Initial object, terminal object, and zero object

Depends on

• Category, object, morphism, domain, codomain, identity, composition, and hom-collection
• Every theorem about categories has a formal dual obtained by reversing morphisms and composition

Used by

• In a cartesian closed category, any initial object is strict Corollary
• A fully faithful left Kan extension that is not pointwise Counterexample
• Additive category Definition
• Freyd's axioms A0, A1, A1*, A2, A2*, A3, and A3* for abelian categories Definition
• Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties Definition
• Subobject classifier Definition
• The end and the coend of a functor CᵒᵖtimesCtoD Definition
• Weakly initial object and jointly weakly initial set Definition
• Zero complex and stalk complex Definition
• Pointed sets are equivalent to sets and partial functions but not isomorphic as categories Example
• The Kleisli adjunction for the maybe monad is monadic but not strictly monadic Example
• Every functor with a left adjoint also has a right adjoint False statement
• FALSE: every Kan extension is pointwise False statement
• FALSE: the Yoneda embedding preserves colimits False statement
• Left adjoints preserve limits False statement
• A cone over an identity diagram is weakly initial, and the identity diagram has a limit exactly when the category has an initial object Lemma
• A zero object supplies a unique compatible system of zero morphisms Proposition
• Initial and terminal objects are exactly the representations of the constant singleton functor Proposition
• Limits of empty diagrams are terminal objects, and colimits of empty diagrams are initial objects Proposition
• The cokernel of the zero map out of the zero object is the target, and dually for kernels Proposition
• The empty biproduct is a zero object Proposition
• The kernel of a monomorphism is zero and the cokernel of an epimorphism is zero Proposition
• A category with finite products is monoidal Theorem
• A locally cartesian closed category has pullbacks, and with a terminal object it has all finite limits Theorem
• A locally cartesian closed category with a terminal object is cartesian closed Theorem
• In a preadditive category, an object is initial exactly when it is terminal Theorem
• Initial and terminal objects are unique up to a unique isomorphism Theorem
• Limits and colimits are Kan extensions along the functor to the terminal category Theorem
• Universal arrows to a functor are initial in comma categories, and universal arrows from a functor are terminal Theorem
• Universal elements are initial in a covariant category of elements and terminal in a presheaf category of elements Theorem

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)