# 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

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 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 .

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.

## 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

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,

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.[2]

### Homotopy colimit

Similarly, one can define a colimit as the *left* adjoint to the diagonal functor Δ_{0} given above. To define a homotopy colimit, we must modify Δ_{0} 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 Δ_{0}.

## 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 , 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

- Hatcher's Algebraic Topology, 4.G.
- 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.