# Categorical algebra

## Sets and Relations

The following APIs implement FinSet, the category of Finite Sets (actually the skeleton of FinSet). The objects of this category are natural numbers where n represents a set with n elements. The morphisms are functions between such sets. We use the skeleton of FinSet in order to ensure that all sets are finite and morphisms can be stored using lists of integers. Finite relations are built out of FinSet and can be used to do some relational algebra.

Catlab.CategoricalAlgebra.Sets.SetFunctionType

Abstract type for morphism in the category Set.

Every instance of SetFunction{<:SetOb{T},<:SetOb{T′}} is callable with elements of type T, returning an element of type T′.

Note: This type would be better called simply Function but that name is already taken by the base Julia type.

Catlab.CategoricalAlgebra.Sets.SetObType

Abstract type for object in the category Set.

The type parameter T is the element type of the set.

Note: This type is more abstract than the built-in Julia types AbstractSet and Set, which are intended for data structures for finite sets. Those are encompassed by the subtype FinSet.

Catlab.CategoricalAlgebra.FinSets.FinSetType

Finite set.

A finite set has abstract type FinSet{S,T}. The second type parameter T is the element type of the set and the first parameter S is the collection type, which can be a subtype of AbstractSet or another Julia collection type. In addition, the skeleton of the category FinSet is the important special case S = Int. The set ${1,…,n}$ is represented by the object FinSet(n) of type FinSet{Int,Int}.

Catlab.CategoricalAlgebra.FinSets.JoinAlgorithmType

Algorithm for limit of cospan or multicospan with feet being finite sets.

In the context of relational databases, such limits are called joins. The trivial join algorithm is NestedLoopJoin, which is algorithmically equivalent to the generic algorithm ComposeProductEqualizer. The algorithms HashJoin and SortMergeJoin are usually much faster. If you are unsure what algorithm to pick, use SmartJoin.

Catlab.CategoricalAlgebra.FinSets.TabularLimitType

Limit of finite sets viewed as a table.

Any limit of finite sets can be canonically viewed as a table (TabularSet) whose columns are the legs of the limit cone and whose rows correspond to elements of the limit object. To construct this table from an already computed limit, call TabularLimit(::AbstractLimit; ...). The column names of the table are given by the optional argument names.

In this tabular form, applying the universal property of the limit is trivial since it is just tupling. Thus, this representation can be useful when the original limit algorithm does not support efficient application of the universal property. On the other hand, this representation has the disadvantage of generally making the element type of the limit set more complicated.

Catlab.CategoricalAlgebra.FinSets.TabularSetType

Finite set whose elements are rows of a table.

The underlying table should be compliant with Tables.jl. For the sake of uniformity, the rows are provided as named tuples, which assumes that the table is not "extremely wide". This should not be a major limitation in practice but see the Tables.jl documentation for further discussion.

## Free Diagrams, Limits, and Colimts

The following modules define free diagrams in an arbitrary category and specify limit and colimt cones over said diagrams. Thes constructions enjoy the fullest support for FinSet and are used below to define presheaf categories as C-Sets. The general idea of these functions is that you set up a limit computation by specifying a diagram and asking for a limit or colimit cone, which is returned as a struct containing the apex object and the leg morphisms. This cone structure can be queried using the functions apex and legs. Julia's multiple dispatch feature is heavily used to specialize limit and colimit computations for various diagram shapes like product/coproduct and equalizer/coequalizer. As a consumer of this API, it is highly recommended that you use multiple dispatch to specialize your code on the diagram shape whenever possible.

Catlab.CategoricalAlgebra.FreeDiagramsModule

Free diagrams in a category.

A free diagram in a category is a diagram whose shape is a free category. Examples include the empty diagram, pairs of objects, discrete diagrams, parallel pairs, composable pairs, and spans and cospans. Limits and colimits are most commonly taken over free diagrams.

Catlab.CategoricalAlgebra.FreeDiagrams.BipartiteFreeDiagramType

A free diagram with a bipartite structure.

Such diagrams include most of the fixed shapes, such as spans, cospans, and parallel morphisms. They are also the generic shape of diagrams for limits and colimits arising from undirected wiring diagrams. For limits, the boxes correspond to vertices in $V₁$ and the junctions to vertices in $V₂$. Colimits are dual.

Catlab.CategoricalAlgebra.FreeDiagrams.BipartiteFreeDiagramMethod

Convert a free diagram to a bipartite free diagram.

