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47-398: The topic of the work is a generalized homotopy theory using higher category theory . The word "stacks" in the title refers to what are nowadays usually called " ∞-groupoids ", one possible definition of which Grothendieck sketches in his manuscript. (The stacks of algebraic geometry, which also go back to Grothendieck, are not the focus of this manuscript.) Among the concepts introduced in

94-564: A CW complex . In the same vein as above, a " map " is a continuous function, possibly with some extra constraints. Often, one works with a pointed space —that is, a space with a "distinguished point", called a basepoint. A pointed map is then a map which preserves basepoints; that is, it sends the basepoint of the domain to that of the codomain. In contrast, a free map is one which needn't preserve basepoints. The Cartesian product of two pointed spaces X , Y {\displaystyle X,Y} are not naturally pointed. A substitute

141-399: A contravariant functor from the category of spaces to the category of abelian groups that satisfies the axioms generalizing ordinary cohomology theory. As it turns out, such a functor may not be representable by a space but it can always be represented by a sequence of (pointed) spaces with structure maps called a spectrum. In other words, to give a generalized cohomology theory is to give

188-595: A topological space X {\displaystyle X} and how the foundations for such a topic could be laid down and relativized using topos theory making way for higher gerbes . Moreover, he was critical of using strict groupoids for laying down these foundations since they would not be sufficient for developing the full theory he envisioned. He laid down his ideas of what such an ∞-groupoid should look like, and gave some axioms sketching out how he envisioned them. Essentially, they are categories with objects, arrows, arrows between arrows, and so on, analogous to

235-423: A CW complex, then so does its loop space Ω X {\displaystyle \Omega X} . Given a topological group G , the classifying space for principal G -bundles ("the" up to equivalence) is a space B G {\displaystyle BG} such that, for each space X , where Brown's representability theorem guarantees the existence of classifying spaces. The idea that

282-445: A Hurewicz fibration can be checked locally on a paracompact space. While a cofibration is injective with closed image, a fibration need not be surjective. There are also based versions of a cofibration and a fibration (namely, the maps are required to be based). A pair of maps i : A → X {\displaystyle i:A\to X} and p : E → B {\displaystyle p:E\to B}

329-489: A classifying space classifies principal bundles can be pushed further. For example, one might try to classify cohomology classes: given an abelian group A (such as Z {\displaystyle \mathbb {Z} } ), where K ( A , n ) {\displaystyle K(A,n)} is the Eilenberg–MacLane space . The above equation leads to the notion of a generalized cohomology theory; i.e.,

376-571: A cofibration is characterized by the existence of a retract, a fibration is characterized by the existence of a section called the path lifting as follows. Let p ′ : N p → B I {\displaystyle p':Np\to B^{I}} be the pull-back of a map p : E → B {\displaystyle p:E\to B} along χ ↦ χ ( 1 ) : B I → B {\displaystyle \chi \mapsto \chi (1):B^{I}\to B} , called

423-406: A fibration is a covering map as it comes with a unique path lifting. If E {\displaystyle E} is a principal G -bundle over a paracompact space, that is, a space with a free and transitive (topological) group action of a ( topological ) group, then the projection map p : E → X {\displaystyle p:E\to X} is a fibration, because

470-574: A homotopy g t : Z → B {\displaystyle g_{t}:Z\to B} such that p ∘ h 0 = g 0 {\displaystyle p\circ h_{0}=g_{0}} , there exists a homotopy h t : Z → X {\displaystyle h_{t}:Z\to X} that extends h 0 {\displaystyle h_{0}} and such that p ∘ h t = g t {\displaystyle p\circ h_{t}=g_{t}} . While

517-408: A homotopy such that s ( e , χ ) ( 0 ) = e , ( p I ∘ s ) ( e , χ ) = χ {\displaystyle s(e,\chi )(0)=e,\,(p^{I}\circ s)(e,\chi )=\chi } where p I : E I → B I {\displaystyle p^{I}:E^{I}\to B^{I}}

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564-422: A path from the map h 0 {\displaystyle h_{0}} to the map h 1 {\displaystyle h_{1}} . Indeed, a homotopy can be shown to be an equivalence relation . When X , Y are pointed spaces, the maps h t {\displaystyle h_{t}} are required to preserve the basepoint and the homotopy h {\displaystyle h}

