Coherence condition

<h2 id="an-illustrative-example-a-monoidal-category">An illustrative example: a monoidal category</h2>
Part of the data of a <a href="/facts/Monoidal_category/F2l5oKMD">monoidal category</a> is a chosen morphism

α
 
 A
 ,
 B
 ,
 C
 
 
 
 
 {\displaystyle \alpha _{A,B,C}}
 
, called the associator:

α
          
            A
            ,
            B
            ,
            C
          
        
        :
        (
        A
        ⊗
        B
        )
        ⊗
        C
        →
        A
        ⊗
        (
        B
        ⊗
        C
        )
      
    
    {\displaystyle \alpha _{A,B,C}\colon (A\otimes B)\otimes C\rightarrow A\otimes (B\otimes C)}

for each triple of <a href="/facts/Object_(category_theory)/7WzxUThf">objects</a> 
 
 
 
 A
 ,
 B
 ,
 C
 
 
 {\displaystyle A,B,C}
 
 in the category. Using compositions of these 
 
 
 
 
 α
 
 A
 ,
 B
 ,
 C
 
 
 
 
 {\displaystyle \alpha _{A,B,C}}
 
, one can construct a morphism

(
        (
        
          A
          
            N
          
        
        ⊗
        
          A
          
            N
            −
            1
          
        
        )
        ⊗
        
          A
          
            N
            −
            2
          
        
        )
        ⊗
        ⋯
        ⊗
        
          A
          
            1
          
        
        )
        →
        (
        
          A
          
            N
          
        
        ⊗
        (
        
          A
          
            N
            −
            1
          
        
        ⊗
        ⋯
        ⊗
        (
        
          A
          
            2
          
        
        ⊗
        
          A
          
            1
          
        
        )
        )
        .
      
    
    {\displaystyle ((A_{N}\otimes A_{N-1})\otimes A_{N-2})\otimes \cdots \otimes A_{1})\rightarrow (A_{N}\otimes (A_{N-1}\otimes \cdots \otimes (A_{2}\otimes A_{1})).}

Actually, there are many ways to construct such a morphism as a composition of various 
 
 
 
 
 α
 
 A
 ,
 B
 ,
 C
 
 
 
 
 {\displaystyle \alpha _{A,B,C}}
 
. One coherence condition that is typically imposed is that these compositions are all equal.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
Typically one proves a coherence condition using a <a href="/facts/Coherence_theorem/ezWC0V7p">coherence theorem</a>, which states that one only needs to check a few equalities of compositions in order to show that the rest also hold. In the above example, one only needs to check that, for all quadruples of objects 
 
 
 
 A
 ,
 B
 ,
 C
 ,
 D
 
 
 {\displaystyle A,B,C,D}
 
, the following diagram commutes.

Any pair of morphisms from 
 
 
 
 (
 (
 ⋯
 (
 
 A
 
 N
 
 
 ⊗
 
 A
 
 N
 −
 1
 
 
 )
 ⊗
 ⋯
 )
 ⊗
 
 A
 
 2
 
 
 )
 ⊗
 
 A
 
 1
 
 
 )
 
 
 {\displaystyle ((\cdots (A_{N}\otimes A_{N-1})\otimes \cdots )\otimes A_{2})\otimes A_{1})}
 
 to 
 
 
 
 (
 
 A
 
 N
 
 
 ⊗
 (
 
 A
 
 N
 −
 1
 
 
 ⊗
 (
 ⋯
 ⊗
 (
 
 A
 
 2
 
 
 ⊗
 
 A
 
 1
 
 
 )
 ⋯
 )
 )
 
 
 {\displaystyle (A_{N}\otimes (A_{N-1}\otimes (\cdots \otimes (A_{2}\otimes A_{1})\cdots ))}
 
 constructed as compositions of various 
 
 
 
 
 α
 
 A
 ,
 B
 ,
 C
 
 
 
 
 {\displaystyle \alpha _{A,B,C}}
 
 are equal.

<h2 id="further-examples">Further examples</h2>
Two simple examples that illustrate the definition are as follows. Both are directly from the definition of a category.

<h3>Identity</h3>
Let f : A → B be a morphism of a category containing two objects A and B. Associated with these objects are the identity morphisms 1A : A → A and 1B : B → B. By composing these with f, we construct two morphisms:

f o 1A : A → B, and
1B o f : A → B.
Both are morphisms between the same objects as f. We have, accordingly, the following coherence statement:

f o 1A = f  = 1B o f.
<h3>Associativity of composition</h3>
Let f : A → B, g : B → C and h : C → D be morphisms of a category containing objects A, B, C and D. By repeated composition, we can construct a morphism from A to D in two ways:

(h o g) o f : A → D, and
h o (g o f) : A → D.
We have now the following coherence statement:

