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| In [[mathematics]], especially in [[algebraic topology]], the '''homotopy limit and colimit''' are variants of the notions of [[limit (category theory)|limit]] and colimit. They are denoted by holim and hocolim, respectively.
| | Let me inroduce myself, my name is Alta although it isn't the name on my birth document. Invoicing has been my profession for your time but soon my husband and Let me start some of our business. As a man what he really loves is to climb and he'll be starting something else along making use of. Illinois has been her space. See what's new on his website here: http://devolro.com/armoring<br><br>Here is my weblog; [http://devolro.com/armoring door] |
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| ==Introductory examples==
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| ===Homotopy pushout===
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| The concept of homotopy colimit is a generalization of ''homotopy pushouts''. This notion is motivated by the following observation: the (ordinary) [[pushout (category theory)|pushout]]
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| :<math>D^n \sqcup_{S^{n-1}} pt</math>
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| is the space obtained by contracting the ''n''-1-sphere (which is the boundary of the ''n''-dimensional disk) to a single point. This space is [[homeomorphic]] to the ''n''-sphere S<sup>''n''</sup>. On the other hand, the pushout
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| :<math>pt \sqcup_{S^{n-1}} pt</math>
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| is a point. Therefore, even though the ([[contractible space|contractible]]) disk ''D''<sup>''n''</sup> was replaced by a point, (which is homotopy equivalent to the disk), the two pushouts are ''not'' [[homotopy equivalence|homotopy]] (or [[weak equivalence|weakly]]) equivalent.
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| Therefore, the pushout is not well-aligned with a principle of homotopy theory, which considers weakly equivalent spaces as carrying the same information: if one (or more) of the spaces used to form the pushout is replaced by a weakly equivalent space, the pushout is not guaranteed to stay weakly equivalent. The homotopy pushout does ''not'' share this defect.
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| The ''homotopy pushout'' of two maps <math>A \leftarrow B \rightarrow C</math> of topological spaces is defined as
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| :<math>A \sqcup B \times [0,1] \sqcup B \sqcup B \times [0,1] \sqcup_0 C</math>,
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| i.e., instead of glueing ''B'' in both ''A'' and ''C'', two copies of a [[cylinder]] on ''B'' are glued together and their ends are glued to ''A'' and ''C''.
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| For example, the homotopy colimit of the diagram (whose maps are projections)
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| :<math>X_0 \leftarrow X_0 \times X_1 \rightarrow X_1</math>
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| is the [[join (topology)|join]] <math>X_0 * X_1</math>.
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| It can be shown that the homotopy pushout does not share the defect of the ordinary pushout: replacing ''A'', ''B'' and / or ''C'' by a homotopic space, the homotopy pushout ''will'' also be homotopic. In this sense, the homotopy pushouts treats homotopic spaces as well as the (ordinary) pushout does with homeomorphic spaces.
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| ===Mapping telescope===
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| The homotopy colimit of a sequence of spaces
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| :<math>X_1 \to X_2 \to \cdots,</math>
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| is the [[mapping telescope]].<ref>Hatcher's Algebraic Topology, 4.G.</ref><!-- For example, the colimit of a sequence of [[cofibration]]s is a homotopy colimit. -->
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| ==General definition==
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| Treating examples such as the mapping telescope and the homotopy pushout on an equal footing can be achieved by considering an ''I''-diagram of spaces, where ''I'' is some "indexing" category. This is nothing but a [[functor]]
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| :<math>X: I \to Spaces,</math> | |
| i.e., to each object i in I, one assigns a space ''X''<sub>''i''</sub> and maps between them, according to the maps in ''i''. The category of such diagrams is denoted ''Spaces''<sup>''I''</sup>.
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| There is a natural functor called the diagonal,
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| :<math>\Delta: Spaces \to Spaces^I</math>
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| which sends any space ''X'' to the diagram which consists of ''X'' everywhere (and the identity of ''X'' as maps between them). In (ordinary) category theory, the [[right adjoint]] to this functor is the [[limit (category theory)|limit]]. The homotopy limit is defined by altering this situation: it is the right adjoint to
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| :<math>\Delta: Spaces \to Spaces^I</math>
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| which sends a space ''X'' to the ''I''-diagram which at some object ''i'' gives
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| :<math>X \times |N(I / i)|</math>
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| Here ''I'' / ''i'' is the [[over-category]] (its objects are arrows ''j'' → ''i'', where ''j'' is any object of ''I''), ''N'' is the [[nerve (category theory)|nerve]] of this category and |-| is the topological realization of this simplicial set.<ref>Bousfield & Kan: ''Homotopy limits, Completions and Localizations'', Springer, LNM 304. Section XI.3.3</ref> | |
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| ==Relation to the (ordinary) colimit and limit==
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| There is always a map
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| :<math>hocolim X_i \to colim X_i.</math>
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| Typically, this map is ''not'' a weak equivalence. For example, the homotopy pushout encountered above always maps to the ordinary pushout. This map is not typically a weak equivalence, for example the join is not weakly equivalent to the pushout of <math>X_0 \leftarrow X_0 \times X_1 \rightarrow X_1</math>, which is a point.
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| ==Further examples and applications==
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| Just as limit is used to [[Completion (ring theory)|complete]] a ring, holim is used to [[completion of a spectrum|complete a spectrum]].
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| == References ==
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| <references />
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| *Hatcher, ''Algebraic Topology''
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| == Further reading ==
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| *http://mathoverflow.net/questions/135462/homotopy-limit-colimit-diagrams-in-stable-model-categories
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| [[Category:Homotopy theory]]
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Let me inroduce myself, my name is Alta although it isn't the name on my birth document. Invoicing has been my profession for your time but soon my husband and Let me start some of our business. As a man what he really loves is to climb and he'll be starting something else along making use of. Illinois has been her space. See what's new on his website here: http://devolro.com/armoring
Here is my weblog; door