Group actions and equivariantization

An action of a finite group consists of tensor autoequivalences $T_g$ and coherent monoidal natural isomorphisms

\[\label{eq:action-compositor} \sigma_{g,h}:T_g\circ T_h\longrightarrow T_{gh}.\]

They obey

\[\label{eq:action-compositor-coherence} \sigma_{gh,k}\circ(\sigma_{g,h}\ast\mathrm{id}_{T_k}) =\sigma_{g,hk}\circ(\mathrm{id}_{T_g}\ast\sigma_{h,k}),\]

where $\ast$ denotes whiskering, that is, horizontal composition with the indicated identity natural transformation, together with the normalized identity data. A categorical action therefore contains more data than an action on simple labels; see (Etingof et al., 2015; Definitions 2.7.1 and 4.15.1).

gtensor_action(C,elems,images,monoidal_structure) stores an ordered list of group elements, their functors, and a dictionary of structure transformations indexed by pairs of positions. The same ordering must be used throughout. The stored format has no separate unit isomorphism, so normalized action data use $T_e=\mathrm{Id}_{\mathcal C}$ and $\sigma_{e,g}=\sigma_{g,e}=\mathrm{id}_{T_g}$. is_tensor_action checks the displayed associativity equation for the stored transformations. Supply elems as the complete group-element list, one tensor autoequivalence in images for each entry, and a natural monoidal isomorphism for every pair, with compatible endpoints and normalized identity data. These properties are input requirements; is_tensor_action evaluates the action associativity equation. action_by_inner_autoequivalences constructs candidates from invertible objects; it does not enumerate all tensor actions.

Equivariant objects

With this direction of $\sigma$, an equivariant object has isomorphisms $u_g:T_g(X)\to X$ with $u_e=\mathrm{id}_X$ and

\[\label{eq:equivariant-object-coherence} u_g\circ T_g(u_h)=u_{gh}\circ(\sigma_{g,h})_X.\]

This is the direction of the structure maps in Etingof et al. (2015), Definition 2.7.2. The constructor is equivariantization(C,T) (or equivariantization(C,G,T)). The action is an argument; equivariantization(C) alone is not the documented constructor. equivariant_induction uses the underlying direct sum $\bigoplus_{g\in G}T_g(X)$ with its coherent equivariant structure. An equivariant object stores one structure map for each entry of T.elements. The low-level constructor expects isomorphisms with $u_e=\mathrm{id}_X$; is_equivariant checks their displayed pairwise compatibility equation. Both is_tensor_action and is_equivariant use represented equality and are currently intended for exact coefficient fields.

In characteristic dividing $|G|$, equivariantization need not be semisimple. Fusion-category algorithms and $F$-symbol extraction therefore require a separate semisimplicity determination in this case.

Crossed products

For a SixJCategory with a supplied action T, call gcrossed_product(C,T) or C ⋊ T. The resulting skeletal category uses

\[\label{eq:crossed-product-tensor} (X,g)\otimes(Y,h)=(X\otimes T_g(Y),gh).\]

This is Etingof et al. (2015), Definition 4.15.5 and Eq. (4.22). Its associator depends on both $\sigma$ and the tensorators of $T_g$. The constructor supplies this crossed product; it does not add the further data of a braided $G$-crossed extension. It assumes that T is a coherent action on C and does not call is_tensor_action or a pentagon check. The result supplies no braiding and no verified pivotal structure; check the action before construction and run pentagon_axiom on the result when coherence matters.

The API reference lists the principal public names for the framework and these constructions. Use Julia help mode or methods(name) for the exact signatures in the installed version.

The category catalogue is the reference guide to the concrete models, skeletal categories, and datasets distributed with the package.