Homotopy colimit and limit
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. The main idea is this: if we have a diagramconsidered as an object in the category of diagrams">Category (mathematics)">category of diagrams,, then the homotopy limit and colimits then correspond to the cone and coconewhich are objects in the homotopy category, where is the category with one object and one morphism. Note this category is equivalent to the standard homotopy category since the latter homotopy functor category has functors which picks out an object in 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 pushoutis the space obtained by contracting the -sphere to a single point. This space is homeomorphic to the n-sphere Sn. On the other hand, the pushout
is a point. Therefore, even though the disk Dn was replaced by a point,, the two pushouts are not homotopy 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 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 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
is the join.
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 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 diagramof topological spaces. There are a number of ways to model this colimit: the first is to consider the spacewhere is the equivalence relation identifyingwhich 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 given above.Mapping telescope
The homotopy colimit of a sequence of spacesis 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 functori.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. In 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, 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 wherewhere 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 is the dual concept of a homotopy pushout. Concretely, given and, it can be constructed asFor example, the homotopy fiber of over a point y is the homotopy pullback of along. The homotopy pullback of along the identity is nothing but the mapping path space of.
The universal property of a homotopy pullback yields the natural map, 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 and a diagram, we can construct the homotopy colimit using a simplicial replacement of the diagram. This is a simplicial space, given by the diagrampg 16-17wheregiven by chains of composable maps in the indexing category. Then, the homotopy colimit of can be constructed as the geometric realization of this simplicial space, soNotice that this agrees with the picture given above for the composition diagram of.
Relation to the (ordinary) colimit and limit
There is always a mapTypically, 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, which is a point.