(h o g) o f = h o (g o f).
In these two particular examples, the coherence statements are theorems for the case of an abstract category, since they follow directly from the axioms; in fact, they are axioms. For the case of a concrete mathematical structure, they can be viewed as conditions, namely as requirements for the mathematical structure under consideration to be a concrete category, requirements that such a structure may meet or fail to meet.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/2-category/YeMVfoAc">2-category</a></li>
<li><a href="/facts/Pseudoalgebra/1cVPBvdp">Pseudoalgebra</a></li>
<li><a href="/facts/Tricategory/MVgl07mH">Tricategory</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Calaque, Damien; Etingof, Pavel (2008). "Lectures on tensor categories". Quantum Groups. IRMA Lectures in Mathematics and Theoretical Physics. Vol. 12. pp. 1–38. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0401246">math/0401246</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4171%2F047-1%2F1">10.4171/047-1/1</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-03719-047-0.</li>
<li>Kelly, G.M (1964). "On MacLane's conditions for coherence of natural associativities, commutativities, etc". Journal of Algebra. 1 (4): 397–402. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0021-8693%2864%2990018-3">10.1016/0021-8693(64)90018-3</a>.</li>
<li>Kelly, G. M.; Laplaza, M.; Lewis, G.; Mac Lane, Saunders (1972). Coherence in Categories. Lecture Notes in Mathematics. Vol. 281. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBFb0059553">10.1007/BFb0059553</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-05963-9.</li>
<li>Im, Geun Bin; Kelly, G.M. (1986). "A universal property of the convolution monoidal structure". Journal of Pure and Applied Algebra. 43: 75–88. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0022-4049%2886%2990005-8">10.1016/0022-4049(86)90005-8</a>.</li>
<li>Kassel, Christian (1995). "Tensor Categories". Quantum Groups. Graduate Texts in Mathematics. Vol. 155. pp. 275–293. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4612-0783-2_11">10.1007/978-1-4612-0783-2_11</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-6900-7.</li>
<li>Laplaza, Miguel L. (1972). "Coherence for distributivity". Coherence in Categories. Lecture Notes in Mathematics. Vol. 281. pp. 29–65. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBFb0059555">10.1007/BFb0059555</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-05963-9.</li>
<li>Lack, Stephen (2000). <a href="https://doi.org/10.1006%2Faima.1999.1881">"A Coherent Approach to Pseudomonads"</a>. Advances in Mathematics. 152 (2): 179–202. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1006%2Faima.1999.1881">10.1006/aima.1999.1881</a>.</li>
<li>MacLane, Saunders (October 1963). <a href="https://hdl.handle.net/1911/62865">"Natural Associativity and Commutativity"</a>. Rice Institute Pamphlet - Rice University Studies. <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/1911%2F62865">1911/62865</a>.</li>
<li><a href="/facts/Saunders_Mac_Lane/rjgUqfkA">Mac Lane, Saunders</a> (1971). <a href="https://link.springer.com/chapter/10.1007/978-1-4612-9839-7_8">"7. Monoids §2 Coherence"</a>. <a href="/facts/Categories_for_the_Working_Mathematician/SLQDat9B">Categories for the working mathematician</a>. Graduate texts in mathematics. Vol. 4. Springer. pp. 161–165. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4612-9839-7_8">10.1007/978-1-4612-9839-7_8</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781461298397.</li>
<li>MacLane, Saunders; Paré, Robert (1985). "Coherence for bicategories and indexed categories". Journal of Pure and Applied Algebra. 37: 59–80. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0022-4049%2885%2990087-8">10.1016/0022-4049(85)90087-8</a>.</li>
<li>Power, A.J. (1989). "A general coherence result". Journal of Pure and Applied Algebra. 57 (2): 165–173. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0022-4049%2889%2990113-8">10.1016/0022-4049(89)90113-8</a>.</li>
<li>Yanofsky, Noson S. (2000). <a href="https://eudml.org/doc/91637">"The syntax of coherence"</a>. Cahiers de Topologie et Géométrie Différentielle Catégoriques. 41 (4): 255–304.</li></ul>

<h2 id="external-links">External links</h2>
<ul><li>Malkiewich, Cary; Ponto, Kate (2021). "Coherence for bicategories, lax functors, and shadows". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/2109.01249">2109.01249</a> [<a href="https://arxiv.org/archive/math.CT">math.CT</a>].</li></ul>

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

<ol>
<li id="fn:1">(Kelly 1964, Introduction) - Kelly, G.M (1964). "On MacLane's conditions for coherence of natural associativities, commutativities, etc". Journal of Algebra. 1 (4): 397–402. doi:10.1016/0021-8693(64)90018-3. <a href="https://doi.org/10.1016%2F0021-8693%2864%2990018-3" target="_blank">https://doi.org/10.1016%2F0021-8693%2864%2990018-3</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Coherence condition open-in-new

Coherence condition