Homotopy colimit and limit

><h2 id="introductory-examples">Introductory examples</h2>
<h3>Homotopy pushout</h3>
The concept of homotopy colimit<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>pg 4-8 is a generalization of homotopy pushouts, such as the <a href="/facts/Mapping_cylinder/kB1yUpLb">mapping cylinder</a> used to define a <a href="/facts/Cofibration/NCWE6Cbt">cofibration</a>. This notion is motivated by the following observation: the (ordinary) <a href="/facts/Pushout_(category_theory)/6wuns0Yo">pushout</a>

D
          
            n
          
        
        
          ⊔
          
            
              S
              
                n
                −
                1
              
            
          
        
        p
        t
      
    
    {\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 <a href="/facts/Homeomorphic/jMfWg3Mn">homeomorphic</a> to the n-sphere Sn. On the other hand, the pushout

p
        t
        
          ⊔
          
            
              S
              
                n
                −
                1
              
            
          
        
        p
        t
      
    
    {\displaystyle pt\sqcup _{S^{n-1}}pt}

is a point. Therefore, even though the (<a href="/facts/Contractible_space/REXYj8uS">contractible</a>) disk Dn was replaced by a point, (which is homotopy equivalent to the disk), the two pushouts are not <a href="/facts/Homotopy_equivalence/1nWrPxdj">homotopy</a> (or <a href="/facts/Weak_equivalence_(homotopy_theory)/KUWfV7rc">weakly</a>) 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
 ←
 B
 →
 C
 
 
 {\displaystyle A\leftarrow B\rightarrow C}
 
 of topological spaces is defined as

A
 
 ⊔
 
 1
 
 
 B
 ×
 [
 0
 ,
 1
 ]
 
 ⊔
 
 0
 
 
 B
 
 ⊔
 
 1
 
 
 B
 ×
 [
 0
 ,
 1
 ]
 
 ⊔
 
 0
 
 
 C
 
 
 {\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 <a href="/facts/Cylinder_(geometry)/0wQPPr3k">cylinder</a> 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
          
        
        ←
        
          X
          
            0
          
        
        ×
        
          X
          
            1
          
        
        →
        
          X
          
            1
          
        
      
    
    {\displaystyle X_{0}\leftarrow X_{0}\times X_{1}\rightarrow X_{1}}

is the <a href="/facts/Join_(topology)/81UQka6h">join</a> 
 
 
 
 
 X
 
 0
 
 
 ∗
 
 X
 
 1
 
 
 
 
 {\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.

<h3>Composition of maps</h3>
Another useful and motivating examples of a homotopy colimit is constructing models for the homotopy colimit of the diagram<blockquote>
 
 
 
 A
 
 →
 
 f
 
 
 X
 
 →
 
 g
 
 
 Y
 
 
 {\displaystyle A\xrightarrow {f} X\xrightarrow {g} Y}
 
</blockquote>of topological spaces. There are a number of ways to model this colimit: the first is to consider the space<blockquote>
 
 
 
 
 [
 
 (
 A
 ×
 I
 )
 ∐
 (
 X
 ×
 I
 )
 ∐
 Y
 
 ]
 
 
 /
 
 ∼
 
 
 {\displaystyle \left[(A\times I)\coprod (X\times I)\coprod Y\right]/\sim }
 
</blockquote>where 
 
 
 
 ∼
 
 
 {\displaystyle \sim }
 
 is the <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a> identifying<blockquote>
 
 
 
 
 
 
 
 (
 a
 ,
 1
 )
 
 
 
 ∼
 (
 f
 (
 a
 )
 ,
 0
 )
 
 
 
 
 (
 x
 ,
 1
 )
 
 
 
 ∼
 g
 (
 x
 )
 
 
 
 
 
 
 {\displaystyle {\begin{aligned}(a,1)&\sim (f(a),0)\\(x,1)&\sim g(x)\end{aligned}}}
 
</blockquote>which can pictorially be described as the picture<blockquote></blockquote>Because we can similarly interpret the diagram above as the <a href="/facts/Commutative_diagram/bdG8xuer">commutative diagram</a>, from properties of categories, we get a commutative diagram<blockquote></blockquote>giving a homotopy colimit. We could guess this looks like<blockquote></blockquote>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<blockquote></blockquote>giving another model of the homotopy colimit which is homotopy equivalent to the original diagram (without the composition of 
 
 
 
 g
 ∘
 f
 
 
 {\displaystyle g\circ f}
 
) given above.
<h3>Mapping telescope</h3>
The homotopy colimit of a sequence of spaces

X
          
            1
          
        
        →
        
          X
          
            2
          
        
        →
        ⋯
        ,
      
    
    {\displaystyle X_{1}\to X_{2}\to \cdots ,}

is the <a href="/facts/Mapping_telescope/kB1yUpLb">mapping telescope</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> One example computation is taking the homotopy colimit of a sequence of <a href="/facts/Cofibration/NCWE6Cbt">cofibrations</a>. The colimit of <a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>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.

