3.3 Special cases: products, coproducts, equalizers, and coequalizers

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:
    1. Form product of two objects at cospan base
    2. 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:
    1. Form coproduct of two objects at span base
    2. 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