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| {{About|fibrations in algebraic topology|fibrations in category theory, as used in descent theory and categorical logic|Fibred category}}
| | == '単なる幸運 == |
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| In [[topology]], a branch of mathematics, a '''fibration''' is a generalization of the notion of a [[fiber bundle]]. A fiber bundle makes precise the idea of one [[topological space]] (called a fiber) being "parameterized" by another topological space (called a base). A fibration is like a fiber bundle, except that the fibers need not be the same space, rather they are just [[homotopy equivalent]]. Fibrations do not necessarily have the local [[Cartesian product]] structure that defines the more restricted fiber bundle case, but something weaker that still allows "sideways" movement from fiber to fiber. Fiber bundles have a particularly simple [[homotopy theory]] that allows topological information about the bundle to be inferred from information about one or both of these constituent spaces. A fibration satisfies an additional condition (the [[homotopy lifting property]]) guaranteeing that it will behave like a fiber bundle from the point of view of homotopy theory.
| | スピードが自然に異常テロとみなされるので、このハードルを破るタイム [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ 時計 価格]。<br>蘇銭を脇に<br>、目はビューの精錬ダン「医学」のポイントが可能な6製品から、またシャオヤン少し驚いて、彼はおそらくシャオヤンは、そのハードルを超えていると思いますが、今このパーティーがそう言うのを聞くことができます心、それはまだ驚異ですので、実際の速度、最初の人でそれを見ているので、多くの年のために彼を呼んだ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ 時計 メンズ]。<br><br>'単なる幸運 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html casio 腕時計 ゴールド]。」シャオヤンは、彼が自然に意図的に何かを隠すことはありませんここでは、微笑んで、すべての人を信頼することができます [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html カシオ 時計 電波 ソーラー]。<br><br>「ハハ、子供は、父親のビジョンは、非プールの事を考えると、あなたは最初から知っている、同じ良い仕事ではありませんでした」李さんKaihuaiが笑顔で、そのように見えますが、彼自身の王よりも戦いを分割することです興奮 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html カシオ gショック 腕時計]。<br>笑顔、すぐにすぐに色あせ、彼の父は李さん、悲しみのシャオヤンの目知覚できない通過ヒントを言及聞く<br> |
| | 相关的主题文章: |
| | <ul> |
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| | <li>[http://www.vibrantsantafe.com/cgi-bin/vibrant/user.cgi http://www.vibrantsantafe.com/cgi-bin/vibrant/user.cgi]</li> |
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| | <li>[http://chuizi.tv/home.php?mod=space&uid=23100 http://chuizi.tv/home.php?mod=space&uid=23100]</li> |
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| | <li>[http://bbs.thinkidea.net/forum.php?mod=viewthread&tid=676852 http://bbs.thinkidea.net/forum.php?mod=viewthread&tid=676852]</li> |
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| | </ul> |
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| == Formal definition == | | == それは、その強力な戦闘天皇杯怖いです == |
| A '''fibration''' (or '''Hurewicz fibration''') is a [[continuous function (topology)|continuous mapping]] {{math|''p'' : ''E'' → ''B''}} satisfying the [[homotopy lifting property]] with respect to any space. [[Fiber bundle]]s (over [[paracompact]] bases) constitute important examples. In [[homotopy theory]] any mapping is 'as good as' a fibration—i.e. any map can be decomposed as a homotopy equivalence into a "mapping path space" followed by a fibration. (See [[homotopy fiber]].)
