In mathematics, especially in algebraic topology, the homotopy limit and colimit[1]pg 52 are variants of the notions of limit and colimit extended to the homotopy category
considered as an object in the homotopy category of diagrams
, (where the homotopy equivalence of diagrams is considered pointwise), then the homotopy limit and colimits then correspond to the cone and cocone
and a natural transformation corresponds to a continuous function of topological spaces.
Another perspective formalizing these kinds of constructions are derivators[2]pg 193 which are a new framework for homotopical algebra.
The concept of homotopy colimit[1]pg 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.
Therefore, even though the (contractible) disk Dn 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.
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
There are a number of ways to model this colimit: the first is to consider the space
which 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
The homotopy colimit of a sequence of spaces is the mapping telescope.
[3] One example computation is taking the homotopy colimit of a sequence of cofibrations.
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.
There is a natural functor called the diagonal, 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.
The homotopy limit is defined by altering this situation: it is the right adjoint to which sends a space X to the I-diagram which at some object i gives Here I/i is the slice category (its objects are arrows j → i, where j is any object of I), N is the nerve of this category and |-| is the topological realization of this simplicial set.
[4] Similarly, one can define a colimit as the left adjoint to the diagonal functor Δ0 given above.
To define a homotopy colimit, we must modify Δ0 in a different way.
A homotopy colimit can be defined as the left adjoint to a functor Δ : Spaces → SpacesI where where Iop is the opposite category of I.
Although this is not the same as the functor Δ above, it does share the property that if the geometric realization of the nerve category (|N(-)|) is replaced with a point space, we recover the original functor Δ0.
The universal property of a homotopy pullback yields the natural map
, we can construct the homotopy colimit using a simplicial replacement of the diagram.
given by chains of composable maps in the indexing category
can be constructed as the geometric realization of this simplicial space, so
Notice that this agrees with the picture given above for the composition diagram of