Standard library of theories

Catlab's standard library of generalized algebraic theories.

The focus is on categories and monoidal categories, but other related structures are also included.

Base type for GAT expressions in categories or other categorical structures.

All symbolic expression types exported by Catlab.Theories are subtypes of this abstract type.

Base type for morphism expressions in categorical structures.

Base type for object expressions in categorical structures.

Theory of abelian bicategories of relations

Unlike ThBicategoryRelations, this theory uses additive notation.

References:

• Carboni & Walters, 1987, "Cartesian bicategories I", Sec. 5
• Baez & Erbele, 2015, "Categories in control"

Theory of additive categories

An additive category is a biproduct category enriched in abelian groups. Thus, it is a semiadditive category where the hom-monoids have negatives.

Theory of lattices as algebraic structures

This is one of two standard axiomatizations of a lattice, the other being ThLattice. Because the partial order is not present, this theory is merely an algebraic theory (no dependent types).

The partial order is recovered as $A ≤ B$ iff $A ∧ B = A$ iff $A ∨ B = B$. This definition could be reintroduced into a generalized algebraic theory using an equality type Eq(lhs::El, rhs::El)::TYPE combined with term constructors `meet_leq(eq::Eq(A∧B, A))::(A ≤ B) and join_leq(eq::Eq(A∨B, B))::(A ≤ B). We do not employ that trick here because at that point it is more convenient to just start with the poset structure, as in ThLattice.

Theory of bicategories of relations

TODO: The 2-morphisms are missing.

References:

• Carboni & Walters, 1987, "Cartesian bicategories I"
• Walters, 2009, blog post, "Categorical algebras of relations", http://rfcwalters.blogspot.com/2009/10/categorical-algebras-of-relations.html

Theory of biproduct categories

Mathematically the same as ThSemiadditiveCategory but written multiplicatively, instead of additively.

Theory of cartesian (monoidal) categories

For the traditional axiomatization of products, see ThCategoryWithProducts.

Theory of cartesian closed categories, aka CCCs

A CCC is a cartesian category with internal homs (aka, exponential objects).

FIXME: This theory should also extend ThClosedMonoidalCategory, but multiple inheritance is not yet supported.

Theory of a cartesian double category

Loosely speaking, a cartesian double category is a double category $D$ such that the underlying catgories $D₀$ and $D₁$ are both cartesian categories, in a compatible way.

Reference: Aleiferi 2018, PhD thesis.

Theory of categories

Note: Throughout Catlab, we compose morphisms in diagrammatic order (from left to right), i.e., if $f:A→B$ and $g:B→C$ then the composite morphism $f⋅g:A→C$ is compose(f,g). Under this convention, functions are applied on the right, e.g., if $a ∈ A$ then $af ∈ B$.

We retain the usual meaning of the symbol $∘$ (\circ), i.e., $g∘f$ is compose(f,g). This usage is too entrenched to overturn. However, we use the symbol $⋅$ (\cdot) for composition in diagrammatic order.

Theory of 2-categories

Theory of a category with (finite) coproducts

Finite coproducts are presented in biased style, via the nullary case (initial objects) and the binary case (binary coproducts). The axioms are dual to those of ThCategoryWithProducts.

For a monoidal category axiomatization of finite coproducts, see ThCocartesianCategory.

Theory of a category with (finite) products

Finite products are presented in biased style, via the nullary case (terminal objects) and the binary case (binary products). The equational axioms are standard, especially in type theory (Lambek & Scott, 1986, Section 0.5 or Section I.3). Strictly speaking, this theory is not of a "category with finite products" (a category in which finite products exist) but of a "category with chosen finite products".

For a monoidal category axiomatization of finite products, see ThCartesianCategory.

Theory of (symmetric) closed monoidal categories

Theory of cocartesian (monoidal) categories

For the traditional axiomatization of coproducts, see ThCategoryWithCoproducts.

