Categorification

<h2 id="examples">Examples</h2>
<p>One form of categorification takes a structure described in terms of sets, and interprets the sets as <a href="/facts/Isomorphism_class/wDUgsJQ8">isomorphism classes</a> of objects in a category.  For example, the set of <a href="/facts/Natural_number/u65og8JE">natural numbers</a> can be seen as the set of <a href="/facts/Cardinality/m6VPojwT">cardinalities</a> of <a href="/facts/Finite_set/za1newlP">finite sets</a> (and any two sets with the same cardinality are isomorphic).  In this case, operations on the set of natural numbers, such as addition and multiplication, can be seen as carrying information about <a href="/facts/Coproduct/0bJ8SCdu">coproducts</a> and <a href="/facts/Product_(category_theory)/evFecz0t">products</a> of the <a href="/facts/Category_of_finite_sets/Ahrjj5hl">category of finite sets</a>.  Less abstractly, the idea here is that manipulating sets of actual objects, and taking coproducts (combining two sets in a union) or products (building arrays of things to keep track of large numbers of them) came first.  Later, the concrete structure of sets was abstracted away – taken "only up to isomorphism", to produce the abstract theory of arithmetic.  This is a "decategorification" – categorification reverses this step.
</p><p>Other examples include <a href="/facts/Homology_theory/VFJmryEW">homology theories</a> in <a href="/facts/Topology/2EqW47cU">topology</a>. <a href="/facts/Emmy_Noether/agcYrVLt">Emmy Noether</a> gave the modern formulation of homology as the <a href="/facts/Rank_of_a_group/tqC9KFwI">rank</a> of certain <a href="/facts/Free_abelian_group/oaSRGmpY">free abelian groups</a> by categorifying the notion of a <a href="/facts/Betti_number/r42Y8k6f">Betti number</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> See also <a href="/facts/Khovanov_homology/jrqc5SlL">Khovanov homology</a> as a <a href="/facts/Knot_invariant/BEvEEpHQ">knot invariant</a> in <a href="/facts/Knot_theory/mb7rtqVP">knot theory</a>.
</p><p>An example in <a href="/facts/Finite_group_theory/vv2tvogB">finite group theory</a> is that the <a href="/facts/Ring_of_symmetric_functions/SmENv9o4">ring of symmetric functions</a> is categorified by the <a href="/facts/Category_of_representations/Lxr3DLnt">category of representations</a> of the <a href="/facts/Symmetric_group/RUksK5l5">symmetric group</a>. The decategorification map sends the <a href="/facts/Specht_module/wNyeJcj6">Specht module</a> indexed by partition 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 to the <a href="/facts/Schur_class/cchBmhw7">Schur function</a> indexed by the same partition,
</p>

S
          
            λ
          
        
        
        
          
            
              
                →
              
              
                φ
              
            
          
        
        
        
          s
          
            λ
          
        
        ,
      
    
    {\displaystyle S^{\lambda }\,{\stackrel {\varphi }{\to }}\;s_{\lambda },}

<p>essentially following the <a href="/facts/Character_(mathematics)/M46SyYUp">character</a> map from a favorite basis of the associated <a href="/facts/Grothendieck_group/HSLHgFvh">Grothendieck group</a> to a representation-theoretic favorite basis of the ring of <a href="/facts/Symmetric_function/IB8N4T3m">symmetric functions</a>. This map reflects how the structures are similar; for example
</p>

[
          
            
              Ind
              
                
                  S
                  
                    m
                  
                
                ⊗
                
                  S
                  
                    n
                  
                
              
              
                
                  S
                  
                    n
                    +
                    m
                  
                
              
            
            ⁡
            (
            
              S
              
                μ
              
            
            ⊗
            
              S
              
                ν
              
            
            )
          
          ]
        
        
        
          and
        
        
        
          s
          
            μ
          
        
        
          s
          
            ν
          
        
      
    
    {\displaystyle \left[\operatorname {Ind} _{S_{m}\otimes S_{n}}^{S_{n+m}}(S^{\mu }\otimes S^{\nu })\right]\qquad {\text{and}}\qquad s_{\mu }s_{\nu }}

<p>have the same decomposition numbers over their respective bases, both given by <a href="/facts/Littlewood%25E2%2580%2593Richardson_rule/8NSx5K36">Littlewood–Richardson coefficients</a>.
</p>
<h2 id="abelian-categorifications">Abelian categorifications</h2>
<p>For a category 
  
