Functoriality of the center of an algebra

TL;DR

The paper develops a rigorous, multi-layered functorial framework for the full center of algebras in monoidal categories, extending the classical center to a center object in the monoidal center and promoting it to a lax 2-functor between appropriate bicategories. Central to the construction are action internal Homs, centralizers of module functors, and a cospan-enriched target category built from commutative algebras in the monoidal center, together with a refined 2-diagram/cell structure that hints at a higher tricategorical lift. The authors identify precise conditions under which the center functor becomes non-lax (e.g., for fusion categories or invertible-domain-morphism scenarios) and show how this formalism naturally captures bulk-defect data in rational conformal field theory, including the bulk-defect OPE structure. The framework unifies algebraic Morita theory with topological field theory insights, providing a powerful language for analyzing domain walls, defect fields, and their categorical centers in RCFT and related settings.

Abstract

The notion of the center of an algebra over a field k has a far reaching generalization to algebras in monoidal categories. The center then lives in the monoidal center of the original category. This generalization plays an important role in the study of bulk-boundary duality of rational conformal field theories. In this paper, we study functorial properties of the center. We show that it gives rise to a 2-functor from the bicategory of semisimple indecomposable module categories over a fusion category to the bicategory of commutative algebras in the monoidal center of this fusion category. Morphism spaces of the latter bicategory are extended from algebra homomorphisms to certain categories of cospans. We conjecture that the above 2-functor arises from a lax 3-functor between tricategories, and that in this setting one can relax the conditions from fusion categories to finite tensor categories. We briefly outline how one is naturally lead to the above 2-functor when studying rational conformal field theory with defects of all codimensions. For example, the cospans of the target bicategory correspond to spaces of defect fields and to the bulk-defect operator product expansions.

Figures (4)

• Figure 1: In these figures we cut out little discs around the marked points and mapped the picture to the plane. Figure a) shows the surface which provides the multiplication on the space $D$. Namely, it gives rise to a morphism $D \otimes D \to D$ in the category $\mathcal{D} = \text{Rep}(V_L)_+ \boxtimes \text{Rep}(V_R)_-$. Figure b) shows the surface that defines the morphism $B_1 \to D$.
• Figure 2: Surfaces can be sewn together using the local coordinates around an in- and out-going marked points. We symbolize this procedure in the above pictures by the dashed circles. For example, figure a) shows a sphere with two in-going and one out-going marked point which has been obtained by sewing two copies of the spheres in Figure \ref{['fig:defect-mult+inclusion']} b to the in-going punctures of the sphere in Figure \ref{['fig:defect-mult+inclusion']} a. The two decompositions at different points in the moduli space shown in figure a) and b) above result in the statement that the morphism $B_1 \to D$ in $\mathcal{D}$ is compatible with the multiplication on $B_1$ and $D$.
• Figure 3: The CFT sewing constraint corresponding to \ref{['diag:comm']} in the conventions used in Section \ref{['sec:intro-3']}: We consider a CFT with three possible colors for the world sheet, given by the three $\mathcal{C}$-modules $\mathcal{L}$, $\mathcal{M}$, $\mathcal{N}$, and four different defect conditions labelled by the functors $F,F',G,G'$. The internal homs $[F,F']$, etc., give the space of defect (changing) fields. The composition of functors amounts to the 'fusion' of topological defect lines. The sewing shown in figure a) corresponds to the upper path in \ref{['diag:comm']}, and the sewing in figure b) to the lower path. In words it says that $[F,F']$ and $[G,G']$ can both be interpreted as subspaces of the space of defect changing fields from the fused defect $GF$ to $G'F'$, and that fields in these subspaces mutually commute.
• Figure 4: Here we use the same conventions as in the figures in Section \ref{['sec:intro-3']}. Figures a), b) and c) above show how the sphere with two in-going and one out-going marked point as shown can be obtained at two different points in moduli space from sewing the building blocks in Figure \ref{['fig:defect-mult+inclusion']}. The resulting identity of morphisms in the category $\mathcal{D} = \mathrm{Rep}(V_L \otimes_\mathbb{C} V_R)$ is the first diagram in \ref{['eq:cospan-central']} with $A = B_1$ and $T = D$.

Theorems & Definitions (132)

• Remark 1.1
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Lemma 2.6
• proof
• Lemma 2.7
• proof
• Lemma 2.8