Theory of a (finitely) cocomplete category

Finite colimits are presented in biased style, via finite coproducts and coequalizers. The axioms are dual to those of ThCompleteCategory.

Theory of compact closed categories

Theory of a (finitely) complete category

Finite limits are presented in biased style, via finite products and equalizers. The equational axioms for equalizers are obscure, but can found in (Lambek & Scott, 1986, Section 0.5), who in turn attribute them to "Burroni's pioneering ideas". Strictly speaking, this theory is not of a "finitely complete category" (a category in which finite limits exist) but of a "category with chosen finite limits".

Theory of copresheaves.

Axiomatized as a covariant category action.

Theory of dagger categories

Theory of dagger compact categories

In a dagger compact category, there are two kinds of adjoints of a morphism f::Hom(A,B), the adjoint mate mate(f)::Hom(dual(B),dual(A)) and the dagger adjoint dagger(f)::Hom(B,A). In the category of Hilbert spaces, these are respectively the Banach space adjoint and the Hilbert space adjoint (Reed-Simon, Vol I, Sec VI.2). In Julia, they would correspond to transpose and adjoint in the official LinearAlegbra module. For the general relationship between mates and daggers, see Selinger's survey of graphical languages for monoidal categories.

FIXME: This theory should also extend ThDaggerCategory, but multiple inheritance is not yet supported.

Theory of dagger symmetric monoidal categories

Also known as a symmetric monoidal dagger category.

FIXME: This theory should also extend ThDaggerCategory, but multiple inheritance is not yet supported.

Theory of a displayed category.

More precisely, this is the theory of a category $C$ (Ob,Hom) together with a displayed category over $C$ (Fib,FibHom). Displayed categories axiomatize lax functors $C → **Span**$, or equivalently objects of a slice category $**Cat**/C$, in a generalized algebraic style.

Reference: Ahrens & Lumsdaine 2019, "Displayed categories", Definition 3.1.

Theory of a distributive bicategory of relations

References:

• Carboni & Walters, 1987, "Cartesian bicategories I", Remark 3.7 (mention in passing only)
• Patterson, 2017, "Knowledge representation in bicategories of relations", Section 9.2

FIXME: Should also inherit ThBicategoryOfRelations, but multiple inheritance is not yet supported.

Theory of a distributive category

A distributive category is a distributive monoidal category whose tensor product is the cartesian product, see ThDistributiveMonoidalCategory.

FIXME: Should also inherit ThCartesianCategory.

Theory of a distributive (symmetric) monoidal category

Reference: Jay, 1992, LFCS tech report LFCS-92-205, "Tail recursion through universal invariants", Section 3.2

FIXME: Should also inherit ThCocartesianCategory.

Theory of a distributive semiadditive category

This terminology is not standard but the concept occurs frequently. A distributive semiadditive category is a semiadditive category (or biproduct) category, written additively, with a tensor product that distributes over the biproduct.

FIXME: Should also inherit ThSemiadditiveCategory

Theory of double categories

A strict double category $D$ is an internal category

$(S,T: D₁ ⇉ D₀, U: D₀ → D₁, *: D₁ ×_{D₀} D₁ → D₁)$

in Cat where

• objects of $D₀$ are objects of $D$
• morphisms of $D₀$ are arrows (vertical morphisms) of $D$
• objects of $D₁$ are proarrows (horizontal morphisms) of $D$
• morphisms of $D₁$ are cells of $D$.

The domain and codomain (top and bottom) of a cell are given by the domain and codomain in $D₁$ and the source and target (left and right) are given by the functors $S,T$.

Theory of a double category with tabulators

A tabulator of a proarrow is a double-categorical limit. It is a certain cell with identity domain to the given proarrow that is universal among all cells of that form. A double category "has tabulators" if the external identity functor has a right adjoint. The values of this right adjoint are the apex objects of its tabulators. The counit of the adjunction provides the universal cells. Tabulators figure in the double-categorical limit construction theorem of Grandis-Pare 1999. In the case where the double category is actually a 2-category, tabulators specialize to cotensors, a more familiar 2-categorical limit.