611-454: A sense the closest possible non-commutative generalization of chain complexes, the homology groups of the chain complex becoming the homotopy groups of the “non-commutative chain complex” or stack. - Grothendieck This is later explained by the intuition provided by the Dold–Kan correspondence : simplicial abelian groups correspond to chain complexes of abelian groups, so a higher stack modeled as

658-730: A simplicial group should correspond to a "non-abelian" chain complex F ∙ {\displaystyle {\mathcal {F}}_{\bullet }} . Moreover, these should have an abelianization given by homology and cohomology, written suggestively as H k ( X , F ∙ ) {\displaystyle H^{k}(X,{\mathcal {F}}_{\bullet })} or R F ∗ ( F ∙ ) {\displaystyle \mathbf {R} F_{*}({\mathcal {F}}_{\bullet })} , since there should be an associated six functor formalism . Moreover, there should be an associated theory of Lefschetz operations, similar to

705-502: A space X {\displaystyle X} , one can just define the homology of X {\displaystyle X} to the homology of the CW approximation of X {\displaystyle X} (the cell structure of a CW complex determines the natural homology, the cellular homology and that can be taken to be the homology of the complex.) A map f : A → X {\displaystyle f:A\to X}

752-501: A spectrum. A K-theory is an example of a generalized cohomology theory. A basic example of a spectrum is a sphere spectrum : S 0 → S 1 → S 2 → ⋯ {\displaystyle S^{0}\to S^{1}\to S^{2}\to \cdots } See also: Characteristic class , Postnikov tower , Whitehead torsion There are several specific theories Homotopy hypothesis Too Many Requests If you report this error to

799-411: A sphere S n {\displaystyle S^{n}} has two cells: one 0-cell and one n {\displaystyle n} -cell, since S n {\displaystyle S^{n}} can be obtained by collapsing the boundary S n − 1 {\displaystyle S^{n-1}} of the n -disk to a point. In general, every manifold has

846-478: A structure was later called a coherator since it keeps track of all higher coherences. This structure has been formally studied by George Malsiniotis making some progress on setting up these foundations and showing the homotopy hypothesis . As a matter of fact, the description is formally analogous, and nearly identical, to the description of the homology groups of a chain complex – and it would seem therefore that that stacks (more specifically, Gr-stacks) are in

893-500: Is a CW pair ( X , A ) {\displaystyle (X,A)} ; many often work only with CW complexes and the notion of a cofibration there is then often implicit. A fibration in the sense of Hurewicz is the dual notion of a cofibration: that is, a map p : X → B {\displaystyle p:X\to B} is a fibration if given (1) a map h 0 : Z → X {\displaystyle h_{0}:Z\to X} and (2)

940-439: Is a neighborhood deformation retract ; that is, X {\displaystyle X} contains a mapping cylinder neighborhood of a closed subspace A {\displaystyle A} and f {\displaystyle f} the inclusion (e.g., a tubular neighborhood of a closed submanifold). In fact, a cofibration can be characterized as a neighborhood deformation retract pair. Another basic example

987-449: Is a geometric realization functor | ⋅ | : M → Spaces {\displaystyle |\cdot |:M\to {\text{Spaces}}} and a weak equivalence M [ W − 1 ] ≃ Hot {\displaystyle M[W^{-1}]\simeq {\text{Hot}}} where Hot denotes the homotopy category . Homotopy theory In mathematics , homotopy theory

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1034-447: Is a map satisfying the RLP for the inclusions i : S n − 1 → D n {\displaystyle i:S^{n-1}\to D^{n}} where S − 1 {\displaystyle S^{-1}} is the empty set. A Hurewicz fibration is a Serre fibration and the converse holds for CW complexes. On the other hand, a cofibration

1081-708: Is a space but a morphism is the homotopy class of a map. Then CW approximation  —  There exist a functor (called the CW approximation functor) from the homotopy category of spaces to the homotopy category of CW complexes as well as a natural transformation where i : Ho ⁡ ( CW ) ↪ Ho ⁡ ( spaces ) {\displaystyle i:\operatorname {Ho} ({\textrm {CW}})\hookrightarrow \operatorname {Ho} ({\textrm {spaces}})} , such that each θ X : i ( Θ ( X ) ) → X {\displaystyle \theta _{X}:i(\Theta (X))\to X}

