nLab
semisimple category

Idea

A semisimple category is a Vect-enriched category in which each object is a direct sum of finitely many simple objects.

Definition

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

• it has finite biproducts (usually called ‘direct sums’),

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

  What does it mean to ‘have subobjects’? (I assume that the ‘direct sums’ are biproducts.) —Toby

  have subobjects = idempotents split and yes, finite biproducts. Simple objects are ones in which End(X) = k.

  Urs: shouldn’t we say something like: a category is semisimple if each object is a direct sum of finitely many simple objects?

  Bruce: Urs, you’re right, and that’s indeed the way one morally thinks about it, but it’s a less canonical way of proceeding. We ask ourselves: given a linear category with direct sums and subobjects, and a chosen maximal collection $\{X_i\}$ of nonisomorphic simple objects, how can we check if its semisimple? In the one way, we have to check whether a certain canonically defined map is an isomorphism. In the other way, we have to check if each object $V$ can be expressed as a direct sum of the $X_i$’s. Actually finding such a decomposition would be a noncanonical operation. So your shorter more snappy definition would force an auditor to perform an evil thing if he actually wanted to check it :-) It’s that old thing about “only that part of a representation which behaves like an irreducible $\rho$ is canonical, the actual break-down of that rep into direct sums of $\rho$’s is noncanonical”. That is, what is canonical is $\mathrm{Hom}(V,X_i)$ and not $V={⨁}_i{n}_i{X}_i$.

  Also, if we just had “a category is semisimple if each object is a direct sum of finitely many simple objects” without the conditions on direct sums and subobjects then we could have someone who nastily removes, say, all three-dimensional vector spaces from $\mathrm{Vect}$. It would still satisfy “each object is a direct sum of finitely many simple objects” but it shouldnt be regarded as a semisimple category since there are “holes”.

  Urs: okay, so “each object is finite sum of simples” is the right idea, while the right defintion is a bit different. I have accoridngly now created a section “Idea” with the former statement. (Every entry should start with a section “Idea”!) See if you like this. Otherwise, feel free to adjust.

  But in any case, it would be nice to have a discussion on how the “right definition” implies that every object is isomorphic to a finite direct sum of representables.

• there exist objects $X_i$ labeled by an index set $I$ such that $\mathrm{Hom}(X_i,X_j)\cong {\delta }_{\mathrm{ij}}k$ where $k$ is the ground field (such objects are called simple) and such that for any two objects $V$ and $W$ in the category, the natural composition map

  (1)$$\underset{i\in I}{⨁}\mathrm{Hom}(V,X_i)\otimes \mathrm{Hom}(X_i,W)\to \mathrm{Hom}(V,W)$$\bigoplus_{i \in I} Hom(V, X_i) \otimes Hom(X_i, W) \rightarrow Hom(V, W)

  is an isomorphism.

Direct sums of simple objects

Note that this definition implies that every object $V$ is a direct sum of simple objects $X_i$. To see this, note that the third item of the definition is equivalent to stipulating that the vector space $\mathrm{Hom}(X_i,V)$ is in canonical duality with the vector space $\mathrm{Hom}(V,X_i)$. Indeed, we have a canonical pairing

(2)$$\mathrm{Hom}(V,X_i)\otimes \mathrm{Hom}(X_i,V)\to k$$Hom(V, X_i) \otimes Hom(X_i, V) \rightarrow k

given by sending $f\otimes g\mapsto ⟨f\circ g⟩$ where the ”$⟨\cdot ⟩$” notation refers to extracting scalars from endomorphisms of simple objects (each such endomorphism is a scalar multiple of the identity). We also have a canonical copairing

(3)$$k\to \mathrm{Hom}(X_i,V)\otimes \mathrm{Hom}(V,X_i)$$k \rightarrow Hom(X_i, V) \otimes Hom(V, X_i)

given by sending ${id}_{X_i}$ to the ”$i$th block” of the image of the identity ${id}_V$ arrow under the isomorphism given in the definition. One can check that this pairing and copairing satisfy the snake equations. Hence if we choose a basis

(4)$$\{a_{i,p}:X_i\to V\}$$\{ a_{i,p} : X_i \rightarrow V \}

for each vector space $\mathrm{Hom}(X_i,V)$, we get a corresponding dual basis

(5)$$\{a_i^p:V\to X_i\}$$\{ a_i^p : V \rightarrow X_i \}

satisfying

(6)$$a_i^p{a}_{j,q}={\delta }_{\mathrm{ij}}{\delta }_p^q\mathrm{and}\sum _{i,p}{a}_{i,p}{a}_i^p={id}_V.$$a_i^p a_{j,q} = \delta_{ij} \delta_p^q \quad and \quad \sum_{i,p} a_{i,p} a_i^p = \id_V.

This says precisely that $V$ has been expressed as a direct sum of the $X_i$.

Remarks

• This definition of semisimple (taken from the reference of Müger below) 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 $\mathrm{Hom}(V,W)$ must at least have the same dimension as$\mathrm{Hom}(W,V)$. This is the easiest way to check if a category will fail to be semisimple. For instance, the category $\mathrm{Rep}(A)$ of representations of an algebra $A$ will rarely be semisimple, precisely because there is no relation between $\mathrm{Hom}(V,W)$ and $\mathrm{Hom}(W,V)$ in general. Again, this can be traced back to the original algebra $A$ 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 $S:C\to C$ from the category to itself with the property that there are canonical isomorphisms

  (7)$$\mathrm{Hom}(V,W)\cong \mathrm{Hom}(W,\mathrm{SV}{)}^{\vee }$$Hom(V, W) \cong Hom(W, SV)^\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 $*:\mathrm{Hom}(V,W)\to \mathrm{Hom}(W,V)$. 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)

Examples

• The archetypical simple example is Vect itself, the category of (finite dimensional!) vector spaces over some ground field $k$. This has a single isomorphism class of simple objects: given by $k$ itself.

• The category of representations of a compact Lie group $G$ is semisimple, with the simple objects being precisely the irreducible representations (this is the content of Schur’s lemma). If $G$ is noncompact, one needs to pass from the concept of ‘direct sum’ to ‘direct integral?’.

• Every fusion category is a semisimple category.

References

• M. Müger, From Subfactors to Categories and Topology I. Frobenius algebras in and Morita equivalence classes of tensor categories.