Reduce a free diagram to a free bipartite diagram with the same limit (the default, colimit=false) or the same colimit (colimit=true). The reduction is essentially the same in both cases, except for the choice of where to put isolated vertices, where we follow the conventions described at cone_objects and cocone_objects. The resulting object is a bipartite free diagram equipped with maps from the vertices of the bipartite diagram to the vertices of the original diagram.

Catlab.CategoricalAlgebra.FreeDiagrams.bundle_legsMethod

Bundle together legs of a multi(co)span.

For example, calling bundle_legs(span, SVector((1,2),(3,4))) on a multispan with four legs gives a span whose left leg bundles legs 1 and 2 and whose right leg bundles legs 3 and 4. Note that in addition to bundling, this function can also permute legs and discard them.

The bundling is performed using the universal property of (co)products, which assumes that these (co)limits exist.

Catlab.CategoricalAlgebra.FreeDiagrams.cone_objectsMethod

Objects in diagram that will have explicit legs in limit cone.

In category theory, it is common practice to elide legs of limit cones that can be computed from other legs, especially for diagrams of certain fixed shapes. For example, when it taking a pullback (the limit of a cospan), the limit object is often treated as having two projections, rather than three. This function encodes such conventions by listing the objects in the diagram that will have corresponding legs in the limit object created by Catlab.

See also: cocone_objects.

Catlab.CategoricalAlgebra.Limits.colimitMethod

Colimit of a diagram.

To define colimits in a category with objects Ob, override the method colimit(::FreeDiagram{Ob}) for general colimits or colimit(::D) with suitable type D <: FixedShapeFreeDiagram{Ob} for colimits of specific shape, such as coproducts or coequalizers.

See also: limit

Catlab.CategoricalAlgebra.Limits.limitMethod

Limit of a diagram.

To define limits in a category with objects Ob, override the method limit(::FreeDiagram{Ob}) for general limits or limit(::D) with suitable type D <: FixedShapeFreeDiagram{Ob} for limits of specific shape, such as products or equalizers.

See also: colimit

Catlab.CategoricalAlgebra.Limits.pullbackMethod

Pullback of a pair of morphisms with common codomain.

To implement for a type T, define the method limit(::Cospan{T}) and/or limit(::Multicospan{T}) or, if you have already implemented products and equalizers, rely on the default implementation.

Catlab.CategoricalAlgebra.Limits.pushoutMethod

Pushout of a pair of morphisms with common domain.

To implement for a type T, define the method colimit(::Span{T}) and/or colimit(::Multispan{T}) or, if you have already implemented coproducts and coequalizers, rely on the default implementation.

Catlab.Theories.coequalizerMethod

Coequalizer of morphisms with common domain and codomain.

To implement for a type T, define the method colimit(::ParallelPair{T}) or colimit(::ParallelMorphisms{T}).

Catlab.Theories.copairMethod

Copairing of morphisms: universal property of coproducts/pushouts.

To implement for coproducts of type T, define the method universal(::BinaryCoproduct{T}, ::Cospan{T}) and/or universal(::Coproduct{T}, ::Multicospan{T}) and similarly for pushouts.

Catlab.Theories.coproductMethod

Coproduct of objects.

To implement for a type T, define the method colimit(::ObjectPair{T}) and/or colimit(::DiscreteDiagram{T}).

Catlab.Theories.createMethod

Unique morphism out of an initial object.

To implement for a type T, define the method universal(::Initial{T}, ::SMulticospan{0,T}).

Catlab.Theories.deleteMethod

Unique morphism into a terminal object.

To implement for a type T, define the method universal(::Terminal{T}, ::SMultispan{0,T}).

Catlab.Theories.equalizerMethod

Equalizer of morphisms with common domain and codomain.

To implement for a type T, define the method limit(::ParallelPair{T}) and/or limit(::ParallelMorphisms{T}).

Catlab.Theories.factorizeMethod

Factor morphism through (co)equalizer, via the universal property.

To implement for equalizers of type T, define the method universal(::Equalizer{T}, ::SMultispan{1,T}). For coequalizers of type T, define the method universal(::Coequalizer{T}, ::SMulticospan{1,T}).

Catlab.Theories.pairMethod

Pairing of morphisms: universal property of products/pullbacks.

To implement for products of type T, define the method universal(::BinaryProduct{T}, ::Span{T}) and/or universal(::Product{T}, ::Multispan{T}) and similarly for pullbacks.

