Basic Examples (8) 

Construct a simple abstract category from an Association of arrows/morphisms:

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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 Association of morphisms:

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Show the list of directed edges:

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Construct the same abstract category, but with new composition and identity symbols ⊗ and ─, and with an equivalence imposed between objects X and Y and morphisms g⊗f and g:

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Show the reduced directed graph, with all equivalences between objects and morphisms imposed and with labels on the morphisms:

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Show the reduced directed graph without labels on the morphisms:

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Show the Association of morphisms, with all equivalences between objects and morphisms imposed:

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Show the list of equivalences between objects:

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Show the list of equivalences between morphisms:

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Show the association form of the abstract category:

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Construct an abstract category from a square-shaped quiver using the resource function AbstractQuiver:

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The corresponding abstract category:

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Show that the resulting square is not commutative:

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Compute the list of equations between morphisms that must hold for the square to be commutative:

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Construct the same abstract category, but with the aforementioned equivalence between morphisms h⊙f and i⊙g imposed:

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Show that the resulting square is now commutative:

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Construct an abstract category from an oblong-shaped quiver:

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The corresponding abstract category:

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Compute the list of equations between morphisms that must hold for the oblong to be commutative:

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Compute the list of equations between morphisms that must hold by virtue of morphism composition being associative:

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Compute the list of equations between morphisms that must hold by virtue of all objects having an identity morphism:

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Construct an abstract category consisting of a pair of parallel morphisms g₁ and g₂ and a third morphism f leading out from them, together with an equivalence between morphisms f⊙g₁ and f⊙g₂:

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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 reduced directed graph, with all equivalences between objects and morphisms imposed and with labels on the morphisms:

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Show the reduced directed graph without labels on the morphisms:

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Show the association of monomorphisms (i.e. left-cancellative/injective morphisms), illustrating that f is not a monomorphism:

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Construct the same abstract category, but with an additional equivalence imposed between morphisms g₁ and g₂, thus forcing f to be a monomorphism:

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Show the reduced directed graph, with all equivalences between objects and morphisms imposed and with labels on the morphisms:

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Show the reduced directed graph without labels on the morphisms:

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Show that f is now a monomorphism:

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Show the association of monomorphisms modulo all equivalences between objects and morphisms:

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Every monomorphism is an epimorphism (i.e. a right-cancellative/surjective morphism) in the dual category:

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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 association of bimorphisms (i.e. left- and right-cancellative/bijective morphisms) in the original category, illustrating that f is not a bimorphism:

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Show the association of bimorphisms modulo all equivalences between objects and morphisms:

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Construct an abstract category consisting of a pair of morphisms f and g leading between objects X and Y in opposite directions:

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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 that neither f nor g is a retraction (i.e. a left inverse) nor a section (i.e. a right inverse), and therefore also neither is an isomorphism (i.e. a left and right inverse):

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Construct the same abstract category, but with an equivalence imposed between morphisms g⊙f and the identity on object X:

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Show the reduced directed graph, with all equivalences between objects and morphisms imposed and with labels on the morphisms:

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Show the reduced labeled graph without labels on the morphisms:

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Show that g is now a retraction and f is now a section, though neither is an isomorphism:

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Show, equivalently, that g is a left inverse of f and f is a right inverse of g, but neither is a full inverse of the other:

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Construct the same abstract category, but with an additional equivalence imposed between morphisms f⊙g and the identity on object Y:

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Show the reduced directed graph, with all equivalences between objects and morphisms imposed and with labels on the morphisms:

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Show the reduced directed graph without labels on the morphisms:

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Show that f is now a retraction and g is now a section, and both are now isomorphisms:

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Show, equivalently, that f is now both a left and right inverse of g, and vice versa:

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Show the associations of retractions, sections and isomorphisms modulo all equivalences between objects and morphisms:

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Show the associations of left inverses, right inverses and "full" inverses modulo all equivalences between objects and morphisms:

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Show that the category is therefore a groupoid (i.e. all morphisms are isomorphisms):

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Construct a new abstract category with two-way arrows between objects X, Y and Z:

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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 that the category is not a groupoid:

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Compute the list of equations between morphisms that must hold for the category to be a groupoid:

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Show the associations of endomorphisms (morphisms from an object to itself) and automorphisms (isomorphisms from an object to itself):

