# Categorification

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{{Short description|Connects set theory with category theory}}
In [mathematics](/source/mathematics), '''categorification''' is the process of replacing [set-theoretic](/source/set_theory) [theorem](/source/theorem)s with [category-theoretic](/source/category_theory) analogues. Categorification, when done successfully, replaces [set](/source/set_(mathematics))s with [categories](/source/category_(mathematics)), [function](/source/function_(mathematics))s with [functor](/source/functor)s, and [equation](/source/equation)s with [natural isomorphisms](/source/natural_transformation) of functors satisfying additional properties.  The term was coined by [Louis Crane](/source/Louis_Crane).<ref>{{Cite journal |last=Crane |first=Louis |last2=Frenkel |first2=Igor B. |date=1994-10-01 |title=Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases |url=https://doi.org/10.1063/1.530746 |journal=Journal of Mathematical Physics |volume=35 |issue=10 |pages=5136–5154 |doi=10.1063/1.530746 |issn=0022-2488|arxiv=hep-th/9405183 }}</ref><ref>{{Cite journal |last=Crane |first=Louis |date=1995-11-01 |title=Clock and category: Is quantum gravity algebraic? |url=https://doi.org/10.1063/1.531240 |journal=Journal of Mathematical Physics |volume=36 |issue=11 |pages=6180–6193 |doi=10.1063/1.531240 |issn=0022-2488|arxiv=gr-qc/9504038 }}</ref>

The reverse of categorification is the process of ''decategorification''. Decategorification is a systematic process by which [isomorphic](/source/isomorphic) objects in a category are identified as [equal](/source/equality_(mathematics)). Whereas decategorification is a straightforward process, categorification is usually much less straightforward. In the [representation theory](/source/representation_theory) of [Lie algebra](/source/Lie_algebra)s, [modules](/source/module_over_a_ring) over specific algebras are the principal objects of study, and there are several frameworks for what a categorification of such a module should be, e.g., so called (weak) abelian categorifications.<ref>{{citation|first1=Mikhail|last1=Khovanov|author1-link=Mikhail Khovanov|first2=Volodymyr|last2=Mazorchuk|first3=Catharina|last3=Stroppel|author3-link= Catharina Stroppel |arxiv=math.RT/0702746 |title=A brief review of abelian categorifications|journal=Theory Appl. Categ.|volume=22|number=19|year=2009|pages=479–508}}</ref>

Categorification and decategorification are not precise mathematical procedures, but rather a class of possible analogues. They are used in a similar way to words like '[generalization](/source/generalization)', and not like '[sheafification](/source/sheafification)'.<ref>{{cite web | url = https://mathoverflow.net/a/4880/38821 | title = What precisely Is "Categorification"? | author = Alex Hoffnung | date = 2009-11-10 }}</ref>

==Examples==
One form of categorification takes a structure described in terms of sets, and interprets the sets as [isomorphism class](/source/isomorphism_class)es of objects in a category.  For example, the set of [natural numbers](/source/Natural_number) can be seen as the set of [cardinalities](/source/cardinality) of [finite set](/source/finite_set)s (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 [coproduct](/source/coproduct)s and [products](/source/product_(category_theory)) of the [category of finite sets](/source/category_of_finite_sets).  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.

Other examples include [homology theories](/source/homology_theory) in [topology](/source/topology). [Emmy Noether](/source/Emmy_Noether) gave the modern formulation of homology as the [rank](/source/Rank_of_a_group) of certain [free abelian groups](/source/Free_abelian_group) by categorifying the notion of a [Betti number](/source/Betti_number).{{sfn|Baez|Dolan|1998}} See also [Khovanov homology](/source/Khovanov_homology) as a [knot invariant](/source/knot_invariant) in [knot theory](/source/knot_theory).

An example in [finite group theory](/source/finite_group_theory) is that the [ring of symmetric functions](/source/ring_of_symmetric_functions) is categorified by the [category of representations](/source/category_of_representations) of the [symmetric group](/source/symmetric_group). The decategorification map sends the [Specht module](/source/Specht_module) indexed by partition <math>\lambda</math> to the [Schur function](/source/schur_class) indexed by the same partition,

:<math>S^\lambda \,\stackrel{\varphi}{\to}\; s_\lambda,</math>

essentially following the [character](/source/character_(mathematics)) map from a favorite basis of the associated [Grothendieck group](/source/Grothendieck_group) to a representation-theoretic favorite basis of the ring of [symmetric function](/source/symmetric_function)s. This map reflects how the structures are similar; for example

:<math>\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 </math>

have the same decomposition numbers over their respective bases, both given by [Littlewood–Richardson coefficients](/source/Littlewood%E2%80%93Richardson_rule).