    
      
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {B}}}
  
, let 
  
    
      
        K
        (
        
          
            B
          
        
        )
      
    
    {\displaystyle K({\mathcal {B}})}
  
 be the <a href="/facts/Grothendieck_group/HSLHgFvh">Grothendieck group</a> of 
  
    
      
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {B}}}
  
.
</p><p>Let 
  
    
      
        A
      
    
    {\displaystyle A}
  
 be a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> which is <a href="/facts/Free_abelian_group/oaSRGmpY">free as an abelian group</a>, and let 
  
    
      
        
          a
        
        =
        {
        
          a
          
            i
          
        
        
          }
          
            i
            ∈
            I
          
        
      
    
    {\displaystyle \mathbf {a} =\{a_{i}\}_{i\in I}}
  
 be a basis of 
  
    
      
        A
      
    
    {\displaystyle A}
  
 such that the multiplication is positive in 
  
    
      
        
          a
        
      
    
    {\displaystyle \mathbf {a} }
  
, i.e.
</p>

a
          
            i
          
        
        
          a
          
            j
          
        
        =
        
          ∑
          
            k
          
        
        
          c
          
            i
            j
          
          
            k
          
        
        
          a
          
            k
          
        
        ,
      
    
    {\displaystyle a_{i}a_{j}=\sum _{k}c_{ij}^{k}a_{k},}
  
 with 
  
    
      
        
          c
          
            i
            j
          
          
            k
          
        
        ∈
        
          
            Z
          
          
            ≥
            0
          
        
        .
      
    
    {\displaystyle c_{ij}^{k}\in \mathbb {Z} _{\geq 0}.}

<p>Let 
  
    
      
        B
      
    
    {\displaystyle B}
  
 be an 
  
    
      
        A
      
    
    {\displaystyle A}
  
-<a href="/facts/Module_(mathematics)/jP0LIhFL">module</a>. Then a (weak) abelian categorification of 
  
    
      
        (
        A
        ,
        
          a
        
        ,
        B
        )
      
    
    {\displaystyle (A,\mathbf {a} ,B)}
  
 consists of an <a href="/facts/Abelian_category/4WN473Yc">abelian category</a> 
  
    
      
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {B}}}
  
, an isomorphism 
  
    
      
        ϕ
        :
        K
        (
        
          
            B
          
        
        )
        →
        B
      
    
    {\displaystyle \phi :K({\mathcal {B}})\to B}
  
, and exact <a href="/facts/Endofunctor/l2u3rIBE">endofunctors</a> 
  
    
      
        
          F
          
            i
          
        
        :
        
          
            B
          
        
        →
        
          
            B
          
        
      
    
    {\displaystyle F_{i}:{\mathcal {B}}\to {\mathcal {B}}}
  
 such that
</p>
<ol><li>the functor 
  
    
      
        
          F
          
            i
          
        
      
    
    {\displaystyle F_{i}}
  
 lifts the action of 
  
    
      
        
          a
          
            i
          
        
      
    
    {\displaystyle a_{i}}
  
 on the module 
  
    
      
        B
      
    
    {\displaystyle B}
  
, i.e. 
  
    
      
        ϕ
        [
        
          F
          
            i
          
        
        ]
        =
        
          a
          
            i
          
        
        ϕ
      
    
    {\displaystyle \phi [F_{i}]=a_{i}\phi }
  
, and</li>
<li>there are isomorphisms 
  
    
      
        
          F
          
            i
          
        
        
          F
          
            j
          
        
        ≅
        
          ⨁
          
            k
          
        
        
          F
          
            k
          
          
            
              c
              
                i
                j
              
              
                k
              
            
          
        
        ,
      
    
    {\displaystyle F_{i}F_{j}\cong \bigoplus _{k}F_{k}^{c_{ij}^{k}},}
  
, i.e. the composition 
  
    
      
        
          F
          
            i
          
        
        
          F
          
            j
          
        
      
    
    {\displaystyle F_{i}F_{j}}
  
 decomposes as the direct sum of functors 
  
    
      
        
          F
          
            k
          
        
      
    
    {\displaystyle F_{k}}
  
 in the same way that the product 
  
    
      
        
          a
          
            i
          
        
        
          a
          
            j
          
        
      
    
    {\displaystyle a_{i}a_{j}}
  
 decomposes as the linear combination of basis elements 
  
    
      
        
          a
          
            k
          
        
      