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| The ''fibers'' are by definition the subspaces of {{mvar|E}} that are the inverse images of points {{mvar|b}} of {{mvar|B}}. If the base space {{mvar|B}} is path connected, it is a consequence of the definition that the fibers of two different points {{math|''b''<sub>1</sub>}} and {{math|''b''<sub>2</sub>}} in {{mvar|B}} are [[homotopy equivalence|homotopy equivalent]]. Therefore one usually speaks of "the fiber" {{mvar|F}}.
| | 魂が本当に晋王朝皇帝皇帝缶バケツレベルへ、そして彼の一人の男が、それはこれらの武将の世界と競争することができることがあれば、シャオヤンは彼らの背中に一つに見えたが、それはため息をため息で、数の強度ものの [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオの時計]。それは、その強力な戦闘天皇杯怖いです [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ 電波ソーラー時計]!<br><br>「すべてのもの、だけ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html カシオ 電波時計 腕時計]...ああ、私のベストを尽くすことができます」<br><br>群衆は単なる連合本部を戻ったが、それが不足している半月のキャンドルくんはまだ紫色の研究が立って、彼の後ろに、会場の外に立っていたことがわかる [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html 電波時計 casio]。<br><br>'そして、実際にそのことを考えて、この男は意味...'ショーが遠く行くのを見た、ろうそくくんも天皇の行為の魂にも同様に敏感明らかに、ため息をついた。<br><br>区元は黙ってうなずいて言った:「今日のカウント、ごく少数の人だけであれば大騒ぎを破ることができる空のよう、魂天国の夢が打ち砕かれている、多関節にしてみてください。 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオ電波ソーラー腕時計レディース] '<br><br>言葉は彼さえ、しかし、言って |
| | 相关的主题文章: |
| | <ul> |
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| | <li>[http://www.sdlrttl.cn/plus/view.php?aid=118633 http://www.sdlrttl.cn/plus/view.php?aid=118633]</li> |
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| | <li>[http://bbs.gameorange.cn/forum.php?mod=viewthread&tid=75160 http://bbs.gameorange.cn/forum.php?mod=viewthread&tid=75160]</li> |
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| | <li>[http://www.europeanwindstorms.org/cgi-bin/storms/storms.cgi http://www.europeanwindstorms.org/cgi-bin/storms/storms.cgi]</li> |
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| | </ul> |
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| == Serre fibrations == | | == シャオヤン勝利した場合、それらは結果にどのようなことでしょう == |
| A continuous mapping with the homotopy lifting property for [[CW complex]]es (or equivalently, just cubes {{math|''I''<sup>''n''</sup>}}) is called a ''Serre fibration'', in honor of the part played by the concept in the thesis of [[Jean-Pierre Serre]]. This thesis firmly established in [[algebraic topology]] the use of [[spectral sequence]]s, and clearly separated the notions of fiber bundles and fibrations from the notion of [[sheaf (mathematics)|sheaf]] (both concepts together having been implicit in the pioneer treatment of [[Jean Leray]]). Because a sheaf (thought of as an [[étalé space]]) can be considered a [[local homeomorphism]], the notions were closely interlinked at the time. One of the main desirable properties of the [[Serre spectral sequence]] is to account for the action of the [[fundamental group]] of the base {{mvar|B}} on the homology of the "total space" {{mvar|E}}.
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| == Examples ==
| | シャオヤン勝利した場合、それらは結果にどのようなことでしょう [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html gps 腕時計 カシオ]。<br><br>'ブーム [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html カシオ 腕時計 ソーラー 電波]!'<br>彼らの心の中<br>パニック [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ スタンダード 腕時計] '混沌'深淵、巨大な姿が再び空、少し恥ずかしいシャオヤンを見て。<br><br>'!魂天、あなたの血の皇帝本体は、それが炎症が天皇の体であってもよいことを私には良さそうです、「シャオヤンはゆっくりと、生体内で血液中の抑制をかき回す笑っている:'そして、あなたはの枯渇に見える? [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオ 腕時計 gps] '<br><br>ソウル天国の顔」色は [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-10.html casio電波腕時計] '非常に暗い、彼はin vivoでの状況を理解し、フロントグリップの後、彼は同じように驚くべき強さだったが、長い間息の観点から、ダン「医学」ので、おそらくそれを借りることができることを見出し、後者を戦うシャオヤン、天皇プラス本体よりもやや小さいことが判明し、別の火災相凝縮物をたくさん持って、専制的パワーが彼の体ディリでさえ血、ジョンソンは実際には少し '色は、計り知れないこれが続けば」、私は本当に、失うのが怖い |
| In the following examples a fibration is denoted
| | 相关的主题文章: |
| :{{bigmath|''F'' → ''E'' → ''B''}}, | | <ul> |
| where the first map is the inclusion of "the" fiber {{mvar|F}} into the total space {{mvar|E}} and the second map is the fibration onto the basis {{mvar|B}}. This is also referred to as a fibration sequence.
