Products, scalar extension, and related constructions

This page lists constructions that transform an existing category or select a category from it. Each operation requires corresponding methods on the input model; the existence of the generic function does not imply support for every category type.

Products, arrows, and opposites

The categorical product and Deligne tensor product are different. For finite split semisimple categories, simple objects of the Deligne product are pairs of simple objects, while simple objects of the additive categorical product are supported in one factor.

The opposite category reverses every arrow. Reversing a braiding is a different operation: reverse_braiding(C) leaves the category and tensor product in place and replaces

\[\label{eq:reverse-braiding} c_{X,Y}\quad\text{by}\quad c^{\mathrm{rev}}_{X,Y}=c_{Y,X}^{-1}.\]

The current reverse_braiding method supports SixJCategory models and transports their existing monoidal data rather than constructing an opposite category.

Coefficients, splitting, and completions

Mathematically, one first forms the Hom-space extension $\mathcal C\otimes_k^{\mathrm{Hom}} L$, with the same objects as $\mathcal C$ and

\[\label{eq:scalar-extension-hom} \operatorname{Hom}_{\mathcal C\otimes_k^{\mathrm{Hom}} L}(X,Y) =\operatorname{Hom}_{\mathcal C}(X,Y)\otimes_k L.\]

New idempotents need not have images in this category. For a Hom-finite Krull–Schmidt category, the idempotent-complete additive scalar extension is the Karoubi completion

\[\label{eq:scalar-extension-karoubi-envelope} \mathcal C_L =\operatorname{Kar}(\mathcal C\otimes_k^{\mathrm{Hom}} L).\]

It records the new indecomposable summands which appear after extending the endomorphism algebras. For a semisimple category over a perfect field, or more generally when the relevant endomorphism algebras are separable, this is again semisimple abelian. For a weak fusion category it agrees with the Deligne product $\mathcal C\boxtimes_k\operatorname{Vec}_L$; see (Etingof and Gelaki, 2012; §3.1) and (López Franco, 2013; Theorem 3 and §5). The splitting chapter explains the distinction between the Hom-space extension and its completion in detail.

For a nonsemisimple abelian category, Karoubi completion alone need not preserve abelianness. The abelian scalar extension, when it exists, is the Deligne product $\mathcal C\boxtimes_k\operatorname{Vec}_L$; for $\mathcal C\simeq A\text{-mod}$ this is $(A\otimes_kL)\text{-mod}$. The two constructions agree when $L/k$ is finite separable.

The function extension_of_scalars realizes this construction according to the category model. For split skeletal input no simple endomorphism algebra acquires new idempotents, so the implementation transports the coefficient arrays directly. For a supported CenterCategory, it extends the known simple central objects and decomposes those whose endomorphism algebras split over the new field. Scalar extension can therefore change the simple objects and their endomorphism algebras. Specify an embedding whenever the source field has more than one embedding into the target.

Scalar extension along a field homomorphism preserves characteristic. When the same formulas are instead reduced to positive characteristic, semisimplicity and every denominator in the structure maps must be checked again. The coefficient-field chapter explains these distinctions, and the Ising-center example exhibits splitting after scalar extension.

For a chosen object or finite list over a finite field, split changes only that family; it does not enumerate the simple objects of the ambient category or the tensor closure of the chosen family. Its result records the extension field, embedding, extended objects, and their decompositions. The default relative-degree bound is max_degree=64. In a nonsemisimple category the construction makes the resulting indecomposable summands absolutely indecomposable; this does not by itself make them simple. The degree construction establishes this result, while check=true recomputes the residue endomorphism algebras afterward.

Idempotent completion and semisimplification solve different problems. The first adds images of idempotents; the second takes a quotient by negligible morphisms. Neither operation by itself chooses a splitting field. The semisimplification chapter gives its hypotheses, trace-pairing convention, and interface.

Generated subcategories and skeletal coordinates

For $k\geq0$, tensor_power(X,k) returns the unit when $k=0$ and otherwise forms the power with the category's tensor product; a negative exponent is not defined. This constructs an object and does not construct a subcategory.

tensor_power_category(X...) begins with the tensor unit and enlarges its stored indecomposable list as tensor words are explored. It does not automatically close under duals, extensions, or subquotients.

For a SixJObject, fusion_subcategory(X) reads the generated subcategory directly from the finite skeletal fusion rules. For a simple object in any semisimple tensor category, a complete proposed simple list can instead be passed as simples=S. The optional matrix products=N records multiplication by the generator: $N_{k,j}$ is the multiplicity of the $k$-th listed simple in $X\otimes S_j$. By default the function checks these decompositions through explicit isomorphisms, as well as the tensor unit, dual closure, and generation from $X$. It returns a concrete FusionSubcategory; set skeletal=true to compute a SixJCategory presentation.

simple_fusion_subcategories(C) considers each simple generator, keeps those generated fusion rings that have no proper fusion subring, and removes repetitions. Its current implementation consequently requires skeletal SixJCategory input. It does not return every subcategory generated by a single simple object.

Skeletonization changes the presentation of a supported split semisimple category; it does not adjoin new half-braidings. Its associator, braiding, and pivotal coefficients are expressed using one shared ordered system of simple representatives and multiplicity-space bases. Thus these data describe one skeletal gauge; the individual pivotal coefficients need not remain unchanged under another choice of bases. The skeletal-model chapter describes the resulting coordinates, while the Drinfeld-center chapter describes the different construction performed by center(C).

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

Continue with Algebra objects and internal modules.