Center (category theory)

Definition

The center of a monoidal category ${\displaystyle \mathcal{C}=(\mathcal{C},\otimes ,I)}$, denoted ${\displaystyle \mathcal{Z}\mathcal{(}\mathcal{C}\mathcal{)}}$, is the category whose objects are pairs (A,u) consisting of an object A of ${\displaystyle \mathcal{C}}$ and an isomorphism ${\displaystyle u_X:A\otimes X\to X\otimes A}$ which is natural in ${\displaystyle X}$ satisfying

${\displaystyle u_{X\otimes Y}=(1\otimes u_Y)(u_X\otimes 1)}$

and

${\displaystyle u_I=1_A}$ (this is actually a consequence of the first axiom).

An arrow from (A,u) to (B,v) in ${\displaystyle \mathcal{Z}\mathcal{(}\mathcal{C}\mathcal{)}}$ consists of an arrow ${\displaystyle f:A\to B}$ in ${\displaystyle \mathcal{C}}$ such that

${\displaystyle v_X(f\otimes 1_X)=(1_X\otimes f)u_X}$.

This definition of the center appears in Joyal & Street (1991). Equivalently, the center may be defined as

${\displaystyle \mathcal{Z}(\mathcal{C})={\mathrm{E}\mathrm{n}\mathrm{d}}_{\mathcal{C}\otimes {\mathcal{C}}^{op}}(\mathcal{C}),}$

i.e., the endofunctors of C which are compatible with the left and right action of C on itself given by the tensor product.

Braiding

The category ${\displaystyle \mathcal{Z}\mathcal{(}\mathcal{C}\mathcal{)}}$ becomes a braided monoidal category with the tensor product on objects defined as

${\displaystyle (A,u)\otimes (B,v)=(A\otimes B,w)}$

where ${\displaystyle w_X=(u_X\otimes 1)(1\otimes v_X)}$, and the obvious braiding.

Higher categorical version

The categorical center is particularly useful in the context of higher categories. This is illustrated by the following example: the center of the (abelian) category ${\displaystyle {\mathrm{M}\mathrm{o}\mathrm{d}}_R}$ of R-modules, for a commutative ring R, is ${\displaystyle {\mathrm{M}\mathrm{o}\mathrm{d}}_R}$ again. The center of a monoidal ∞-category C can be defined, analogously to the above, as

${\displaystyle Z(\mathcal{C}):={\mathrm{E}\mathrm{n}\mathrm{d}}_{\mathcal{C}\otimes {\mathcal{C}}^{op}}(\mathcal{C})}$.

Now, in contrast to the above, the center of the derived category of R-modules (regarded as an ∞-category) is given by the derived category of modules over the cochain complex encoding the Hochschild cohomology, a complex whose degree 0 term is R (as in the abelian situation above), but includes higher terms such as ${\displaystyle Hom(R,R)}$ (derived Hom).

The notion of a center in this generality is developed by Lurie (2017, §5.3.1). Extending the above-mentioned braiding on the center of an ordinary monoidal category, the center of a monoidal ∞-category becomes an ${\displaystyle E_2}$-monoidal category. More generally, the center of a ${\displaystyle E_k}$-monoidal category is an algebra object in ${\displaystyle E_k}$-monoidal categories and therefore, by Dunn additivity, an ${\displaystyle E_{k+1}}$-monoidal category.