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| | | <li>[http://www.ebookabc.net/bbs/forum.php?mod=viewthread&tid=42702 http://www.ebookabc.net/bbs/forum.php?mod=viewthread&tid=42702]</li> |
| *The projection map from a product space is very easily seen to be a fibration.
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| *[[Fiber bundle]]s have ''local trivializations,'' i.e. Cartesian product structures exist [[locally]] on {{mvar|B}}, and this is usually enough to show that a fiber bundle is a fibration. More precisely, if there are local trivializations over a "numerable open cover" of {{mvar|B}}, the bundle is a fibration. Any open cover of a [[paracompact]] space is numerable. For example, any open cover of a metric space has a locally finite refinement, so any bundle over such a space is a fibration. The local triviality also implies the existence of a [[well-defined]] ''fiber'' ([[up to]] [[homeomorphism]]), at least on each [[connected space|connected component]] of {{mvar|B}}.
| | <li>[http://ppnnlife.com/forum.php?mod=viewthread&tid=28255&fromuid=19002 http://ppnnlife.com/forum.php?mod=viewthread&tid=28255&fromuid=19002]</li> |
| * The [[Hopf fibration]] {{math|''S''<sup>1</sup> → ''S''<sup>3</sup> → ''S''<sup>2</sup>}} was historically one of the earliest non-trivial examples of a fibration.
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| * Over [[complex projective space]], there is a fibration {{math|''S''<sup>1</sup> → ''S''<sup>2''n''+1</sup> → [[complex projective space|'''CP'''<sup>''n''</sup>]]}}.(Notice that the first example ; that of the Hopf fibration, is a special case of this fibration for n=1, since [[ complex projective space| '''CP'''<sup>''1''</sup>]] is homeomorphic to {{math| ''S'' <sup> 2</sup>}} )
| | <li>[http://www.cypp.cn/plus/feedback.php?aid=273 http://www.cypp.cn/plus/feedback.php?aid=273]</li> |
| * The Serre fibration {{math|SO(2) → SO(3) → ''S''<sup>2</sup>}} comes from the action of the [[rotation group SO(3)|rotation group {{math|SO(3)}}]] on the [[sphere|2-sphere]] {{math|''S''<sup>2</sup>}}.
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| * The previous example can also be generalized to a fibration {{math|SO(''n'') → SO(''n''+1) → ''S''<sup>''n''</sup>}} for any non-negative integer {{mvar|n}} (though they only have a fiber that isn't just a point when {{math|''n'' > 1}}) that comes from the action of the [[orthogonal group|special orthogonal group {{math|SO(''n''+1)}}]] on the {{mvar|n}}-sphere.