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Construct an abstract category consisting of exactly one morphism from object X to every morphism in the category, together with a single morphism to X, modulo all equivalences between morphisms:

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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 reduced directed graph, with all equivalences between objects and morphisms imposed and with labels on the morphisms:

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Show the reduced directed graph without labels on the morphisms:

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Show the association of initial objects (i.e. objects with exactly one outgoing morphism to every object):

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Show that X is an initial object, modulo all equivalences between morphisms:

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Show that X is also a strict initial object (i.e. every incoming morphism to X is an isomorphism) when one mods out by all equivalences between morphisms:

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Every initial object is a terminal object (i.e. an object with exactly one incoming morphism to every object) in the dual category:

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Likewise with strictly initial objects being strictly terminal objects (i.e. terminal objects for which every outgoing morphism is an isomorphism) in the dual category:

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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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In both cases, X is not a zero object (i.e. it is not both initial and terminal) nor a strict zero object (i.e. it is not both strictly initial and strictly terminal):

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Show, therefore, that the category is not pointed (i.e. it does not contain a zero object):

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Construct an abstract category consisting of a pair of parallel morphisms g₁ and g₂ and a third morphism f leading out from them:

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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 that f is not a constant morphism (i.e. does not yield the same result when composed on the right by either member of a parallel pair of morphisms):

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Construct the same abstract category, but with an additional equivalence imposed between morphisms f⊙g₁ and f⊙g₂, thus forcing f to be constant:

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Show the reduced directed graph, with all equivalences between objects and morphisms imposed and with labels on the morphisms:

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Show the reduced directed graph without labels on the morphisms:

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Show that f is now a constant morphism:

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Show the association of constant morphisms modulo all equivalences between objects and morphisms:

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Every constant morphism is a co-constant morphism (i.e. a morphism that yields the same result when composed on the left by either member of a parallel pair of morphisms) in the dual category:

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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 association of zero morphisms (i.e. morphisms that are both constant and co-constant), illustrating that f is also zero:

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Show the association of zero morphisms modulo all equivalences between objects and morphisms:

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Scope (4) 

Abstract categories can be constructed directly from abstract quiver objects (by enforcing the composition and identity axioms on the arrows and objects of the underlying quiver):

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Additional arguments can be used to specify the composition and identity symbols:

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Alternatively, additional arguments can be used to specify equivalences between objects and morphisms:

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All four properties (composition symbol, identity symbol, object equivalences and morphism equivalences) may be specified simultaneously:

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Abstract categories can also be constructed directly from directed graph objects (with automatically-selected arrow/morphism names):

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In a similar way to the above, additional arguments can be used to specify composition and identity symbols, as well as equivalences between the resulting objects and morphisms:

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Abstract categories can be constructed from lists of arrows (with automatically-selected arrow/morphism names):

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From associations of arrows/morphisms (with explicitly provided arrow/morphism names):

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With explicit lists of objects specified:

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Composition and identity symbols can be specified for any abstract category:

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Likewise, lists of object equivalences and arrow equivalences can be specified for any abstract category:

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New composition and identity symbols can be specified for any existing abstract category:

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Likewise for new object and morphism equivalences:

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From an explicit association:

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Construct an abstract category from an association of arrows/morphisms with additional object and morphism equivalences:

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Show the list of properties:

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Show the list of objects in the abstract category:

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Show the number of objects in the abstract category:

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Show the association of morphisms in the abstract category:

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Show the list of names of morphisms in the abstract category:

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Show the list of directed edges corresponding to morphisms in the abstract category:

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Show the number of morphisms in the abstract category:

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Show the list of equivalences between objects in the abstract category:

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Show the number of equivalences between objects in the abstract category:

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Show the list of equivalences between morphisms in the abstract category:

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Show the number of equivalences between morphisms in the abstract category:

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Show the list of reduced objects (modded out by all object equivalences) in the abstract category:

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Show the number of reduced objects (modded out by all object equivalences) in the abstract category:

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Show the association of reduced morphisms (modded out by all object and morphism equivalences) in the abstract category:

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Show the list of names of reduced morphisms (modded out by all morphism equivalences) in the abstract category:

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Show the list of directed edges corresponding to reduced morphisms (modded out by all object and morphism equivalences) in the abstract category:

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Show the number of reduced morphisms (modded out by all object and morphism equivalences) in the abstract category:

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Show the association of simple morphisms (with all self-loops and multiedges removed) in the abstract category:

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Show the list of names of simple morphisms (with all self-loops and multiedges removed) in the abstract category:

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Show the list of directed edges corresponding to simple morphisms (with all self-loops and multiedges removed) in the abstract category:

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Show the number of simple morphisms (with all self-loops and multiedges removed) in the abstract category:

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Show the association of reduced simple morphisms (with all self-loops and multiedges removed, plus all object and morphism equivalences modded out) in the abstract category:

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Show the list of names of reduced simple morphisms (with all self-loops and multiedges removed, plus all object and morphism equivalences modded out) in the abstract category:

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Show the list of directed edges corresponding to reduced simple morphisms (with all self-loops and multiedges removed, plus all object and morphism equivalences modded out) in the abstract category:

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Show the number of reduced simple morphisms (with all self-loops and multiedges removed, plus all object and morphism equivalences modded out) in the abstract category:

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Show the (binary) symbol used to denote morphism composition in the abstract category:

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Show the (unary) symbol used to denote identity morphisms on objects in the abstract category:

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Compute the dual category (obtained by swapping the source and target objects of each morphism and reversing the direction of morphism composition):

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Show the abstract quiver forming the underlying structure of the abstract category:

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Show the list of equations imposed implicitly by the axiom that morphism composition in the abstract category must be associative:

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Show the list of equations imposed implicitly by the axiom that morphism composition in the abstract category must be associative, modulo all equivalences between objects and morphisms:

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Show the list of equations imposed implicitly by the axiom that every object in the abstract category must have an identity morphism:

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Show the list of equations imposed implicitly by the axiom that every object in the abstract category must have an identity morphism, modulo all equivalences between objects and morphisms:

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Determine whether the abstract category is a commutative diagram (i.e. all directed paths through the graph with the same source and target object yield the same morphism):

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Show the list of equations between morphisms required to force the abstract category to be a commutative diagram:

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Show the list of equations between morphisms required to force the abstract category to be a commutative diagram, modulo all equivalences between objects and morphisms:

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Show the association of monomorphisms (i.e. left-cancellative/injective morphisms) in the abstract category:

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Show the association of monomorphisms (i.e. left-cancellative/injective morphisms) in the abstract category, modulo all equivalences between objects and morphisms:

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Show the association of epimorphisms (i.e. right-cancellative/surjective morphisms) in the abstract category:

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Show the association of epimorphisms (i.e. right-cancellative/surjective morphisms) in the abstract category, modulo all equivalences between objects and morphisms:

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Show the association of bimorphisms (i.e. left- and right-cancellative/bijective morphisms) in the abstract category:

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Show the association of bimorphisms (i.e. left- and right-cancellative/bijective morphisms) in the abstract category, modulo all equivalences between objects and morphisms:

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Show the association of retractions (i.e. left inverse morphisms) in the abstract category:

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Show the association of retractions (i.e. left inverse morphisms) in the abstract category, modulo all equivalences between objects and morphisms:

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Show the association of left inverse morphisms (i.e. retractions) for each morphism in the abstract category:

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Show the association of left inverse morphisms (i.e. retractions) for each morphism in the abstract category, modulo all equivalences between objects and morphisms:

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Show the association of sections (i.e. right inverse morphisms) in the abstract category:

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Show the association of sections (i.e. right inverse morphisms) in the abstract category, modulo all equivalences between objects and morphisms:

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Show the association of right inverse morphisms (i.e. sections) for each morphism in the abstract category:

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Show the association of right inverse morphisms (i.e. sections) for each morphism in the abstract category, modulo all equivalences between objects and morphisms:

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Show the association of isomorphisms (i.e. left and right inverse morphisms) in the abstract category:

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Show the association of isomorphisms (i.e. left and right inverse morphisms) in the abstract category, modulo all equivalences between objects and morphisms:

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Show the association of inverse morphisms (i.e. isomorphisms) for each morphism in the abstract category:

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Show the association of inverse morphisms (i.e. isomorphisms) for each morphism in the abstract category, modulo all equivalences between objects and morphisms:

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Show the association of endomorphisms (i.e. morphisms from an object to itself) in the abstract category:

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Show the association of endomorphisms (i.e. morphisms from an object to itself) in the abstract category, modulo all equivalences between objects and morphisms:

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Show the association of automorphisms (i.e. isomorphisms from an object to itself) in the abstract category:

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Show the association of automorphisms (i.e. isomorphisms from an object to itself) in the abstract category, modulo all equivalences between objects and morphisms:

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Show the association of initial objects (i.e. objects with exactly one outgoing morphism to each object) in the abstract category:

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Show the association of initial objects (i.e. objects with exactly one outgoing morphism to each object) in the abstract category, modulo all equivalences between objects and morphisms:

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Show the association of strict initial objects (i.e. initial objects whose incoming morphisms are all isomorphisms) in the abstract category:

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Show the association of strict initial objects (i.e. initial objects whose incoming morphisms are all isomorphisms) in the abstract category, modulo all equivalences between objects and morphisms:

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Show the association of terminal objects (i.e. objects with exactly one incoming morphism from each object) in the abstract category:

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Show the association of terminal objects (i.e. objects with exactly one incoming morphism from each object) in the abstract category, modulo all equivalences between objects and morphisms:

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Show the association of strict terminal objects (i.e. terminal objects whose outgoing morphisms are all isomorphisms) in the abstract category:

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Show the association of strict terminal objects (i.e. terminal objects whose outgoing morphisms are all isomorphisms) in the abstract category, modulo all equivalences between objects and morphisms:

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Show the association of zero objects (i.e. objects with exactly one outgoing morphism and one incoming morphism to/from each object) in the abstract category:

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Show the association of zero objects (i.e. objects with exactly one outgoing morphism and one incoming morphism to/from each object) in the abstract category, modulo all equivalences between objects and morphisms:

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Show the association of strict zero objects (i.e. zero objects whose incoming and outgoing morphisms are all isomorphisms) in the abstract category:

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Show the association of strict zero objects (i.e. zero objects whose incoming and outgoing morphisms are all isomorphisms) in the abstract category, modulo all equivalences between objects and morphisms:

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Determine whether the abstract category is pointed (i.e. contains a zero object):

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Show the association of constant morphisms (i.e. morphisms which yield the same result when composed on the right with either member of a parallel pair of morphisms) in the abstract category:

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Show the association of constant morphisms (i.e. morphisms which yield the same result when composed on the right with either member of a parallel pair of morphisms) in the abstract category, modulo all equivalences between objects and morphisms:

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Show the association of co-constant morphisms (i.e. morphisms which yield the same result when composed on the left with either member of a parallel pair of morphisms) in the abstract category:

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Show the association of co-constant morphisms (i.e. morphisms which yield the same result when composed on the left with either member of a parallel pair of morphisms) in the abstract category, modulo all equivalences between objects and morphisms:

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Show the association of zero morphisms (i.e. morphisms which yield the same result when composed on the left or right with either member of a parallel pair of morphisms) in the abstract category:

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Show the association of zero morphisms (i.e. morphisms which yield the same result when composed on the left or right with either member of a parallel pair of morphisms) in the abstract category, modulo all equivalences between objects and morphisms:

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Determine whether the abstract category is a groupoid (i.e. all morphisms are isomorphisms):

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Show the list of equations between morphisms required to force the abstract category to be a groupoid (may not necessarily exist):

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Show the list of equations between morphisms required to force the abstract category to be a groupoid, modulo all equivalences between objects and morphisms (may not necessarily exist):

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Show the full directed graph with labels on the morphisms of the abstract category:

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Show the full directed graph without labels on the morphisms of the abstract category:

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Show the reduced directed graph, with all equivalences between objects and morphisms imposed and with labels on the morphisms of the abstract category:

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Show the reduced directed graph, with all equivalences between objects and morphisms imposed and without labels on the morphisms of the abstract category:

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Show the simple directed graph, with all self-loops and multiedges removed and with labels on the morphisms of the abstract category:

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Show the simple directed graph, with all self-loops and multiedges removed and without labels on the morphisms of the abstract category:

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Show the reduced simple directed graph, with all self-loops and multiedges removed, plus all object and morphism equivalences modded out and with labels on the morphisms of the abstract category:

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Show the reduced simple directed graph, with all self-loops and multiedges removed, plus all object and morphism equivalences modded out and without labels on the morphisms of the abstract category:

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Show the explicit association form of the category:

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