1128-402: Is a space that has a filtration X ⊃ ⋯ ⊃ X n ⊃ X n − 1 ⊃ ⋯ ⊃ X 0 {\displaystyle X\supset \cdots \supset X^{n}\supset X^{n-1}\supset \cdots \supset X^{0}} whose union is X {\displaystyle X} and such that For example,

1175-571: Is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology , but nowadays is learned as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as: In homotopy theory and algebraic topology, the word "space" denotes a topological space . In order to avoid pathologies , one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated weak Hausdorff or

1222-453: Is a weak homotopy equivalence. Similar statements also hold for pairs and excisive triads. Explicitly, the above approximation functor can be defined as the composition of the singular chain functor S ∗ {\displaystyle S_{*}} followed by the geometric realization functor; see § Simplicial set . The above theorem justifies a common habit of working only with CW complexes. For example, given

1269-571: Is an isomorphism for each n ≥ 0 {\displaystyle n\geq 0} and each choice of a base point. A homotopy equivalence is a weak homotopy equivalence but the converse need not be true. Through the adjunction a homotopy h : X × I → Y {\displaystyle h:X\times I\to Y} is sometimes viewed as a map X → Y I = Map ⁡ ( I , Y ) {\displaystyle X\to Y^{I}=\operatorname {Map} (I,Y)} . A CW complex

1316-598: Is called a based homotopy . A based homotopy is the same as a (based) map X ∧ I + → Y {\displaystyle X\wedge I_{+}\to Y} where I + {\displaystyle I_{+}} is I {\displaystyle I} together with a disjoint basepoint. Given a pointed space X and an integer n ≥ 0 {\displaystyle n\geq 0} , let π n X = [ S n , X ] {\displaystyle \pi _{n}X=[S^{n},X]} be

1363-506: Is called a cofibration if given: such that h 0 ∘ f = g 0 {\displaystyle h_{0}\circ f=g_{0}} , there exists a homotopy h t : X → Z {\displaystyle h_{t}:X\to Z} that extends h 0 {\displaystyle h_{0}} and such that h t ∘ f = g t {\displaystyle h_{t}\circ f=g_{t}} . An example

1410-512: Is exactly a map satisfying the LLP for evaluation maps p : B I → B {\displaystyle p:B^{I}\to B} at 0 {\displaystyle 0} . On the category of pointed spaces, there are two important functors: the loop functor Ω {\displaystyle \Omega } and the (reduced) suspension functor Σ {\displaystyle \Sigma } , which are in

1457-457: Is given by p {\displaystyle p} . This s {\displaystyle s} is called the path lifting associated to p {\displaystyle p} . Conversely, if there is a path lifting s {\displaystyle s} , then p {\displaystyle p} is a fibration as a required homotopy is obtained via s {\displaystyle s} . A basic example of

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1504-421: Is said to satisfy the right lifting property or the RLP if p {\displaystyle p} satisfies the above lifting property for each i {\displaystyle i} in c {\displaystyle {\mathfrak {c}}} . Similarly, a map i : A → X {\displaystyle i:A\to X} is said to satisfy the left lifting property or

1551-442: Is said to satisfy the lifting property if for each commutative square diagram there is a map λ {\displaystyle \lambda } that makes the above diagram still commute. (The notion originates in the theory of model categories .) Let c {\displaystyle {\mathfrak {c}}} be a class of maps. Then a map p : E → B {\displaystyle p:E\to B}

1598-434: Is the smash product X ∧ Y {\displaystyle X\wedge Y} which is characterized by the adjoint relation that is, a smash product is an analog of a tensor product in abstract algebra (see tensor-hom adjunction ). Explicitly, X ∧ Y {\displaystyle X\wedge Y} is the quotient of X × Y {\displaystyle X\times Y} by

1645-586: Is the exact sequence where F f {\displaystyle Ff} is the homotopy fiber of f {\displaystyle f} ; i.e., a fiber obtained after replacing f {\displaystyle f} by a (based) fibration. The cofibration sequence generated by f {\displaystyle f} is X → Y → C f → Σ X → ⋯ , {\displaystyle X\to Y\to Cf\to \Sigma X\to \cdots ,} where C f {\displaystyle Cf}