Catlab.Theories.productMethod

Product of objects.

To implement for a type T, define the method limit(::ObjectPair{T}) and/or limit(::DiscreteDiagram{T}).

## Categories

Catlab.CategoricalAlgebra.CategoriesModule

2-category of categories, functors, and natural transformations.

Categories in mathematics appear in the large, often as categories of sets with extra structure, and in the small, as algebraic structures that generalize groups, monoids, preorders, and graphs. This division manifests in Catlab as well. Large categories (in spirit, if not in the technical sense) occur throughout Catlab as @instances of the theory of categories. For computational reasons, small categories are usually presented by generators and relations.

This module provides a minimal interface to accomodate both situations. Category instances are supported through the wrapper type TypeCat. Finitely presented categories are provided by another module, FinCats.

Catlab.CategoricalAlgebra.Categories.CatType

Abstract base type for a category.

The objects and morphisms in the category have Julia types Ob and Hom, respectively. Note that these types do not necessarily form an @instance of the theory of categories, as they may not meaningfully form a category outside the context of this object. For example, a finite category regarded as a reflexive graph with a composition operation might have type Cat{Int,Int}, where the objects and morphisms are numerical identifiers for vertices and edges in the graph.

The basic operations available in any category are: dom, codom, id, compose.

Catlab.CategoricalAlgebra.Categories.FunctorType

Abstract base type for a functor between categories.

A functor has a domain and a codomain (dom and codom), which are categories, and object and morphism maps, which can be evaluated using ob_map and hom_map. The functor object can also be called directly when the objects and morphisms have distinct Julia types. This is sometimes but not always the case (see Cat), so when writing generic code one should prefer the ob_map and hom_map functions.

Catlab.CategoricalAlgebra.Categories.TransformationType

Abstract base type for a natural transformation between functors.

A natural transformation $α: F ⇒ G$ has a domain $F$ and codomain $G$ (dom and codom), which are functors $F,G: C → D$ having the same domain $C$ and codomain $D$. The transformation consists of a component $αₓ: Fx → Gx$ in $D$ for each object $x ∈ C$, accessible using component or indexing notation (Base.getindex).

Catlab.CategoricalAlgebra.FinCatsModule

2-category of finitely presented categories.

This module is for the 2-category Cat what the module FinSets is for the category Set: a finitary, combinatorial setting where explicit calculations can be carried out. We emphasize that the prefix Fin means "finitely presented," not "finite," as finite categories are too restrictive a notion for many purposes. For example, the free category on a graph is finite if and only if the graph is DAG, which is a fairly special condition. This usage of Fin is also consistent with FinSet because for sets, being finite and being finitely presented are equivalent.

Catlab.CategoricalAlgebra.FinCats.FinCatGraphEqType

Category presented by a finite graph together with path equations.

The objects of the category are vertices in the graph and the morphisms are paths, quotiented by the congruence relation generated by the path equations. See (Spivak, 2014, Category theory for the sciences, §4.5).

Catlab.CategoricalAlgebra.FinCats.FinCatPresentationType

Category defined by a Presentation object.

The presentation type can, of course, be a category (Theories.Category). It can also be a schema (Theories.Schema). In this case, the schema's objects and attribute types are regarded as the category's objects and the schema's morphisms, attributes, and attribute types as the category's morphisms (where the attribute types are identity morphisms). When the schema is formalized as a profunctor whose codomain category is discrete, this amounts to taking the collage of the profunctor.

Catlab.CategoricalAlgebra.FinCats.FinTransformationType

A natural transformation whose domain category is finitely generated.

This type is for natural transformations $α: F ⇒ G: C → D$ such that the domain category $C$ is finitely generated. Such a natural transformation is given by a finite amount of data (one morphism in $D$ for each generating object of $C$) and its naturality is verified by finitely many equations (one equation for each generating morphism of $C$).

Catlab.CategoricalAlgebra.FinCats.is_functorialMethod

Is the purported functor on a presented category functorial?

This function checks that functor preserves domains and codomains. When check_equations is true (the default), it also checks that the functor preserves all equations in the domain category. Note that in some cases this may not be possible.

See also: is_natural.

Catlab.CategoricalAlgebra.FinCats.is_naturalMethod

Is the transformation between FinDomFunctors a natural transformation?