| | </ul> |
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| == Properties ==
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| ===Long exact sequence in homotopy groups===
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| Choose a base point {{math|''b''<sub>0</sub> ∈ ''B''}}. Let {{mvar|F}} refer to the fiber over {{math|''b''<sub>0</sub>}}, i.e. {{math|F {{=}} ''p''<sup>-1</sup>({''b''<sub>0</sub>})}}; and let {{mvar|i}} be the inclusion {{math|''F'' → ''E''}}. Choose a base point {{math|''f''<sub>0</sub> ∈ ''F''}} and let {{math|''e''<sub>0</sub> {{=}} ''i''(''f''<sub>0</sub>)}}. In terms of these base points, we have a [[long exact sequence]]
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| :<math>\cdots\to\pi_n(F)\to\pi_n(E)\to\pi_n(B)\to\pi_{n-1}(F)\to\cdots
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| </math>
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| constructed from the [[homotopy group]]s of the fiber {{mvar|F}}, total space {{mvar|E}}, and base space {{mvar|B}}. The homomorphisms {{math|''π''<sub>''n''</sub>(''F'') → ''π''<sub>''n''</sub>(''E'')}} and {{math|''π''<sub>''n''</sub>(''E'') → ''π''<sub>''n''</sub>(''B'')}} are just the induced homomorphisms from {{mvar|i}} and {{mvar|p}}, respectively.
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| The third set of homomorphisms {{math|''β''<sub>''n''</sub> : ''π''<sub>''n''</sub>(''B'') → ''π''<sub>''n''−1</sub>(''F'')}} (called the "connecting homomorphisms" (in reference to the [[snake lemma]]) or the "boundary maps") can be defined with the following steps.
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| # First, a little terminology: let {{math|''δ''<sub>''n''</sub> : ''S''<sup>''n''</sup> → ''D''<sup>''n''+1</sup>}} be the inclusion of the boundary [[n-sphere|{{mvar|n}}-sphere]] into the [[n-sphere#n-ball|{{math|(''n''+1)}}-ball]]. Let {{math|''γ''<sub>''n''</sub> : ''D''<sup>''n''</sup> → ''S''<sup>''n''</sup>}} be the map that collapses the image of {{math|''δ''<sub>''n''−1</sub>}} in {{math|''D''<sup>''n''</sup>}} to a point.
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| # Let {{math|''φ'' : ''S''<sup>''n''</sup> → ''B''}} be a representing map for an element of {{math|''π''<sub>''n''</sub>(''B'')}}.
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| # Because {{math|''D''<sup>''n''</sup>}} is homeomorphic to the {{mvar|n}}-dimensional cube, we can iteratively apply the homotopy lifting property to construct a lift {{math|''λ'' : ''D''<sup>''n''</sup> → ''E''}} of {{math|''φ'' ∘ ''γ''<sub>''n''</sub>}} (i.e., a map {{mvar|λ}} such that {{math|''p'' ∘ ''λ'' {{=}} ''φ'' ∘ ''γ''<sub>''n''</sub>}}).
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| # Because {{math|''γ''<sub>''n''</sub> ∘ ''δ''<sub>''n''−1</sub>}} is a point map (hereafter referred to as "{{math|{{=}} pt}}"), {{math|pt {{=}} ''φ'' ∘ ''γ''<sub>''n''</sub> ∘ ''δ''<sub>''n''−1</sub> {{=}} ''p'' ∘ ''λ'' ∘ ''δ''<sub>''n''−1</sub>}}, which implies that the image of {{math|''λ'' ∘ ''δ''<sub>''n''−1</sub>}} is in {{mvar|F}}. Therefore, there exists a map {{math|''ψ'' : ''S''<sup>''n''−1</sup> → ''F''}} such that {{math|''i'' ∘ ''ψ'' {{=}} ''λ'' ∘ ''δ''<sub>''n''−1</sub>}}.
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| # We define {{math|''β''<sub>''n''</sub> [''φ''] {{=}} [''ψ'']}}.
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| The above is summarized in the following [[commutative diagram]]:
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| :[[File:Fibration homotopy groups LES connecting morphism diagram.svg|300px]]
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| Repeated application of the homotopy lifting property is used to prove that {{math|''β''<sub>''n''</sub>}} is a well-defined homomorphism and that this sequence is exact.
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| === Euler characteristic ===
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| {{Main|Euler characteristic}}
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| The [[Euler characteristic]] {{mvar|χ}} is multiplicative for [[fibrations]] with certain conditions.
