# Standard library of theories

Through the module Catlab.Theories, Catlab provides a standard library of generalized algebraic theories for categories, monoidal categories, and other categorical structures. The theories correspond, in most cases, to standard definitions in category theory and they are used throughout Catlab and the AlgebraicJulia ecosystem to structure programs and provide a common interface for applied category theory. The module also provides default syntax systems for many of the theories.

Categorical structures for which theories are provided include:

• categories
• monoidal and symmetric monoidal categories
• cartesian and cocartesian categories
• hypergraph categories
• bicategories of relations
• categories with two monoidal products, such as distributive monoidal categories

The contents of this module can be supplemented by the user, and it is even possible to use many parts of Catlab without using this module. The user is free to create new syntax systems for the theories defined here and also to define entirely new theories.

Catlab.TheoriesModule

Catlab's standard library of generalized algebraic theories.

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

Catlab.Theories.CategoryExprType

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.

Catlab.Theories.ThAlgebraicLatticeType

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.

Catlab.Theories.ThBicategoryRelationsType

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
Catlab.Theories.ThCartesianClosedCategoryType

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.

Catlab.Theories.ThCartesianDoubleCategoryType

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.

Catlab.Theories.ThCategoryType

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.

Catlab.Theories.ThCategoryWithProductsType

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.

Catlab.Theories.ThCompleteCategoryType

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".

Catlab.Theories.ThDaggerCompactCategoryType

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.

Catlab.Theories.ThDisplayedCategoryType

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.

Catlab.Theories.ThDistributiveBicategoryRelationsType

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.

Catlab.Theories.ThDistributiveSemiadditiveCategoryType

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

Catlab.Theories.ThDoubleCategoryType

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 $S,T$.

Catlab.Theories.ThEquipmentType

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
Catlab.Theories.ThHypergraphCategoryType

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.

Catlab.Theories.ThIndexedCategoryType

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.

Catlab.Theories.ThIndexedMonoidalCategoryType

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!]
Catlab.Theories.ThMCategoryType

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.

Catlab.Theories.ThMonoidalCategoryType

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.

Catlab.Theories.ThMonoidalCategoryWithBidiagonalsType

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.

Catlab.Theories.ThMonoidalCategoryWithDiagonalsType

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"
Catlab.Theories.ThMonoidalDoubleCategoryType

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.

Catlab.Theories.ThPreorderType

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.

Catlab.Theories.ThRigCategoryType

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.

Catlab.Theories.ThSchemaType

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.

Catlab.Theories.ThSymmetricMonoidalCopresheafType

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.