<h2 id="general-definition">General definition</h2>
<h3>Homotopy limit</h3>
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" <a href="/facts/Category_(mathematics)/7WzxUThf">category</a>. This is a <a href="/facts/Functor/l2u3rIBE">functor</a>

X
        :
        I
        →
        S
        p
        a
        c
        e
        s
        ,
      
    
    {\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,

Δ
          
            0
          
        
        :
        S
        p
        a
        c
        e
        s
        →
        S
        p
        a
        c
        e
        
          s
          
            I
          
        
      
    
    {\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 <a href="/facts/Right_adjoint/bJ1P7AN2">right adjoint</a> to this functor is the <a href="/facts/Limit_(category_theory)/JoLugLs3">limit</a>. The homotopy limit is defined by altering this situation: it is the right adjoint to

Δ
        :
        S
        p
        a
        c
        e
        s
        →
        S
        p
        a
        c
        e
        
          s
          
            I
          
        
      
    
    {\displaystyle \Delta :Spaces\to Spaces^{I}}

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

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

Here I/i is the <a href="/facts/Slice_category/wTa3tURp">slice category</a> (its objects are arrows j → i, where j is any object of I), N is the <a href="/facts/Nerve_(category_theory)/VceKkFGP">nerve</a> of this category and |-| is the topological realization of this <a href="/facts/Simplicial_set/6DaqXMLT">simplicial set</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h3>Homotopy colimit</h3>
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 <a href="/facts/Opposite_category/swDy8NrJ">opposite category</a> 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.

<h2 id="examples">Examples</h2>
A homotopy pullback (or homotopy fiber-product) is the dual concept of a homotopy pushout.It satisfies the universal property of a pullback up to homotopy. Concretely, given 
 
 
 
 f
 :
 X
 →
 Z
 
 
 {\displaystyle f:X\to Z}
 
 and 
 
 
 
 g
 :
 Y
 →
 Z
 
 
 {\displaystyle g:Y\to Z}
 
, it can be constructed as

X
 
 ×
 
 Z
 
 
 h
 
 
 Y
 :=
 X
 
 ×
 
 Z
 
 
 
 Z
 
 I
 
 
 
 ×
 
 Z
 
 
 Y
 =
 {
 (
 x
 ,
 γ
 ,
 y
 )
 
 |
 
 f
 (
 x
 )
 =
 γ
 (
 0
 )
 ,
 g
 (
 y
 )
 =
 γ
 (
 1
 )
 }
 .
 
 
 {\displaystyle X\times _{Z}^{h}Y:=X\times _{Z}Z^{I}\times _{Z}Y=\{(x,\gamma ,y)|f(x)=\gamma (0),g(y)=\gamma (1)\}.}
 
<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
For example, the <a href="/facts/Homotopy_fiber/hH4omFFQ">homotopy fiber</a> of 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 over a point y is the homotopy pullback of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 along 
 
 
 
 y
 ↪
 Y
 
 
 {\displaystyle y\hookrightarrow Y}
 
.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> The homotopy pullback of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 along the identity is nothing but the <a href="/facts/Mapping_path_space/7JFfGWQx">mapping path space</a> of 
 
 
 
 f
 
 
 {\displaystyle f}
 
.
The universal property of a homotopy pullback yields the natural map 
 
 
 
 X
 
 ×
 
 Z
 
 
 Y
 →
 X
 
 ×
 
 Z
 
 
 h
 
 
 Y
 
 
 {\displaystyle X\times _{Z}Y\to X\times _{Z}^{h}Y}
 
, a special case of a natural map from a limit to a homotopy limit. In the case of a homotopy fiber, this map is an inclusion of a fiber to a homotopy fiber.

<h2 id="construction-of-colimits-with-simplicial-replacements">Construction of colimits with simplicial replacements</h2>
Given a small category 
 
 
 
 I
 
 
 {\displaystyle I}
 
 and a diagram 
 
 
 
 D
 :
 I
 →
 
 
 Top
 
 
 
 
 {\displaystyle D:I\to {\textbf {Top}}}
 
, we can construct the homotopy colimit using a simplicial replacement of the diagram. This is a simplicial space, 
 
 
 
 
 srep
 
 (
 D
 
 )
 
 ∙
 
 
 
 
 {\displaystyle {\text{srep}}(D)_{\bullet }}
 
 given by the diagram<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>pg 16-17<blockquote></blockquote>where<blockquote>
 
 
 
 
 srep
 
 (
 D
 
 )
 
 n
 
 
 =
 
 
 ∐
 
 
 i
 
 0
 
 
 ←
 
 i
 
 1
 
 
 ←
 ⋯
 ←
 
 i
 
 n
 
 
 
 
 
 D
 (
 
 i
 
 n
 
 
 )
 
 
 {\displaystyle {\text{srep}}(D)_{n}={\underset {i_{0}\leftarrow i_{1}\leftarrow \cdots \leftarrow i_{n}}{\coprod }}D(i_{n})}
 
</blockquote>given by chains of composable maps in the indexing category 
 
 
 
 I
 
 
 {\displaystyle I}
 
. Then, the homotopy colimit of 
 
 
 
