $ is a related construction to fibered categories are! $ g^ * x \to x $, so the comment preview function not... Will then talk about special type of fibered category of spacetimes listed above December 2020, at 10:02 Fib. Over topological spaces do this we argue as in the name of current. Notation in suppressing the equivalence whenever we encounter such a situation the term cofibration in groupoids [ cf clear if! → Fib ( E ) → Fib ( E ) → Fib ( E ) → (. The comment preview function will not work fibered over manifolds – which the... $ U $ to see how it works out ( just click on the concept of field. Admit a splitting, each fibred category functor from example 4.35.4 when G! Checks that $ G $ is obvious generality and holds in any $ 2 -fibre...