Homotopy colimit and limit

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

${\displaystyle D^{n}\sqcup _{S^{n-1}}pt}$

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 Sⁿ. On the other hand, the pushout

${\displaystyle pt\sqcup _{S^{n-1}}pt}$

is a 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. The homotopy pushout rectifies this defect.

The homotopy pushout of two maps ${\displaystyle A\leftarrow B\rightarrow C}$ of topological spaces is defined as

${\displaystyle A\sqcup _{1}B\times [0,1]\sqcup _{0}B\sqcup _{1}B\times [0,1]\sqcup _{0}C}$,

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)

${\displaystyle X_{0}\leftarrow X_{0}\times X_{1}\rightarrow X_{1}}$

is the join ${\displaystyle X_{0}*X_{1}}$.

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.

Another useful and motivating examples of a homotopy colimit is constructing models for the homotopy colimit of the diagram

${\displaystyle A\xrightarrow {f} X\xrightarrow {g} Y}$

of topological spaces. There are a number of ways to model this colimit: the first is to consider the space

${\displaystyle \left[(A\times I)\coprod (X\times I)\coprod Y\right]/\sim }$

where ${\displaystyle \sim }$ is the equivalence relation identifying

${\displaystyle {\begin{aligned}(a,1)&\sim (f(a),0)\\(x,1)&\sim g(x)\end{aligned}}}$

which can pictorially be described as the picture

Because we can similarly interpret the diagram above as the commutative diagram, from properties of categories, we get a commutative diagram

giving a homotopy colimit. We could guess this looks like

but 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 as

giving another model of the homotopy colimit which is homotopy equivalent to the original diagram (without the composition of ${\displaystyle g\circ f}$) given above.

The homotopy colimit of a sequence of spaces

${\displaystyle X_{1}\to X_{2}\to \cdots ,}$

is the mapping telescope.[3] One example computation is taking the homotopy colimit of a sequence of cofibrations. The colimit of [1]^{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.

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 a functor

${\displaystyle X:I\to Spaces,}$

i.e., to each object i in I, one assigns a space X_i and maps between them, according to the maps in I. The category of such diagrams is denoted Spaces^I.

There is a natural functor called the diagonal,

${\displaystyle \Delta _{0}:Spaces\to Spaces^{I}}$

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

${\displaystyle \Delta :Spaces\to Spaces^{I}}$

which sends a space X to the I-diagram which at some object i gives

${\displaystyle X\times |N(I/i)|}$

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 → Spaces^I where

Δ(X)(i) = Hom_Spaces (|N(I^op /i)|, X),

where I^op 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.