 D
 
 
 {\displaystyle D}
 
 can be constructed as the geometric realization of this simplicial space, so<blockquote>
 
 
 
 
 
 hocolim
 →
 
 
 D
 =
 
 |
 
 
 srep
 
 (
 D
 
 )
 
 ∙
 
 
 
 |
 
 
 
 {\displaystyle {\underset {\to }{\text{hocolim}}}D=|{\text{srep}}(D)_{\bullet }|}
 
</blockquote>Notice that this agrees with the picture given above for the composition diagram of 
 
 
 
 A
 →
 X
 →
 Y
 
 
 {\displaystyle A\to X\to Y}
 
.
<h2 id="relation-to-the-ordinary-colimit-and-limit">Relation to the (ordinary) colimit and limit</h2>
There is always a map

h
          o
          c
          o
          l
          i
          m
        
        
          X
          
            i
          
        
        →
        
          c
          o
          l
          i
          m
        
        
          X
          
            i
          
        
        .
      
    
    {\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 
 
 
 
 
 X
 
 0
 
 
 ←
 
 X
 
 0
 
 
 ×
 
 X
 
 1
 
 
 →
 
 X
 
 1
 
 
 
 
 {\displaystyle X_{0}\leftarrow X_{0}\times X_{1}\rightarrow X_{1}}
 
, which is a point.

<h2 id="further-examples-and-applications">Further examples and applications</h2>
Just as limit is used to <a href="/facts/Completion_(ring_theory)/Q0vjA9JI">complete</a> a ring, holim is used to complete a spectrum.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Derivator/Rk0rcvKS">Derivator</a></li>
<li><a href="/facts/Homotopy_fiber/hH4omFFQ">Homotopy fiber</a></li>
<li><a href="/facts/Homotopy_cofiber/a32m1wAa">Homotopy cofiber</a></li>
<li>Cohomology of categories</li>
<li>Spectral sequence of homotopy colimits</li></ul>

<ul><li><a href="https://pages.uoregon.edu/ddugger/hocolim.pdf">A Primer on Homotopy Colimits</a></li>
<li><a href="https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/homotopy-colimits-in-the-category-of-small-categories/32ACB453B263410CE1643E682D7DE335">Homotopy colimits in the category of small categories</a></li>
<li>Categories and Orbispaces</li>
<li>Hatcher, Allen (2002), <a href="https://pi.math.cornell.edu/~hatcher/AT/ATpage.html">Algebraic Topology</a>, Cambridge: Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-79540-0.</li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li><a href="https://mathoverflow.net/q/135462">Homotopy limit-colimit diagrams in stable model categories</a></li>
<li><a href="https://www.math.ucla.edu/~mikehill/Research/surv-douglas2-201.pdf">pg.80 Homotopy Colimits and Limits</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Dugger, Daniel. "A Primer on Homotopy Colimits" (PDF). Archived (PDF) from the original on 3 Dec 2020. <a href="https://pages.uoregon.edu/ddugger/hocolim.pdf" target="_blank">https://pages.uoregon.edu/ddugger/hocolim.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Grothendieck. "Pursuing Stacks". thescrivener.github.io. Archived (PDF) from the original on 30 Jul 2020. Retrieved 2020-09-17. <a href="https://thescrivener.github.io/PursuingStacks/" target="_blank">https://thescrivener.github.io/PursuingStacks/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Dugger, Daniel. "A Primer on Homotopy Colimits" (PDF). Archived (PDF) from the original on 3 Dec 2020. <a href="https://pages.uoregon.edu/ddugger/hocolim.pdf" target="_blank">https://pages.uoregon.edu/ddugger/hocolim.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Hatcher's Algebraic Topology, 4.G. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Dugger, Daniel. "A Primer on Homotopy Colimits" (PDF). Archived (PDF) from the original on 3 Dec 2020. <a href="https://pages.uoregon.edu/ddugger/hocolim.pdf" target="_blank">https://pages.uoregon.edu/ddugger/hocolim.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Bousfield & Kan: Homotopy limits, Completions and Localizations, Springer, LNM 304. Section XI.3.3 <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Math 527 - Homotopy Theory Homotopy pullbacks
 <a href="https://www.home.uni-osnabrueck.de/mfrankland/Math527/Math527_0308.pdf" target="_blank">https://www.home.uni-osnabrueck.de/mfrankland/Math527/Math527_0308.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Math 527 - Homotopy Theory Homotopy pullbacks
 <a href="https://www.home.uni-osnabrueck.de/mfrankland/Math527/Math527_0308.pdf" target="_blank">https://www.home.uni-osnabrueck.de/mfrankland/Math527/Math527_0308.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Dugger, Daniel. "A Primer on Homotopy Colimits" (PDF). Archived (PDF) from the original on 3 Dec 2020. <a href="https://pages.uoregon.edu/ddugger/hocolim.pdf" target="_blank">https://pages.uoregon.edu/ddugger/hocolim.pdf</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
</ol>

Homotopy colimit and limit open-in-new

Homotopy colimit and limit