Homotopy colimit and limit

Homotopy pushout The concept of homotopy colimit One example computation is taking the homotopy colimit of a sequence of cofibrations. The colimit of '''pg 62 this diagram gives a homotopy colimit. This implies we could compute the homotopy colimit of any mapping telescope by replacing the maps with cofibrations. ==General definition==

Homotopy limit Treating examples such as the mapping telescope and the homotopy pushout on an equal footing can be achieved by considering an -diagram of spaces, where is some "indexing" category. This is a functor :X: I \to Spaces, i.e., to each object in , one assigns a space and maps between them, according to the maps in . The category of such diagrams is denoted . There is a natural functor called the diagonal, :\Delta_0: Spaces \to Spaces^I which sends any space to the diagram which consists of everywhere (and the identity of as maps between them). In (ordinary) category theory, the right adjoint to this functor is the limit. The homotopy limit is defined by altering this situation: it is the right adjoint to :\Delta: Spaces \to Spaces^I which sends a space to the -diagram which at some object gives :X \times |N(I / i)| Here is the slice category (its objects are arrows , where is any object of ), is the nerve of this category and |-| is the topological realization of this simplicial set. Homotopy colimit Similarly, one can define a colimit as the left adjoint to the diagonal functor given above. To define a homotopy colimit, we must modify in a different way. A homotopy colimit can be defined as the left adjoint to a functor where :, where is the opposite category of . Although this is not the same as the functor above, it does share the property that if the geometric realization of the nerve category () is replaced with a point space, we recover the original functor . == Examples ==

A homotopy pullback (or homotopy fiber-product) is the dual concept of a homotopy pushout. Concretely, given f : X \to Z and g : Y \to Z, it can be constructed as :X \times^h_Z Y := X \times_Z Z^I \times_Z Y = \{ (x, \gamma, y) | f(x) = \gamma(0), g(y) = \gamma(1) \}. For example, the homotopy fiber of f : X \to Y over a point y is the homotopy pullback of f along y \hookrightarrow Y. The homotopy pullback of f along the identity is nothing but the mapping path space of f. The universal property of a homotopy pullback yields the natural map X \times_Z Y \to X \times^h_Z Y, a special case of a natural map from a limit to a homotopy limit. In the case of a homotopy fiber, this map is an inclusion of a fiber to a homotopy fiber. == Construction of colimits with simplicial replacements ==

Given a small category I and a diagram D:I \to \textbf{Top}, we can construct the homotopy colimit using a simplicial replacement of the diagram. This is a simplicial space, \text{srep}(D)_\bullet given by the diagram''''''pg 16-17 where\text{srep}(D)_n = \underset{i_0 \leftarrow i_1 \leftarrow \cdots \leftarrow i_n}{\coprod}D(i_n)given by chains of composable maps in the indexing category I. Then, the homotopy colimit of D can be constructed as the geometric realization of this simplicial space, so\underset{\to}{\text{hocolim}}D = |\text{srep}(D)_\bullet|Notice that this agrees with the picture given above for the composition diagram of A \to X \to Y. ==Relation to the (ordinary) colimit and limit==

There is always a map :\mathrm{hocolim} X_i \to \mathrm{colim} X_i. 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 X_0 \leftarrow X_0 \times X_1 \rightarrow X_1, which is a point. ==Further examples and applications==

Just as limit is used to complete a ring, holim is used to complete a spectrum. == See also ==