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Instant-use add-on functions for the Wolfram Language
Function Repository Resource:
Represent the abstract coproduct of an arbitrary collection of objects in an abstract category
ResourceFunction["AbstractCoproduct"][ob,coprod,inj,comp,id] represents the abstract coproduct of the collection of objects ob, with coproduct symbol coprod, injection morphism names inj, composition symbol comp and identity symbol id. | |
ResourceFunction["AbstractCoproduct"][…,obuniv,morphuniv,morphuniq] represents an abstract coproduct with a universal property characterized by the universal object obuniv, universal morphisms morphuniv and unique morphism morphuniq. | |
ResourceFunction["AbstractCoproduct"][assoc] represents an abstract coproduct using the association of object list, coproduct symbol, injection morphism list, composition symbol, identity symbol, universal object, universal morphism list and unique morphism assoc. | |
ResourceFunction["AbstractCoproduct"][…][AbstractCategory[…]] embeds an abstract coproduct into the specified abstract category. | |
ResourceFunction["AbstractCoproduct"][ResourceFunction["AbstractCoproduct"][…],coprod,comp,id] makes a new abstract coproduct from an old coproduct by imposing new coproduct symbol coprod, new composition symbol comp and new identity symbol id. |
"Objects" | list of objects in the abstract coproduct |
"ObjectCount" | number of objects in the abstract coproduct |
"MorphismAssociation" | association of morphisms/edges in the abstract coproduct |
"MorphismNames" | list of names of morphisms in the abstract coproduct |
"MorphismEdges" | list of directed edges associated with morphisms in the abstract coproduct |
"MorphismCount" | number of morphisms in the abstract coproduct |
"SimpleMorphismAssociation" | association of morphism names/edges in the abstract coproduct with self-loops removed |
"SimpleMorphismNames" | list of names of morphisms in the abstract coproduct with self-loops removed |
"SimpleMorphismEdges" | list of directed edges associated with morphisms in the abstract coproduct with self-loops removed |
"SimpleMorphismCount" | number of morphisms in the abstract coproduct with self-loops removed |
"UniversalObjects" | list of objects in the abstract coproduct characterizing the universal property |
"UniversalObjectCount" | number of objects in the abstract coproduct characterizing the universal property |
"UniversalMorphismAssociation" | association of morphism names/edges in the abstract coproduct characterizing the universal property |
"UniversalMorphismNames" | list of names of morphisms in the abstract coproduct characterizing the universal property |
"UniversalMorphismEdges" | list of directed edges associated with morphisms in the abstract coproduct characterizing the universal property |
"UniversalMorphismCount" | number of morphisms in the abstract coproduct characterizing the universal property |
"UniversalReducedMorphismAssociation" | association of morphism names/edges the abstract coproduct characterizing the universal property, modulo morphism equivalences |
"UniversalReducedMorphismNames" | list of names of morphisms in the abstract coproduct characterizing the universal property, modulo morphism equivalences |
"UniversalReducedMorphismEdges" | list of directed edges associated with morphisms in the abstract coproduct characterizing the universal property, modulo morphism equivalences |
"UniversalReducedMorphismCount" | number of morphisms in the abstract coproduct characterizing the universal property, modulo morphism equivalences |
"UniversalSimpleMorphismAssociation" | association of morphism names/edges in the abstract coproduct characterizing the universal property, with self-loops and multiedges removed |
"UniversalSimpleMorphismNames" | list of names of morphisms in the abstract coproduct characterizing the universal property, with self-loops and multiedges removed |
"UniversalSimpleMorphismEdges" | list of edges associated with morphisms in the abstract coproduct characterizing the universal property, with self-loops and multiedges removed |
"UniversalSimpleMorphismCount" | number of morphisms in the abstract coproduct characterizing the universal property, with self-loops and multiedges removed |
"UniversalReducedSimpleMorphismAssociation" | association of morphism names/edges in the abstract coproduct characterizing the universal property, with self-loops and multiedges removed, and modulo morphism equivalences |
"UniversalReducedSimpleMorphismNames" | list of names of morphisms in the abstract coproduct characterizing the universal property, with self-loops and multiedges removed, and modulo morphism equivalences |
"UniversalReducedSimpleMorphismEdges" | list of directed edges associated with morphisms in the abstract coproduct characterizing the universal property, with self-loops and multiedges removed, and modulo morphism equivalences |
"UniversalReducedSimpleMorphismCount" | number of morphisms in the abstract coproduct characterizing the universal property, with self-loops and multiedges removed, and modulo morphism equivalences |
"CoproductSymbol" | symbol used to denote coproducts of objects (should be an operator of arbitrary arity) |
"CompositionSymbol" | symbol used to denote morphism composition (should be a binary operator) |
"IdentitySymbol" | symbol used to denote identity morphisms (should be a unary operator) |
"CoproductCategory" | abstract category with morphisms representing the abstract coproduct |
"UniversalCoproductCategory" | abstract category with morphisms characterizing the universal property of the abstract coproduct |
"UniversalCoproductEquations" | list of equations implicitly imposed by the requirement that the universal property of the abstract coproduct is satisfied |
"FullLabeledGraph" | directed graph form of the abstract coproduct with labels on the morphisms |
"FullUnlabeledGraph" | directed graph form of the abstract coproduct with no labels on the morphisms |
"SimpleLabeledGraph" | directed graph form of the abstract coproduct, with self-loops removed, with labels on the morphisms |
"SimpleUnlabeledGraph" | directed graph form of the abstract coproduct, with self-loops removed, with no labels on the morphisms |
"UniversalFullLabeledGraph" | directed graph form characterizing the universal property of the abstract coproduct, with labels on the morphisms |
"UniversalFullUnlabeledGraph" | directed graph form characterizing the universal property of the abstract coproduct, with no labels on the morphisms |
"UniversalReducedLabeledGraph" | directed graph form characterizing the universal property of the abstract coproduct, modulo morphism equivalences, with labels on the morphisms |
"UniversalReducedUnlabeledGraph" | directed graph form characterizing the universal property of the abstract coproduct, modulo morphism equivalences, with no labels on the morphisms |
"UniversalSimpleLabeledGraph" | directed graph form characterizing the universal property of the abstract coproduct, with self-loops and multiedges removed, with labels on the morphisms |
"UniversalSimpleUnlabeledGraph" | directed graph form characterizing the universal property of the abstract coproduct, with self-loops and multiedges removed, with no labels on the morphisms |