Theory of a proarrow equipment, or equipment for short

Equipments have also been called "framed bicategories," "fibrant double categories," and "gregarious double categories" (?!).

References:

• Shulman, 2008: Framed bicategories and monoidal fibrations
• Cruttwell & Shulman, 2010: A unified framework for generalized multicategories

Theory of groupoids.

Theory of hypergraph categories

Hypergraph categories are also known as "well-supported compact closed categories" and "spidered/dungeon categories", among other things.

FIXME: Should also inherit ThClosedMonoidalCategory and ThDaggerCategory, but multiple inheritance is not yet supported.

Theory of hypergraph categories, in additive notation

Mathematically the same as HypergraphCategory but with different notation.

Theory of a (covariantly) indexed category.

An indexed category is a Cat-valued pseudofunctor. For simplicitly, we assume that the functor is strict.

Just as a copresheaf, or Set-valued functor, can be seen as a category action of a family of sets, an indexed category can be seen as a category action on a family of categories. This picture guides our axiomatization of an indexed category as a generalized algebraic theory.

Theory of a (covariantly) indexed monoidal category.

An indexed monoidal category is a pseudofunctor into MonCat, the 2-category of monoidal categories, lax monoidal functor, and monoidal natural transformations. As usual, we take both the pseudofunctor and the monoidal categories to be strict. However, unlike the most common definition of an indexed monoidal category (see nLab), we allow the transition functors between monoidal categories to be lax monoidal. This follows the usage in (Hofstra & De Marchi 2006).

References:

• Hofstra & De Marchi, 2006: Descent for monads
• Moeller & Vasilakopoulou, 2020: Monoidal Grothendieck construction, Remark 3.18 [this paper is about a different notion!]

Theory of lattices as posets

A (bounded) lattice is a poset with all finite meets and joins. Viewed as a thin category, this means that the category has all finite products and coproducts, hence the names for the inequality constructors in the theory. Compare with ThCartesianCategory and ThCocartesianCategory.

This is one of two standard axiomatizations of a lattice, the other being ThAlgebraicLattice.

Theory of an ℳ-category.

The term "ℳ-category", used on the nLab is not very common, but the concept itself shows up commonly. An ℳ-category is a category with a distinguished wide subcategory, whose morphisms are suggestively called tight; for contrast, a generic morphism is called loose. Equivalently, an ℳ-category is a category enriched in the category ℳ of injections, the full subcategory of the arrow category of Set spanned by injections.

In the following GAT, tightness is axiomatized as a property of morphisms: a dependent family of sets over the hom-sets, each having at most one inhabitant.

Theory of monoidal categories

To avoid associators and unitors, we assume that the monoidal category is strict. By the coherence theorem this involves no loss of generality, but we might add a theory for weak monoidal categories later.

Theory of monoidal categories, in additive notation

Mathematically the same as ThMonoidalCategory but with different notation.

Theory of monoidal categories with bidiagonals

The terminology is nonstandard (is there any standard terminology?) but is supposed to mean a monoidal category with coherent diagonals and codiagonals. Unlike in a biproduct category, the naturality axioms need not be satisfied.

Theory of monoidal categories with bidiagonals, in additive notation

Mathematically the same as MonoidalCategoryWithBidiagonals but written additively, instead of multiplicatively.

Theory of monoidal categories with codiagonals

A monoidal category with codiagonals is a symmetric monoidal category equipped with coherent collections of merging and creating morphisms (monoids). Unlike in a cocartesian category, the naturality axioms need not be satisfied.

For references, see ThMonoidalCategoryWithDiagonals.

Theory of monoidal categories with diagonals

A monoidal category with diagonals is a symmetric monoidal category equipped with coherent operations of copying and deleting, also known as a supply of commutative comonoids. Unlike in a cartesian category, the naturality axioms need not be satisfied.