    
    {\displaystyle a_{k}}
  
.</li></ol>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Combinatorial_proof/EmUsc3H1">Combinatorial proof</a>, the process of replacing <a href="/facts/Number_theory/zhoa7zrc">number theoretic</a> theorems by set-theoretic analogues.</li>
<li><a href="/facts/Higher_category_theory/7Pp6a0VN">Higher category theory</a></li>
<li><a href="/facts/Higher-dimensional_algebra/vyMHqRp4">Higher-dimensional algebra</a></li>
<li><a href="/facts/Categorical_ring/N5g2NPCQ">Categorical ring</a></li></ul>

<ul><li><a href="/facts/John_Baez/pjL45P57">Baez, John</a>; Dolan, James (1998), "Categorification", in Getzler, Ezra; Kapranov, Mikhail (eds.), <i>Higher Category Theory</i>, Contemp. Math., vol. 230, Providence, Rhode Island: American Mathematical Society, pp. 1–36, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math.QA/9802029">math.QA/9802029</a></li>
<li>Crane, Louis; Yetter, David N. (1998), <a href="http://www.numdam.org/item/CTGDC_1998__39_1_3_0">"Examples of categorification"</a>, <i><a href="/facts/Cahiers_de_Topologie_et_G%25C3%25A9om%25C3%25A9trie_Diff%25C3%25A9rentielle_Cat%25C3%25A9goriques/kdkn6cpn">Cahiers de Topologie et Géométrie Différentielle Catégoriques</a></i>, 39 (1): 3–25</li>
<li>Mazorchuk, Volodymyr (2010), <i>Lectures on Algebraic Categorification</i>, QGM Master Class Series, European Mathematical Society, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1011.0144">1011.0144</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2010arXiv1011.0144M">2010arXiv1011.0144M</a></li>
<li>Savage, Alistair (2014), <i>Introduction to Categorification</i>, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1401.6037">1401.6037</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2014arXiv1401.6037S">2014arXiv1401.6037S</a></li>
<li><a href="/facts/Mikhail_Khovanov/gHU9aPJQ">Khovanov, Mikhail</a>; Mazorchuk, Volodymyr; <a href="/facts/Catharina_Stroppel/H7FKpV3t">Stroppel, Catharina</a> (2009), "A brief review of abelian categorifications", <i>Theory Appl. Categ.</i>, 22 (19): 479–508, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math.RT/0702746">math.RT/0702746</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>A blog post by one of the above authors (Baez): <a href="https://golem.ph.utexas.edu/category/2008/10/what_is_categorification.html">https://golem.ph.utexas.edu/category/2008/10/what_is_categorification.html</a>.</li></ul>

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

<ol>
<li id="fn:1"><p>Crane, Louis; Frenkel, Igor B. (1994-10-01). "Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases". Journal of Mathematical Physics. 35 (10): 5136–5154. arXiv:hep-th/9405183. doi:10.1063/1.530746. ISSN 0022-2488. <a href="https://doi.org/10.1063/1.530746" target="_blank">https://doi.org/10.1063/1.530746</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Crane, Louis (1995-11-01). "Clock and category: Is quantum gravity algebraic?". Journal of Mathematical Physics. 36 (11): 6180–6193. arXiv:gr-qc/9504038. doi:10.1063/1.531240. ISSN 0022-2488. <a href="https://doi.org/10.1063/1.531240" target="_blank">https://doi.org/10.1063/1.531240</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Khovanov, Mikhail; Mazorchuk, Volodymyr; Stroppel, Catharina (2009), "A brief review of abelian categorifications", Theory Appl. Categ., 22 (19): 479–508, arXiv:math.RT/0702746 <a href="/wiki/Mikhail_Khovanov" target="_blank">/wiki/Mikhail_Khovanov</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Alex Hoffnung (2009-11-10). "What precisely Is "Categorification"?". <a href="https://mathoverflow.net/a/4880/38821" target="_blank">https://mathoverflow.net/a/4880/38821</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Baez & Dolan 1998. - Baez, John; Dolan, James (1998), "Categorification", in Getzler, Ezra; Kapranov, Mikhail (eds.), Higher Category Theory, Contemp. Math., vol. 230, Providence, Rhode Island: American Mathematical Society, pp. 1–36, arXiv:math.QA/9802029 <a href="https://arxiv.org/abs/math.QA/9802029" target="_blank">https://arxiv.org/abs/math.QA/9802029</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Categorification open-in-new

Categorification