"UniversalReducedSimpleLabeledGraph" | directed graph form characterizing the universal property of the abstract coproduct, with self-loops and multiedges removed, modulo morphism equivalences, with labels on the morphisms |
"UniversalReducedSimpleUnlabeledGraph" | directed graph form characterizing the universal property of the abstract coproduct, with self-loops and multiedges removed, modulo morphism equivalences, with no labels on the morphisms |
"AssociationForm" | abstract coproduct represented as an association of a list of objects, a coproduct symbol, a list of injection morphism names, a composition symbol, an identity symbol, a universal object name, a list of universal morphism names and a unique morphism name |
"Properties" | list of properties |
Construct a simple abstract coproduct of objects A and B:
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Show the full directed graph with labels on the morphisms:
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Show the full directed graph without labels on the morphisms:
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Show the full directed graph characterizing the universal property, with labels on the morphisms:
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Show the full directed graph characterizing the universal property, without labels on the morphisms:
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Show the Association of morphisms:
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Show the Association of (quantified) morphisms characterizing the universal property:
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Show the list of directed edges characterizing the universal property:
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Show the list of (quantified) equations characterizing the universal property:
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Construct an abstract coproduct of objects U, V and W, but with coproduct symbol ⊕, injection morphisms j1, j2 and j3, composition and identity symbols ⊗ and ─, and with a universal property characterized by universal object Q, universal morphisms m1, m2 and m3 and unique morphism u:
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Show the full directed graph characterizing the universal property, with labels on the morphisms:
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Show the full directed graph characterizing the universal property, without labels on the morphisms:
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Show the reduced directed graph characterizing the universal property, with all equivalences between morphisms imposed and with labels on the morphisms:
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Show the reduced directed graph characterizing the universal property, without labels on the morphisms:
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Show the Association of (quantified) morphisms characterizing the universal property, with all equivalences between morphisms imposed:
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Show the list of (quantified) equations characterizing the universal property:
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Show the association form of the abstract coproduct:
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Construct an abstract coproduct of objects X and Y:
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Construct an abstract category consisting of morphisms from X and Y to some common object Z:
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Embed the abstract coproduct into the abstract category (in such a way that the universal property is satisfied):
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Validate that the universal property is indeed satisfied:
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Construct a more complicated abstract category consisting of additional morphisms from X and Y to a further common object W:
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Embed the abstract coproduct into the more complicated abstract category and validate that the universal property is still satisfied:
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Compute minimal abstract category representations of the abstract coproduct and its defining universal property, respectively:
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Abstract coproducts can be constructed from a list of objects, a coproduct symbol, a list of names of injection morphisms, a composition symbol and an identity symbol:
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A universal object symbol, a list of universal morphisms and a unique morphism symbol can also be specified, in order to characterize the universal property fully:
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Abstract coproducts can also be constructed directly from a list of objects and a list of names of injection morphisms (plus composition and identity symbols):
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Or from an object list with coproduct, composition and identity symbols alone:
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If any of these arguments is not fully specified, the appropriate defaults are assumed automatically:
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New coproduct, composition and identity symbols can be specified for any existing abstract coproduct:
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From an explicit association:
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Construct an abstract coproduct of objects X, Y, Z and W:
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Show the list of properties:
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Show the list of objects in the abstract coproduct:
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Show the number of objects in the abstract coproduct:
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Show the association of morphisms in the abstract coproduct:
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Show the list of names of morphisms in the abstract coproduct:
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Show the list of directed edges corresponding to morphisms in the abstract coproduct:
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Show the number of morphisms in the abstract coproduct:
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Show the association of simple morphisms (with all self-loops removed) in the abstract coproduct:
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Show the list of names of simple morphisms (with all self-loops removed) in the abstract coproduct:
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Show the list of directed edges corresponding to simple morphisms (with all self-loops removed) in the abstract coproduct:
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Show the number of simple morphisms (with all self-loops removed) in the abstract coproduct:
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Show the list of objects characterizing the universal property of the abstract coproduct:
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Show the number of objects characterizing the universal property of the abstract coproduct:
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Show the association of morphisms characterizing the universal property of the abstract coproduct:
In[58]:= | ![]() |
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Show the list of names of morphisms characterizing the universal property of the abstract coproduct:
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Show the list of directed edges corresponding to morphisms characterizing the universal property of the abstract coproduct:
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Show the number of morphisms characterizing the universal property of the abstract coproduct:
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Show the association of reduced morphisms (modded out by all morphism equivalences) characterizing the universal property of the abstract coproduct:
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Show the list of names of reduced morphisms (modded out by all morphism equivalences) characterizing the universal property of the abstract coproduct:
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Show the list of directed edges corresponding to reduced morphisms (modded out by all morphism equivalences) characterizing the universal property of the abstract coproduct:
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Show the number of reduced morphisms (modded out by all morphism equivalences) characterizing the universal property of the abstract coproduct:
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Show the association of simple morphisms (with all self-loops and multiedges removed) characterizing the universal property of the abstract coproduct:
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Show the list of names of simple morphisms (with all self-loops and multiedges removed) characterizing the universal property of the abstract coproduct:
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Show the list of directed edges corresponding to simple morphisms (with all self-loops and multiedges removed) characterizing the universal property of the abstract coproduct:
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Show the number of simple morphisms (with all self-loops and multiedges removed) characterizing the universal property of the abstract coproduct:
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Show the association of reduced simple morphisms (with all self-loops and multiedges removed, plus all morphism equivalences modded out) characterizing the universal property of the abstract coproduct:
In[70]:= | ![]() |
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Show the list of names of reduced simple morphisms (with all self-loops and multiedges removed, plus all morphism equivalences modded out) characterizing the universal property of the abstract coproduct:
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Show the list of directed edges corresponding to reduced simple morphisms (with all self-loops and multiedges removed, plus all morphism equivalences modded out) characterizing the universal property of the abstract coproduct:
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Show the number of reduced simple morphisms (with all self-loops and multiedges removed, plus all morphism equivalences modded out) characterizing the universal property of the abstract coproduct:
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Show the (arbitrary arity) symbol used to denote the abstract coproduct operation:
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Show the (binary) symbol used to denote morphism composition in the abstract coproduct:
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Show the (unary) symbol used to denote identity morphisms on objects in the abstract coproduct:
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Compute the abstract category whose morphisms represent the abstract coproduct:
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Compute the abstract category whose morphisms characterize the universal property of the abstract coproduct:
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Show the list of equations imposed implicitly by the requirement that the universal property of the abstract coproduct is satisfied:
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Show the full directed graph with labels on the morphisms of the abstract coproduct:
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Show the full directed graph without labels on the morphism of the abstract coproduct:
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Show the simple directed graph, with all self-loops removed and with labels on the morphisms of the abstract coproduct:
In[82]:= | ![]() |
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Show the simple directed graph, with all self-loops removed and without labels on the morphisms of the abstract coproduct:
In[83]:= | ![]() |
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Show the full directed graph characterizing the universal property of the abstract coproduct, with labels on the morphisms:
In[84]:= | ![]() |
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Show the full directed graph characterizing the universal property of the abstract coproduct, without labels on the morphisms:
In[85]:= | ![]() |
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Show the reduced directed graph characterizing the universal property of the abstract coproduct, with all equivalences between morphisms imposed and with labels on the morphisms:
In[86]:= | ![]() |
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Show the reduced directed graph characterizing the universal property of the abstract coproduct, with all equivalences between morphisms imposed and without labels on the morphisms:
In[87]:= | ![]() |
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Show the simple directed graph characterizing the universal property of the abstract coproduct, with all self-loops and multiedges removed and with labels on the morphisms:
In[88]:= | ![]() |
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Show the simple directed graph characterizing the universal property of the abstract coproduct, with all self-loops and multiedges removed and without labels on the morphisms:
In[89]:= | ![]() |
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Show the reduced simple directed graph characterizing the universal property of the abstract coproduct, with all self-loops and multiedges removed, plus all equivalences between morphisms modded out and with labels on the morphisms:
In[90]:= | ![]() |
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Show the reduced simple directed graph characterizing the universal property of the abstract coproduct, with all self-loops and multiedges removed, plus all equivalences between morphisms modded out and without labels on the morphisms:
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Show the explicit association form of the coproduct:
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