Limits and colimits are fundamental concepts in category theory, shaping how objects and morphisms interact. This section dives into special cases like products, coequalizers, pullbacks, and pushouts, exploring their construction and universal properties.
Understanding these special cases is crucial for grasping the broader concept of limits and colimits. We'll examine how they're computed in different categories and use commutative diagrams to verify their properties.
Special Cases of Limits and Colimits
Products and coequalizers
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Products
Limit of diagram consisting of objects without morphisms between them terminates in category of cones over diagram
Cartesian product of sets exemplifies in Set category
Product topology demonstrates in Top category
Coproducts
Colimit of diagram consisting of objects without morphisms between them initiates in category of cocones under diagram
Disjoint union of sets illustrates in Set category
Free product of groups showcases in Grp category
Equalizers
Limit of parallel morphism pair terminates in category of equalizing cones
Subset of elements where two functions agree represents in Set category
Kernel of difference between two group homomorphisms exemplifies in Grp category
Coequalizers
Colimit of parallel morphism pair initiates in category of coequalizing cocones
Quotient set by equivalence relation generated by function pair demonstrates in Set category
Quotient ring by ideal generated by image differences illustrates in Rng category
Construction of pullbacks and pushouts
Pullbacks
Limit of cospan diagram constructs using products and equalizers:
Form product of two objects at cospan base
Take equalizer of two compositions from product to apex
Fiber product in algebraic geometry serves as example
Pushouts
Colimit of span diagram constructs using coproducts and coequalizers:
Form coproduct of two objects at span base
Take coequalizer of two compositions from apex to coproduct
Amalgamated free product in group theory demonstrates concept
Universal properties of limits
Universal property of pullbacks
Terminates in category of cones over cospan diagram
Uniquely factorizes through any other cone
Preserves monomorphisms (injective functions)
Universal property of pushouts
Initiates in category of cocones under span diagram
Uniquely factorizes from any other cocone
Preserves epimorphisms (surjective functions)
Computation of categorical limits
Set category exercises
Products compute as Cartesian products (ordered pairs)
Coproducts calculate as disjoint unions (tagged unions)
Equalizers determine as subsets (common elements)
Coequalizers find as quotient sets (equivalence classes)
Group category exercises
Products construct as direct products (element-wise operation)
Coproducts compute as free products (words with alternating elements)
Equalizers calculate as kernels (elements mapping to identity)
Coequalizers determine as quotient groups (cosets)
Topological space category exercises
Products compute with product topology (box topology)
Coproducts calculate with disjoint union topology (pasting topology)
Equalizers determine as subspaces (induced topology)
Coequalizers find as quotient spaces (identification topology)
Commutative diagram techniques
Commutativity of pullback and pushout squares verify through path equality
Universal properties apply to prove uniqueness of constructions
Functoriality of limits and colimits utilize for preservation under functors