Key Concepts of Universal Properties to Know for Category Theory

Universal properties in category theory highlight key relationships between objects and morphisms. They provide a framework for understanding initial and terminal objects, products, coproducts, limits, and colimits, forming the backbone of many constructions in this mathematical field.

1. Initial object

   • An initial object in a category is an object such that there is a unique morphism from it to any other object in the category.
   • It serves as a "starting point" for constructing other objects and morphisms.
   • Every category has at most one initial object, which is crucial for defining limits and other constructions.

2. Terminal object

   • A terminal object is an object in a category such that there is a unique morphism from any other object to it.
   • It acts as an "end point" in the category, facilitating the definition of co-limits.
   • Similar to initial objects, every category can have at most one terminal object.

3. Product

   • The product of two objects is an object that represents their "combined" structure, with projection morphisms to each factor.
   • It satisfies a universal property: for any object with morphisms to the factors, there exists a unique morphism to the product.
   • Products can be seen as a way to capture the notion of "pairs" or "tuples" in category theory.

4. Coproduct

   • The coproduct is the dual concept to the product, representing the "disjoint union" of objects.
   • It has a universal property where for any object with morphisms from the components, there exists a unique morphism from the coproduct.
   • Coproducts are useful for constructing new objects from existing ones, akin to summing sets.

5. Equalizer

   • An equalizer is an object that captures the idea of "equality" between two morphisms, providing a way to identify when two morphisms agree.
   • It satisfies a universal property: for any object mapping to the codomain of the two morphisms, there exists a unique morphism to the equalizer.
   • Equalizers are essential for defining kernels in algebraic structures.

6. Coequalizer

   • A coequalizer is the dual of the equalizer, identifying objects based on the equivalence of morphisms.
   • It satisfies a universal property where for any object mapping from the two morphisms, there exists a unique morphism from the coequalizer.
   • Coequalizers are important for defining cokernels and quotient structures.

7. Pullback

   • A pullback is a limit that combines two morphisms with a common codomain, creating a new object that "pulls back" to both sources.
   • It satisfies a universal property: for any object mapping to the common codomain, there exists a unique morphism to the pullback.
   • Pullbacks are useful for constructing fibered products and understanding relationships between objects.

8. Pushout

   • A pushout is the dual of the pullback, combining two morphisms with a common source to create a new object that "pushes out" to both targets.
   • It has a universal property where for any object mapping from the common source, there exists a unique morphism from the pushout.
   • Pushouts are important for constructing colimits and understanding cofibered structures.

9. Limits

   • Limits generalize the concepts of products and equalizers, capturing the idea of "convergence" of diagrams in a category.
   • They satisfy a universal property that allows for the construction of a limit object from a diagram of objects and morphisms.
   • Limits are foundational for understanding how objects relate to one another in a categorical context.

10. Colimits

    • Colimits are the dual concept to limits, generalizing coproducts and coequalizers, representing the "coherent union" of diagrams.
    • They satisfy a universal property that allows for the construction of a colimit object from a diagram of objects and morphisms.
    • Colimits are essential for understanding how to combine and relate objects in a category, particularly in the context of homotopy theory and algebraic topology.