Products, equalizers, and pullbacks are key constructions in category theory. They represent different ways of combining or comparing objects and morphisms, each with its own that defines its behavior across categories.
These constructions appear in various mathematical contexts, from theory to group theory and topology. Understanding their universal properties helps us see common patterns across different mathematical structures and solve complex problems more efficiently.
Products, Equalizers, and Pullbacks
Products, equalizers, and pullbacks
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Top images from around the web for Products, equalizers, and pullbacks
Introduction to Category Theory/Products and Coproducts of Sets - Wikiversity View original
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Abstract Algebra/Group Theory/Products and Free Groups - Wikibooks, open books for an open world View original
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consists of an object A×B and morphisms π1:A×B→A, π2:A×B→B satisfying a universal property
For any object C and morphisms f:C→A, g:C→B, there exists a unique ⟨f,g⟩:C→A×B such that π1∘⟨f,g⟩=f and π2∘⟨f,g⟩=g
consists of an object E and a morphism e:E→A satisfying f∘e=g∘e for given morphisms f,g:A→B
For any object C and morphism h:C→A such that f∘h=g∘h, there exists a unique morphism u:C→E such that e∘u=h
consists of an object P and morphisms p1:P→A, p2:P→B satisfying f∘p1=g∘p2 for given morphisms f:A→C, g:B→C
For any object Q and morphisms q1:Q→A, q2:Q→B such that f∘q1=g∘q2, there exists a unique morphism u:Q→P such that p1∘u=q1 and p2∘u=q2
Construction in various categories
Products
In Set, the product of sets A and B is the Cartesian product A×B={(a,b)∣a∈A,b∈B} with projection functions (π1, π2)
In [Grp](https://www.fiveableKeyTerm:grp), the product of groups G and H is the direct product G×H={(g,h)∣g∈G,h∈H} with componentwise group operation and projection homomorphisms (π1, π2)
Equalizers
In Set, the equalizer of functions f,g:A→B is the subset E={a∈A∣f(a)=g(a)} with the inclusion map e:E→A
In Grp, the equalizer of group homomorphisms f,g:G→H is the subgroup E={g∈G∣f(g)=g(g)} with the inclusion homomorphism e:E→G
Pullbacks
In Set, the pullback of functions f:A→C and g:B→C is the subset P={(a,b)∈A×B∣f(a)=g(b)} with projection maps p1:P→A and p2:P→B
In Grp, the pullback of group homomorphisms f:G→K and g:H→K is the subgroup P={(g,h)∈G×H∣f(g)=g(h)} with projection homomorphisms p1:P→G and p2:P→H
Universal properties
Products
Define the unique morphism ⟨f,g⟩:C→A×B by ⟨f,g⟩(c)=(f(c),g(c)) for any c∈C
Prove uniqueness: If h:C→A×B satisfies π1∘h=f and π2∘h=g, then h(c)=(π1(h(c)),π2(h(c)))=(f(c),g(c))=⟨f,g⟩(c) for any c∈C
Equalizers
Define the unique morphism u:C→E by u(c)=h(c) for any c∈C
Prove uniqueness: If v:C→E satisfies e∘v=h, then v(c)=e(v(c))=h(c)=u(c) for any c∈C
Pullbacks
Define the unique morphism u:Q→P by u(q)=(q1(q),q2(q)) for any q∈Q
Prove uniqueness: If v:Q→P satisfies p1∘v=q1 and p2∘v=q2, then v(q)=(p1(v(q)),p2(v(q)))=(q1(q),q2(q))=u(q) for any q∈Q
Existence and uniqueness problems
Existence
Products, equalizers, and pullbacks may not exist in every category (Pos, Mon)
A category is complete if it has all small limits, including products, equalizers, and pullbacks (Set, Grp, [Ab](https://www.fiveableKeyTerm:ab), [Top](https://www.fiveableKeyTerm:top))
Uniqueness up to unique isomorphism
Products: If A×B and A×′B are both products, there exists a unique isomorphism φ:A×B→A×′B such that π1′∘φ=π1 and π2′∘φ=π2
Equalizers: If e:E→A and e′:E′→A are both equalizers of f,g:A→B, there exists a unique isomorphism φ:E→E′ such that e′∘φ=e
Pullbacks: If P and P′ are both pullbacks of f:A→C and g:B→C, there exists a unique isomorphism φ:P→P′ such that p1′∘φ=p1 and p2′∘φ=p2