==Abelian categorifications==
For a category <math>\mathcal{B}</math>, let <math>K(\mathcal{B})</math> be the [Grothendieck group](/source/Grothendieck_group) of <math>\mathcal{B}</math>.

Let <math>A</math> be a [ring](/source/ring_(mathematics)) which is [free as an abelian group](/source/Free_abelian_group), and let <math>\mathbf{a} = \{a_i\}_{i \in I}</math> be a basis of <math>A</math> such that the multiplication is positive in <math>\mathbf{a}</math>, i.e.

:<math>a_i a_j = \sum_{k} c_{ij}^k a_k,</math> with <math> c_{ij}^k \in \mathbb{Z}_{\geq 0}.</math>

Let <math>B</math> be an <math>A</math>-[module](/source/module_(mathematics)). Then a (weak) abelian categorification of <math>(A, \mathbf{a}, B)</math> consists of an [abelian category](/source/abelian_category) <math>\mathcal{B}</math>, an isomorphism <math> \phi: K(\mathcal{B}) \to B</math>, and exact [endofunctor](/source/endofunctor)s <math>F_i: \mathcal{B} \to \mathcal{B}</math> such that

# the functor <math>F_i</math> lifts the action of <math>a_i</math> on the module <math>B</math>, i.e. <math>\phi [F_i] = a_i \phi</math>, and
# there are isomorphisms <math>F_i F_j \cong \bigoplus_{k} F_k^{c_{ij}^k},</math>, i.e. the composition <math>F_i F_j</math> decomposes as the direct sum of functors <math>F_k</math> in the same way that the product <math>a_i a_j</math> decomposes as the linear combination of basis elements <math>a_k</math>.

==See also==
* [Combinatorial proof](/source/Combinatorial_proof), the process of replacing [number theoretic](/source/number_theory) theorems by set-theoretic analogues.
* [Higher category theory](/source/Higher_category_theory)
* [Higher-dimensional algebra](/source/Higher-dimensional_algebra)
* [Categorical ring](/source/Categorical_ring)

==References==
{{Reflist}}
{{Refbegin|}}
*{{citation|first1=John|last1=Baez|author1-link=John Baez|first2=James|last2=Dolan|contribution=Categorification|arxiv=math.QA/9802029 |title=Higher Category Theory|editor1-first=Ezra|editor1-last=Getzler|editor2-first=Mikhail|editor2-last=Kapranov|series=Contemp. Math.|volume=230|publisher=American Mathematical Society|location=Providence, Rhode Island|year=1998|pages=1–36}}
*{{citation|first1=Louis|last1=Crane|first2=David N.|last2=Yetter|title=Examples of categorification|journal=[Cahiers de Topologie et Géométrie Différentielle Catégoriques](/source/Cahiers_de_Topologie_et_G%C3%A9om%C3%A9trie_Diff%C3%A9rentielle_Cat%C3%A9goriques)|volume=39|year=1998|issue=1|pages=3–25|url=http://www.numdam.org/item/CTGDC_1998__39_1_3_0}}
*{{citation|first1=Volodymyr|last1=Mazorchuk|title=Lectures on Algebraic Categorification|series=QGM Master Class Series|year=2010|publisher=European Mathematical Society|arxiv=1011.0144|bibcode=2010arXiv1011.0144M}}
*{{citation|first1=Alistair|last1=Savage|title=Introduction to Categorification|year=2014|arxiv=1401.6037|bibcode=2014arXiv1401.6037S}}
*{{citation|first1=Mikhail|last1=Khovanov|author1-link=Mikhail Khovanov|first2=Volodymyr|last2=Mazorchuk|first3=Catharina|last3=Stroppel|author3-link= Catharina Stroppel |arxiv=math.RT/0702746 |title=A brief review of abelian categorifications|journal=Theory Appl. Categ.|volume=22|number=19|year=2009|pages=479–508}}
{{Refend}}

==Further reading== 
* A blog post by one of the above authors (Baez): https://golem.ph.utexas.edu/category/2008/10/what_is_categorification.html.

{{Category theory}}

Category:Category theory
Category:Algebraic topology

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