nLab semisimple category




A semisimple category is a category in which each object is a direct sum of finitely many simple objects, and all such direct sums exist.



(semisimple abelian category)
An abelian category is called semisimple if every object is a semisimple object, hence a direct sum of finitely many simple objects.

Alternatively, at least over an algebraically closed ground field kk:


(semisimple linear category)
A linear category (that is, a category enriched over Vect, cf. tensor category) is called semisimple if:

  1. it has finite biproducts

    (usually called ‘direct sums’),

  2. idempotents split

    (so we say that it ‘has subobjects’ or, perhaps better, ‘has retracts’),

  3. there exist objects X iX_i labeled by an index set II such that

    1. for any pair i,ji, j of indices we have an isomorphism

      (1)Hom(X i,X j)δ ijk, Hom(X_i, X_j) \;\cong\; \delta_{i j} k \,,

      where δ\delta denotes the Kronecker delta and kk the ground field

      (such objects are called simple),

    2. for any pair of objects V,WV,\,W, the natural composition map

      (2) iIHom(V,X i)Hom(X i,W)Hom(V,W) \bigoplus_{i \in I} \, Hom(V, X_i) \,\otimes\, Hom(X_i, W) \longrightarrow Hom(V, W)

      is an isomorphism.

    (If the field kk is not algebraically closed, a simple object XX can have a finite-dimensional division algebra other than kk itself as its endomorphism algebra.)

(e.g. Müger (2003, p. 6))


(direct sum decomposition into simple objects)
Definition implies that every object VV is a direct sum of simple objects X iX_i.


The third item of the definition is equivalent to stipulating that the vector space Hom(X i,V)Hom(X_i, V) is in canonical duality with the vector space Hom(V,X i)Hom(V, X_i). Indeed, we have:

  1. a pairing

    Hom(V,X i)Hom(X i,V) Hom(X i,X i) k fg fg, \begin{array}{ccc} Hom(V, X_i) \,\otimes\, Hom(X_i, V) & \longrightarrow & Hom(X_i, X_i) &\simeq& k \\ f \,\otimes\, g &\mapsto& f \,\circ\, g \mathrlap{\,,} \end{array}

    where on the right we used (1),

  2. a copairing

    kId BHom(V,V)Hom(X i,V)Hom(V,X i), k \cdot Id_B \;\hookrightarrow\; Hom(V,V) \longrightarrow Hom(X_i, V) \otimes Hom(V, X_i) \mathrlap{\,,}

    where on the right we used (2),

and one checks that these satisfy the triangle identities and as such exhibit dual vector spaces.

Hence if we choose a linear basis

{a i,p:X iV} \big\{ a_{i,p} \,\colon\, X_i \rightarrow V \big\}

for each vector space Hom(X i,V)Hom(X_i, V), we get a corresponding dual basis

{a i p:VX i} \big\{ a_i^p \,\colon\, V \rightarrow X_i \big\}


a i pa j,q=δ ijδ p qand i,pa i,pa i p=id V. a_i^p a_{j,q} \;=\; \delta_{i j} \delta_p^q \;\;\; \text{and} \;\;\; \sum_{i,p} a_{i,p} a_i^p \;=\; \id_V \mathrlap{\,.}

This says precisely that VV has been expressed as a direct sum of the X iX_i.


Def. (of semisimple linear categories) does not use the concept of abelian category. This is because the concepts that one thinks about with abelian categories such as kernels and cokernels do not play an important conceptual role in semisimple categories, being replaced by the more important concepts of biproduct and retract. Hence it is best to give a streamlined definition from first principles without going through the language of abelian categories which would have muddied the waters.


For a category to be semisimple, it needs to have a certain directional symmetry in its hom-sets, namely that Hom(V,W)Hom(V, W) must at least have the same dimension as Hom(W,V)Hom(W,V). This is the easiest way to check if a category will fail to be semisimple. For instance, the category Rep(A)Rep(A) of representations of an algebra AA will rarely be semisimple, precisely because there is no relation between Hom(V,W)Hom(V, W) and Hom(W,V)Hom(W,V) in general. Again, this can be traced back to the original algebra AA not having any ‘symmetry’ like the inverse operation in a group.


As far as ‘duality’ on the hom-sets is concerned, one might have a S:CCS \colon C \rightarrow C from the category to itself with the property that there are canonical isomorphisms

Hom(V,W)Hom(W,SV) Hom(V, W) \cong Hom(W, S V)^\vee

where “\vee” denotes the ordinary linear dual of a vector space. Such a functor is called a Serre functor in algebraic geometry, and indeed there is precisely such a functor on the derived category of coherent sheaves on a complex manifold — it is given by tensoring with the canonical line bundle.


For 2-Hilbert spaces, there is an antilinear **-operation on the hom-sets *:Hom(V,W)Hom(W,V)* : Hom(V, W) \rightarrow Hom(W,V). The presence of this duality in fact forces the category to be semisimple (this comes down to the fact that a finite-dimensional **-algebra, such as the hom’s between a bunch of objects in the category, must be a full matrix algebra)



Various alternative definitions of semisimple category appear in the literature, and a number are compared here:

  • James Bailie, Semisimple categories (pdf.

For example, he compares a definition of “abelian semisimple category”(an abelian category where every object is a direct sum of a possibly infinite number of simple objects) with “Müger semisimple” categories as discussed here:

There is a related discussion on the nForum and a discussion on MathOverflow.

Last revised on June 18, 2023 at 18:01:07. See the history of this page for a list of all contributions to it.