nLab center of an additive category




Where the center of a category is in general just a commutative monoid (the endomorphism monoid of its identity functor formed in the functor category), for additive categories this commutative monoid carries the further structure of a commutative ring: the endomorphism ring of its identity functor. (In fact this is also true for Ab-enriched categories, which are more general.)



For π’ž\mathcal{C} an Ab-enriched category (e.g. an additive category) its center is

Z(π’ž)≔[π’ž,π’ž](id π’ž,id π’ž)∈CRing, Z(\mathcal{C}) \;\coloneqq\; [\mathcal{C}, \mathcal{C}] \big(id_{\mathcal{C}}, id_{\mathcal{C}}\big) \;\; \in \; CRing \,,


By Ab-enrichment, this means that Z(π’ž)Z(\mathcal{C}) carries the structure of a commutative monoid object internal to Ab, hence: the structure of a commutative ring.



Let R Mod R Mod denote the category of left modules of a ring RR, and let Z(R)Z(R) be the center of RR. Then Z(RMod)β‰…Z(R)Z(R Mod) \cong Z(R).


First note that for any r∈Z(R)r \in Z(R), multiplication by rr acts as an endomorphism of each RR-module, and this endomorphism is natural. This gives a ring homomorphism from Z(R)Z(R) to Z(RMod)Z(R Mod) which is injective because distinct elements of rr act differently as multiplication on the RR-module given by RR itself. To see that it is also surjective and hence bijective, suppose Ξ±\alpha is a natural transformation of the identity functor on RModR \Mod. Then Ξ± R:Rβ†’R\alpha_R \colon R \to R must be right multiplication by some r∈Rr \in R, since every endomorphism of R∈RModR \in R Mod is given by right multiplication by some r∈Rr \in R. Because Ξ± R\alpha_R is natural and right multiplication by any s∈Rs \in R gives an endomorphism of R∈RModR \in R Mod, we have

(xs)r=Ξ± R(xs)=Ξ± R(x)s=(xr)s (x s) r \, = \, \alpha_R(x s) \, = \, \alpha_R(x)s \, = \, (x r) s

for all x,s∈Rx, s \in R, so r∈Z(R)r \in Z(R). More generally, for any module MM and any m∈Mm \in M there is a module homomorphism f:Rβ†’Mf \colon R \to M with f(1)=mf(1) = m, which by naturality implies

Ξ± M(m)=Ξ± M(f(1))=f(Ξ± R(1))=f(r)=rm, \alpha_M (m) \,=\, \alpha_M\big(f(1)\big) \,=\, f\big(\alpha_R(1)\big) \,=\, f(r) \,=\, r m \,,

which shows that Ξ± M\alpha_M is multiplication by rr.


Two rings whose categories of modules are equivalent as Ab-enriched categories are said to be Morita equivalent. As a consequence of Prop. , Morita equivalent commutative rings are already isomorphic.

As a further illustration of these ideas we show how the topology on a compact Hausdorff space is determined by the category of vector bundles over this space. For any compact Hausdorff space XX let Vect ( X ) Vect(X) denote the category of (finite-rank complex) vector bundles over XX. This category is β„‚ \mathbb{C} -linear, i.e. enriched over the category of complex vector spaces. Thus, the center of Vect(X)Vect(X) is a commutative algebra over β„‚\mathbb{C}. Moreover:


If XX is a compact Hausdorff space then the center of Vect ( X ) Vect(X) is C(X)C(X), the function algebra of complex-valued continuous functions on XX.


For any field kk suppose AA is a commutative kk-algebra. Let AProjA Proj be the category of finitely generated projective A A -modules. This is a kk-linear category, and a straightforward extension of the proof of Prop. shows that the center of AProjA Proj is isomorphic to AA, not merely as a commutative ring, but as a commutative kk-algebra.

Let XX be a compact Hausdorff space. By Swan's theorem, Vect ( X ) Vect(X) is equivalent, as a β„‚\mathbb{C}-linear category, to C(X)ProjC(X) Proj. Thus the center of Vect(X)Vect(X) is isomorphic to C(X)C(X).


Suppose XX and YY are compact Hausdorff spaces such that Vect ( X ) Vect(X) and Vect ( Y ) Vect(Y) are equivalent as β„‚\mathbb{C}-linear categories. Then XX and YY are homeomorphic.


By Prop. , if Vect(X)Vect(X) and Vect(Y)Vect(Y) are equivalent as β„‚\mathbb{C}-linear categories then C(X)C(X) and C(Y)C(Y) are isomorphic as complex algebras. The Gelfand-Naimark theorem implies that when XX is compact Hausdorff, it is homeomorphic to the set of algebra homomorphisms C(X)β†’β„‚C(X) \to \mathbb{C}, given the topology of pointwise convergence. Thus the isomorphism of algebras C(X)β‰…C(Y)C(X) \cong C(Y) implies that XX and YY are homeomorphic. (Note that we did not need to show C(X)β‰…C(Y)C(X) \cong C(Y) as C *C^\ast-algebras here, but this follows.)


Note that it is much easier to recover C(X)C(X) and thus XX starting from Vect(X)\Vect(X) as a symmetric monoidal β„‚\mathbb{C}-linear category, since then the endomorphism algebra of the unit object, the trivial line bundle over XX, is C(X)C(X).


An analogue of Prop. also holds for real vector bundles: the center of the ℝ\mathbb{R}-linear category of real vector bundles over XX is the algebra of continuous real-valued functions on XX, and from this we can recover XX, either by using the real version of the Gelfand-Naimark theorem, or by complexifying this algebra and using the usual complex version of the Gelfand-Naimark theorem.


  • In general, the construction of centers is not functorial (except with respect to equivalence of categories); but it is functorial in some important special circumstances, such as certain reconstruction theorems.



Early occurrence of the definition of the center of an additive category:

Last revised on May 18, 2023 at 14:42:20. See the history of this page for a list of all contributions to it.