This function uses the fact that to check whether a transformation is natural, it suffices to check the naturality equations on a generating set of morphisms of the domain category. In some cases, checking the equations may be expensive or impossible. When the keyword argument check_equations is false, only the domains and codomains of the components are checked.

See also: is_functorial.

Catlab.CategoricalAlgebra.FinCats.ob_generatorsFunction

Object generators of finitely presented category.

The object generators are almost always the same as the objects. In principle, however, it is possible to have equations between objects, so that there are fewer objects than object generators.

## Acsets

### Overview and Theory

For an in depth look into the theory behind acsets, we refer the reader to our paper on the subject, which also gives some sense as to how the implementation works. Here, we give a brief tutorial before the the API documentation.

The most essential part of the acset machinery is the schema. The schema parameterizes the acset: each schema corresponds to a different type of acset. Schemas consist of four parts.

• Objects $X,Y$ ((X,Y,Z)::Ob)
• Homomorphisms $f \colon X \to Y$ (f :: Hom(X,Y)), which go from objects to objects
• Attribute types $\mathtt{T}$ (T :: AttrType)
• Attributes $a \colon X \to \mathtt{T}$ (a :: Attr(X,T)), which go from objects to data types

For those with a categorical background, a schema is a presentation of a category $|S|$ along with a functor $S$ from $|S|$ to the arrow category $0 \to 1$, such that $S^{-1}(1)$ is discrete.

An acset $F$ on a schema consists of...

• a set $F(X)$ of "primary keys" for each object
• a function $F(f) \colon F(X) \to F(Y)$ for each morphism
• a Julia data type $F(\mathtt{T})$ for each data type $\mathtt{T}$
• a function $F(a) \colon F(X) \to F(\mathtt{T})$ for each attribute $a$.

For those with a categorical background, an acset on a schema $S$ consists of a functor from $S$ to $\mathsf{Set}$, such that objects in $S^{-1}(0)$ map to finite sets, and objects in $S^{-1}(1)$ map to sets that represent types. For any particular functor $K \colon S^{-1}(1) \to \mathsf{Set}$, we can also take the category of acsets that restrict to this map on $S^{-1}$.

We can also add relations to this presentation, but we currently do nothing with those relations in the implementation; they mostly serve as documentation.

We will now give an example of how this all works in practice.

using Catlab, Catlab.CategoricalAlgebra

# Write down the schema for a weighted graph
@present TheoryWeightedGraph(FreeSchema) begin
V::Ob
E::Ob
src::Hom(E,V)
tgt::Hom(E,V)
T::AttrType
weight::Attr(E,T)
end

# Construct the type used to store acsets on the previous schema
# We *index* src and tgt, which means that we store not only
# the forwards map, but also the backwards map.
@acset_type WeightedGraph(TheoryWeightedGraph, index=[:src,:tgt])

# Construct a weighted graph, with floats as edge weights
g = @acset WeightedGraph{Float64} begin
V = 4
E = 5
src = [1,1,1,2,3]
tgt = [2,3,4,4,4]
weight = [7.2, 9.3, 9.4, 0.1, 42.0]
end
Main.##ex-#312.WeightedGraph{Float64} with elements V = 1:4, E = 1:5
E src tgt weight
1 1 2 7.2
2 1 3 9.3
3 1 4 9.4
4 2 4 0.1
5 3 4 42.0

### API

We first give an overview of the data types used in the acset machinery.

FreeSchema A finite presentation of a category that will be used as the schema of a database in the algebraic databases conception of categorical database theory. Functors out of a schema into FinSet are combinatorial structures over the schema. Attributes in a schema allow you to encode numerical (any julia type) into the database. You can find several examples of schemas in Catlab.Graphs where they define categorical versions of graph theory.

CSet/AttributedCSet is a struct/constructors whose values (tables, indices) are parameterized by a CatDesc/AttrDesc. These are in memory databases over the schema equiped with ACSetTranformations as natural transformations that encode relationships between database instances.

CSetType/AttributedCSetTypeprovides a function to construct a julia type for ACSet instances, parameterized by CatDesc/AttrDesc. This function constructs the new type at runtime. In order to have the interactive nature of Julia, and to dynamically construct schemas based on runtime values, we need to define new Julia types at runtime. This function converts the schema spec to the corresponding Julia type.

CatDesc/AttrDesc the encoding of a schema into a Julia type. These exist because Julia only allows certain kinds of data in the parameter of a dependent type. Thus, we have to serialize a schema into those primitive data types so that we can use them to parameterize the ACSet type over the schema. This is an implementation detail subject to complete overhaul.