References:

• Fong & Spivak, 2019, "Supplying bells and whistles in symmetric monoidal categories" (arxiv:1908.02633)
• Selinger, 2010, "A survey of graphical languages for monoidal categories", Section 6.6: "Cartesian center"
• Selinger, 1999, "Categorical structure of asynchrony"

Theory of monoidal double categories

To avoid associators and unitors, we assume that the monoidal double category is fully strict: both the double category and its monoidal product are strict. Apart from assuming strictness, this theory agrees with the definition of a monoidal double category in (Shulman 2010) and other recent works.

In a monoidal double category $(D,⊗,I)$, the underlying categories $D₀$ and $D₁$ are each monoidal categories, $(D₀,⊗₀,I₀)$ and $(D₁,⊗₁,I₁)$, subject to further axioms such as the source and target functors $S, T: D₁ → D₀$ being strict monoidal functors.

Despite the apparent asymmetry in this setup, the definition of a monoidal double category unpacks to be nearly symmetric with respect to arrows and proarrows, except that the monoidal unit $I₀$ of $D₀$ induces the monoidal unit of $D₁$ as $I₁ = U(I₀)$.

References:

• Shulman, 2010: Constructing symmetric monoidal bicategories

FIXME: Should also inherit ThMonoidalCategory{Ob,Hom} but multiple inheritance is not supported.

Theory of partial orders (posets)

Theory of preorders

The generalized algebraic theory of preorders encodes inequalities $A≤B$ as dependent types `$Leq(A,B)$ and the axioms of reflexivity and transitivity as term constructors.

Theory of presheaves.

Axiomatized as a contravariant category action.

Theory of a rig category, also known as a bimonoidal category

Rig categories are the most general in the hierarchy of distributive monoidal structures.

Question: Do we also want the distributivty and absorption isomorphisms? Usually we ignore coherence isomorphisms such as associators and unitors.

FIXME: This theory should also inherit ThMonoidalCategory, but multiple inheritance is not supported.

The GAT that parameterizes Attributed C-sets A schema is comprised of a category C, a discrete category D, and a profunctor Attr : C^op x D → Set. In GAT form, this is given by extending the theory of categories with two extra types, AttrType for objects of D, and Attr, for elements of the sets given by the profunctor.

Theory of semiadditive categories

Mathematically the same as BiproductCategory but written additively, instead of multiplicatively.

Theory of (strict) symmetric monoidal categories

Theory of symmetric monoidal categories, in additive notation

Mathematically the same as ThSymmetricMonoidalCategory but with different notation.

Theory of a symmetric monoidal copresheaf

The name is not standard but refers to a lax symmetric monoidal functor into Set. This can be interpreted as an action of a symmetric monoidal category, just as a copresheaf (set-valued functor) is an action of a category. The theory is simpler than that of a general lax monoidal functor because (1) the domain is a strict monoidal category and (2) the codomain is fixed to the cartesian monoidal category Set.

FIXME: This theory should also extend ThCopresheaf but multiple inheritance is not yet supported.

Theory of symmetric monoidal double categories

Unlike the classical notion of strict double categories, symmetric monoidal double categories do not treat the two directions on an equal footing, even when everything (except the braiding) is strict. See MonoidalDoubleCategory for references.

FIXME: Should also inherit ThSymmetricMonoidalCategory{Ob,Hom} but multiple inheritance is not supported.

Theory of a symmetric rig category

FIXME: Should also inherit ThSymmetricMonoidalCategory.

Theory of thin categories

Thin categories have at most one morphism between any two objects and are isomorphic to preorders.

Theory of thin symmetric monoidal category

Thin SMCs are isomorphic to commutative monoidal prosets.

Theory of traced monoidal categories

Collect generators of object in monoidal category as a vector.

Number of "dimensions" of object in monoidal category.