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

\( 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

p\( 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 \( A\leftarrow B\rightarrow C \) 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)

\( X_{0}\leftarrow X_{0}\times X_{1}\rightarrow X_{1} \)

is the join \( X_{0}*X_{1}. \)

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

\( X_{1}\to X_{2}\to \cdots , \)

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

\( X:I\to Spaces, \)

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,

\( \Delta_0: Spaces \to Spaces^I \)

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

\( \Delta :Spaces\to Spaces^{I} \)

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

\( X\times |N(I/i)| \)

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

\( {\mathrm {hocolim}}X_{i}\to {\mathrm {colim}}X_{i}. \)

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 \( X_{0}\leftarrow X_{0}\times X_{1}\rightarrow X_{1} \) , 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

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.