Catlab.CategoricalAlgebra.CSets.ACSetTransformationType

Transformation between attributed C-sets.

Homomorphisms of attributed C-sets generalize homomorphisms of C-sets (CSetTransformation), which the user should understand before reading further.

A homomorphism of attributed C-sets with schema S: C ↛ A (a profunctor) is a natural transformation between the corresponding functors col(S) → Set, where col(S) is the collage of S. When the components on attribute types, indexed by objects of A, are all identity functions, the morphism is called tight; in general, it is called loose. The terms "tight" and "loose" come from what the nLab calls an "M-category". The category of acsets on a fixed schema S is an M-category. Calling ACSetTransformation will construct a tight or loose morphism as appropriate, depending on which components are specified.

Since every tight morphism can be considered a loose one, the distinction between tight and loose may seem an unimportant technicality, but it can have important consequences because choosing one or the other greatly affects limits and colimits of acsets. In practice, the tight morphisms suffice for most purposes, including computing colimits. However, when computing limits of acsets, the loose morphism are usually preferable.

Catlab.CategoricalAlgebra.CSets.BacktrackingSearchType

Find attributed $C$-set homomorphisms using backtracking search.

This procedure uses the classic backtracking search algorithm for a combinatorial constraint satisfaction problem (CSP). As is well known, the homomorphism problem for relational databases is reducible to CSP. Since the C-set homomorphism problem is "the same" as the database homomorphism problem (insofar as attributed C-sets are "the same" as relational databases), it is also reducible to CSP. Backtracking search for CSP is described in many computer science textbooks, such as (Russell & Norvig 2010, Artificial Intelligence, Third Ed., Chapter 6: Constraint satisfaction problems, esp. Algorithm 6.5). In our implementation, the search tree is ordered using the popular heuristic of "minimum remaining values" (MRV), also known as "most constrained variable.

Catlab.CategoricalAlgebra.CSets.CSetTransformationType

Transformation between C-sets.

Recall that a C-set homomorphism is a natural transformation: a transformation between functors C → Set satisfying the naturality axiom for every (generating) morphism in C.

This data type records the data of a C-set transformation. Naturality is not strictly enforced but is expected to be satisfied. It can be checked using the function is_natural.

Catlab.CategoricalAlgebra.CSets.HomomorphismQueryType

Find attributed $C$-set homomorphisms using a conjunctive query.

This algorithm evaluates a conjunctive query (limit in FinSet) to find all homomorphisms between two $C$-sets. In fact, conjunctive queries are exactly the representable functors from $C$-sets to sets, so every conjunctive query arises in this way, with the caveat that conjunctive queries may correspond to to infinite $C$-sets when $C$ is infinite (but possibly finitely presented).

Catlab.CategoricalAlgebra.CSets.homomorphismMethod

Find a homomorphism between two attributed $C$-sets.

Returns nothing if no homomorphism exists. For many categories $C$, the $C$-set homomorphism problem is NP-complete and thus this procedure generally runs in exponential time. It works best when the domain object is small.

To restrict to monomorphisms, or homomorphisms whose components are all injective functions, set the keyword argument monic=true. To restrict only certain components to be injective or bijective, use monic=[...] or iso=[...]. For example, setting monic=[:V] for a graph homomorphism ensures that the vertex map is injective but imposes no constraints on the edge map.

To restrict the homomorphism to a given partial assignment, set the keyword argument initial. For example, to fix the first source vertex to the third target vertex in a graph homomorphism, set initial=(V=Dict(1 => 3),).

Use the keyword argument alg to set the homomorphism-finding algorithm. By default, a backtracking search algorithm is used (BacktrackingSearch).

See also: homomorphisms, isomorphism.

Catlab.CategoricalAlgebra.StructuredCospans.StructuredCospanType

Structured cospan.

The first type parameter L encodes a functor L: A → X from the base category A, often FinSet, to a category X with "extra structure." An L-structured cospan is then a cospan in X whose feet are images under L of objects in A. The category X is assumed to have pushouts.

Structured cospans form a double category with no further assumptions on the functor L. To obtain a symmetric monoidal double category, L must preserve finite coproducts. In practice, L usually has a right adjoint R: X → A, which implies that L preserves all finite colimits. It also allows structured cospans to be constructed more conveniently from an object x in X plus a cospan in A with apex R(x).

See also: StructuredMulticospan`.