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| If {{math|''p'' : ''E'' → ''B''}} is a fibration with fiber {{mvar|F}}, with the base {{mvar|B}} [[path-connected]], and the fibration is orientable over a field {{mvar|K}}, then the Euler characteristic with coefficients in the field {{mvar|K}} satisfies the product property:<ref>{{citation
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| |title=Algebraic Topology
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| |first=Edwin Henry
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| |last=Spanier
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| |authorlink=Edwin Spanier
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| |publisher=Springer
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| |year=1982
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| |isbn=978-0-387-94426-5
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| |url=http://books.google.com/?id=h-wc3TnZMCcC
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| }}, [http://books.google.com/books?id=h-wc3TnZMCcC&pg=PA481 Applications of the homology spectral sequence, p. 481]</ref>
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| :{{bigmath|''χ''(''E'') {{=}} ''χ''(''F'') · ''χ''(''B'')}}.
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| This includes product spaces and covering spaces as special cases,
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| and can be proven by the [[Serre spectral sequence]] on homology of a fibration.
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| For fiber bundles, this can also be understood in terms of a [[transfer map]] {{math|''τ'' : ''H''<sub>∗</sub>(''B'') → ''H''<sub>∗</sub>(''E'')}}—note that this is a lifting and goes "the wrong way"—whose composition with the projection map {{math|''p''<sub>∗</sub> : ''H''<sub>∗</sub>(''E'') → ''H''<sub>∗</sub>(''B'')}} is multiplication by the Euler characteristic of the fiber:<ref>{{citation
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| |title=Fibre bundles and the Euler characteristic
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| |first=Daniel Henry
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| |last=Gottlieb
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| |journal=Journal of Differential Geometry
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| |volume=10
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| |year=1975
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| |pages=39–48
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| |url=http://www.math.purdue.edu/~gottlieb/Bibliography/17FibreBundlesAndtheEulerCharacteristic.pdf
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| |issue=1
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| }}</ref>
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| {{math|''p''<sub>∗</sub> ∘ ''τ'' {{=}} ''χ''(''F'') · 1}}.
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| ==Fibrations in closed model categories==
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| Fibrations of topological spaces fit into a more general framework, the so-called [[closed model category|closed model categories]]. In such categories, there are distinguished classes of morphisms, the so-called ''fibrations'', ''[[cofibration]]s'' and ''[[weak equivalence (homotopy theory)|weak equivalence]]s''. Certain [[axiom]]s, such as stability of fibrations under composition and [[pullback (category theory)|pullbacks]], factorization of every morphism into the composition of an acyclic cofibration followed by a fibration or a cofibration followed by an acyclic fibration, where the word "acyclic" indicates that the corresponding arrow is also a weak equivalence, and other requirements are set up to allow the abstract treatment of homotopy theory. (In the original treatment, due to [[Daniel Quillen]], the word "trivial" was used instead of "acyclic.")
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| It can be shown that the category of topological spaces is in fact a model category, where (abstract) fibrations are just the Serre fibrations introduced above and weak equivalences are weak [[homotopy equivalence]]s.<ref>{{citation
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| |title=Handbook of algebraic topology
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| |first1=William G.
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| |last1=Dwyer
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| |first2=J.
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| |last2=Spaliński
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| |authorlink=William Dwyer
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| |publisher=North-Holland
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| |location=Amsterdam
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| |year=1995
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| |chapter=Homotopy theories and model categories
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| |pages=73–126
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| |mr=1361887
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| |url=http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Spalinski/theories
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| }}</ref>
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| ==See also==
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| * [[Homotopy fiber]]
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| == References ==
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| <references/> | |
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| [[Category:Algebraic topology]]
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| [[Category:Homotopy theory]]
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| [[Category:Differential topology]]
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| [[Category:Category theory]]
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