Homotopy colimit

In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category formula_1. The main idea is this: if we have a diagramformula_2considered as an object in the homotopy category of diagrams formula_3, (where the homotopy equivalence of diagrams is considered pointwise), then the homotopy limit and colimits then correspond to the cone and coconeformula_4which are objects in the homotopy category formula_5, where formula_6 is the category with one object and one morphism. Note this category is equivalent to the standard homotopy category formula_1 since the latter homotopy functor category has functors which picks out an object in formula_8 and a natural transformation corresponds to a continuous function of topological spaces. Note this construction can be generalized to model categories, which give techniques for constructing homotopy limits and colimits in terms of other homotopy categories, such as derived categories. Another perspective formalizing these kinds of constructions are derivatorspg 193 which are a new framework for homotopical algebra.
Introductory examples.
Homotopy pushout.
The concept of homotopy colimitpg 4-8 is a generalization of "homotopy pushouts", such as the mapping cylinder used to define a cofibration. This notion is motivated by the following observation: the (ordinary) pushout 
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"n". On the other hand, the pushout
is a point. Therefore, even though the (contractible) disk "D""n" was replaced by a point, (which is homotopy equivalent to the disk), the two pushouts are "not" homotopy (or weakly) equivalent.
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 rectifies this defect.
The "homotopy pushout" of two maps formula_11 of topological spaces is defined as
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". 
For example, the homotopy colimit of the diagram (whose maps are projections)
is the join formula_14.
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.
Composition of maps.
Another useful and motivating examples of a homotopy colimit is constructing models for the homotopy colimit of the diagramformula_15of topological spaces. There are a number of ways to model this colimit: the first is to consider the spaceformula_16where formula_17 is the equivalence relation identifyingformula_18which can pictorially be described as the pictureBecause we can similarly interpret the diagram above as the commutative diagram, from properties of categories, we get a commutative diagramgiving a homotopy colimit. We could guess this looks likebut notice we have introduced a new cycle to fill in the new data of the composition. This creates a technical problem which can be solved using simplicial techniques: giving a method for constructing a model for homotopy colimits. The new diagram, forming the homotopy colimit of the composition diagram pictorially is represented asgiving another model of the homotopy colimit which is homotopy equivalent to the original diagram (without the composition of formula_19) given above.
Mapping telescope.
The homotopy colimit of a sequence of spaces 
is the mapping telescope. 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
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,
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 
which sends a space to the -diagram which at some object gives 
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 .
Construction of colimits with simplicial replacements.
Given a small category formula_25 and a diagram formula_26, we can construct the homotopy colimit using a simplicial replacement of the diagram. This is a simplicial space, formula_27 given by the diagrampg 16-17 whereformula_28given by chains of composable maps in the indexing category formula_25. Then, the homotopy colimit of formula_30 can be constructed as the geometric realization of this simplicial space, soformula_31Notice that this agrees with the picture given above for the composition diagram of formula_32.
Relation to the (ordinary) colimit and limit.
There is always a map 
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 formula_13, which is a point.
Further examples and applications.
Just as limit is used to complete a ring, holim is used to complete a spectrum.