1692-418: Is the homotooy cofiber of f {\displaystyle f} constructed like a homotopy fiber (use a quotient instead of a fiber.) The functors Ω , Σ {\displaystyle \Omega ,\Sigma } restrict to the category of CW complexes in the following weak sense: a theorem of Milnor says that if X {\displaystyle X} has the homotopy type of

1739-397: The adjoint relation . Precisely, they are defined as Because of the adjoint relation between a smash product and a mapping space, we have: These functors are used to construct fiber sequences and cofiber sequences . Namely, if f : X → Y {\displaystyle f:X\to Y} is a map, the fiber sequence generated by f {\displaystyle f}

1786-425: The mapping path space of p {\displaystyle p} . Viewing p ′ {\displaystyle p'} as a homotopy N p × I → B {\displaystyle Np\times I\to B} (see § Homotopy ), if p {\displaystyle p} is a fibration, then p ′ {\displaystyle p'} gives

1833-582: The wedge sum X ∨ Y {\displaystyle X\vee Y} . Let I denote the unit interval [ 0 , 1 ] {\displaystyle [0,1]} . A map is called a homotopy from the map h 0 {\displaystyle h_{0}} to the map h 1 {\displaystyle h_{1}} , where h t ( x ) = h ( x , t ) {\displaystyle h_{t}(x)=h(x,t)} . Intuitively, we may think of h {\displaystyle h} as

1880-491: The LLP if it satisfies the lifting property for each p {\displaystyle p} in c {\displaystyle {\mathfrak {c}}} . For example, a Hurewicz fibration is exactly a map p : E → B {\displaystyle p:E\to B} that satisfies the RLP for the inclusions i 0 : A → A × I {\displaystyle i_{0}:A\to A\times I} . A Serre fibration

1927-628: The homotopy classes of based maps S n → X {\displaystyle S^{n}\to X} from a (pointed) n -sphere S n {\displaystyle S^{n}} to X . As it turns out, Every group is the fundamental group of some space. A map f {\displaystyle f} is called a homotopy equivalence if there is another map g {\displaystyle g} such that f ∘ g {\displaystyle f\circ g} and g ∘ f {\displaystyle g\circ f} are both homotopic to

Pursuing Stacks - Misplaced Pages Continue

1974-407: The homotopy type of a CW complex; in fact, Morse theory implies that a compact manifold has the homotopy type of a finite CW complex. Remarkably, Whitehead's theorem says that for CW complexes, a weak homotopy equivalence and a homotopy equivalence are the same thing. Another important result is the approximation theorem. First, the homotopy category of spaces is the category where an object

2021-533: The identities. Two spaces are said to be homotopy equivalent if there is a homotopy equivalence between them. A homotopy equivalence class of spaces is then called a homotopy type . There is a weaker notion: a map f : X → Y {\displaystyle f:X\to Y} is said to be a weak homotopy equivalence if f ∗ : π n ( X ) → π n ( Y ) {\displaystyle f_{*}:\pi _{n}(X)\to \pi _{n}(Y)}

2068-414: The objects C 0 {\displaystyle C_{0}} and the inclusion functors C n → C n + 1 {\displaystyle C_{n}\to C_{n+1}} , where the categories C n {\displaystyle C_{n}} keep track of the higher homotopical information up to level n {\displaystyle n} . Such

2115-551: The situation for higher homotopies. It's conjectured this could be accomplished by looking at a successive sequence of categories and functors C 0 → C 1 → ⋯ → C n → C n + 1 → ⋯ {\displaystyle C_{0}\to C_{1}\to \cdots \to C_{n}\to C_{n+1}\to \cdots } that are universal with respect to any kind of higher groupoid. This allows for an inductive definition of an ∞-groupoid that depends on

2162-459: The thesis of Raynaud . Because Grothendieck envisioned an alternative formulation of higher stacks using globular groupoids, and observed there should be a corresponding theory using cubical sets , he came up with the idea of test categories and test functors. Essentially, test categories should be categories M {\displaystyle M} with a class of weak equivalences W {\displaystyle W} such that there

2209-548: The work are derivators and test categories . Some parts of the manuscript were later developed in: Pursuing stacks started out as a letter from Grothendieck to Daniel Quillen. In this letter he discusses Quillen's progress on the foundations for homotopy theory and remarked on the lack of progress since then. He remarks how some of his friends at Bangor university , including Ronald Brown , were studying higher fundamental groupoids Π n ( X ) {\displaystyle \Pi _{n}(X)} for

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