# Homotopy colimit

In mathematics, especially in algebraic topology, the homotopy limit and colimit are variants of the notions of limit and colimit. They are denoted by holim and hocolim, respectively.

## Introductory examples

### Homotopy pushout

The concept of homotopy colimit 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.

### Mapping telescope

The homotopy colimit of a sequence of spaces

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

is the mapping telescope.[1]

## 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 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 Xi and maps between them, according to the maps in I. The category of such diagrams is denoted SpacesI.

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.[2]

### Homotopy colimit

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

Δ(X)(i) = HomSpaces (|N(Iop /i)|, X),

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.

## Relation to the (ordinary) colimit and limit

There is always a map

${\displaystyle \mathrm {hocolim} X_{i}\to \mathrm {colim} X_{i}.}$

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 ${\displaystyle X_{0}\leftarrow X_{0}\times X_{1}\rightarrow X_{1}}$, which is a point.

## Further examples and applications

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

## References

1. Hatcher's Algebraic Topology, 4.G.
2. Bousfield & Kan: Homotopy limits, Completions and Localizations, Springer, LNM 304. Section XI.3.3
• Hatcher, Allen (2002), Algebraic Topology, Cambridge: Cambridge University Press, ISBN 0-521-79540-0.