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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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
Key Terms to Review (15)
Ab: In category theory, 'ab' often refers to the category of abelian groups, which are groups that are also abelian under addition. This concept is crucial as it encapsulates both algebraic structure and properties related to limits and colimits, essential for understanding various functorial behaviors and categorical limits.
Adjunction: Adjunction is a fundamental concept in category theory where two functors stand in a specific relationship, known as an adjoint pair. This relationship allows one functor to be thought of as providing a best approximation of the other, capturing the essence of universal properties. Adjunctions are critical in many areas of mathematics, including the study of limits, colimits, and various constructions that connect different categories.
Commutative Diagram: A commutative diagram is a visual representation in category theory that illustrates how various objects and morphisms relate to one another through a series of paths that yield the same result regardless of the path taken. This concept serves as a powerful tool to express relationships between mathematical structures, showing how different compositions and mappings can lead to consistent outcomes.
Epimorphism: An epimorphism is a type of morphism in category theory that is considered a surjective map, meaning it can be thought of as a morphism that 'covers' its codomain. This concept plays a crucial role in understanding how different structures can be related within various mathematical frameworks, linking to essential notions like composition of morphisms and the properties of different kinds of mappings.
Equalizer: An equalizer is a concept in category theory that captures the idea of a universal construction which 'equalizes' two morphisms from the same object. It is an object equipped with morphisms that provide a way to identify elements that are mapped to the same image under two different morphisms, showcasing relationships between objects in a category. This concept ties into broader themes like uniqueness up to unique isomorphism and forms a foundational aspect in understanding products and pullbacks.
Functor: A functor is a mapping between categories that preserves the structure of those categories, specifically the objects and morphisms. It consists of two main components: a function that maps objects from one category to another, and a function that maps morphisms in a way that respects composition and identity morphisms.
Grp: In category theory, a 'grp' represents the category of groups, where the objects are groups and the morphisms are group homomorphisms. This concept serves as a fundamental example of a category that encapsulates both algebraic structures and their relationships, connecting to various mathematical contexts such as functors, limits, and colimits.
Monomorphism: A monomorphism is a type of morphism in category theory that can be thought of as an injection or one-to-one function. It preserves distinctness in the sense that different elements in the source category remain distinct when mapped to the target category, which allows for meaningful interpretations across various mathematical structures.
Morphism: A morphism is a structure-preserving map between two objects in a category, reflecting the relationships between those objects. Morphisms can represent functions, arrows, or transformations that connect different mathematical structures, serving as a foundational concept in category theory that emphasizes relationships rather than individual elements.
Product: In category theory, a product is a construction that captures the idea of combining multiple objects into a single object that contains all the information from the original objects. It involves a pair of morphisms from the product object to each of the original objects, satisfying a universal property which makes it unique up to isomorphism. The concept connects closely to other categorical constructions like equalizers and pullbacks, as well as principles related to duality in opposite categories.
Pullback: A pullback is a specific type of limit in category theory that allows for the construction of a new object, capturing the idea of 'pulling back' along two morphisms that share a common codomain. This construction not only provides a way to combine structures from different categories but also reflects essential properties of morphisms, uniqueness, and limits in category theory.
Set: A set is a well-defined collection of distinct objects, considered as an object in its own right. In category theory, sets serve as the fundamental building blocks for constructing more complex mathematical structures, allowing for the exploration of relationships and mappings between different sets through functions.
String diagram: A string diagram is a graphical representation used in category theory to illustrate the relationships between objects and morphisms within a category. It uses simple shapes and lines to depict objects as points and morphisms as arrows connecting these points, enabling a visual approach to understand complex categorical structures like products, equalizers, and pullbacks.
Top: In category theory, a 'top' typically refers to a terminal object in a category, which is an object such that there is a unique morphism from any object in the category to this terminal object. The existence of terminal objects helps in defining limits and colimits, playing a crucial role in understanding the structure of categories.
Universal Property: A universal property is a fundamental concept in category theory that describes an object in terms of its relationships with other objects through morphisms. It serves as a characterization of objects that can uniquely determine them via certain properties, often in the context of limits and colimits, making them essential for understanding constructions like products, coproducts, and adjoint functors.