CatDat

initial object

An initial object is an object that has a unique morphism to every object in the category. This property refers to the existence of an initial object.

• Dual property: terminal object
• Related properties: finite coproducts
• nLab Link

Relevant implications

• pointed is equivalent to   zero morphisms and  initial object

  easy
• strict initial object implies   initial object

  by definition
• left cancellative and  initial object implies   strict initial object

  It suffices to prove that in general any monomorphism $f : A \to 0$ into an initial object is an isomorphism. If $g : 0 \to A$ is the unique morphism, then $f \circ g = \mathrm{id}_0$ since $0$ is initial. But then $f$ is a split epimorphism and a monomorphism, hence an isomorphism.
• right cancellative and  initial object implies   strict initial object

  Let $f : A \to 0$ be a morphism. Let $g : 0 \to A$ be the unique morphism. It is an epimorphism by assumption. Also, $f \circ g = \mathrm{id}_0$ since $0$ is initial. But then $g$ is a split monomorphism and an epimorphism, hence an isomorphism.
• cartesian closed and  initial object implies   strict initial object

  See the nLab.
• groupoid and  initial object implies   trivial

  easy
• finite coproducts is equivalent to   initial object and  binary coproducts

  [dualized] The non-trivial direction follows since finite products can be constructed recursively via $X_1 \times \cdots \times X_{n+1} = (X_1 \times \cdots \times X_n) \times X_{n+1}$.
• pushouts and  initial object implies   binary coproducts

  [dualized] If $1$ is a terminal object, then $X \times_1 Y = X \times Y$.
• initial object implies   connected

  [dualized] If $1$ denotes the terminal object, then for any two objects $A,B$ we have the zig-zag $A \to 1 \leftarrow B$.
• wide pushouts and  initial object implies   cocomplete

  [dualized] See the nLab.
• self-dual and  initial object implies   terminal object

  trivial by self-duality
• self-dual and  terminal object implies   initial object

  trivial by self-duality

Examples

• category of abelian groups
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative rings
• category of finite abelian groups
• category of finite orders
• category of finite sets
• category of finite sets and injections
• category of finitely generated abelian groups
• category of free abelian groups
• category of groups
• category of left R-modules
• category of locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with ∞ allowed
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of monoids
• category of pointed sets
• category of posets
• category of rings
• category of rngs
• category of schemes
• category of sets
• category of sets and relations
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of vector spaces
• category of Z-functors
• partial order [0,1]
• partial order of extended natural numbers
• partial order of natural numbers
• partial order of ordinal numbers
• preorder of integers w.r.t. divisiblity
• trivial category
• walking isomorphism
• walking morphism

Counterexamples

• category of fields
• category of finite sets and bijections
• category of finite sets and surjections
• category of non-empty sets
• delooping of a non-trivial finite group
• delooping of an infinite group
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• discrete category on two objects
• empty category
• walking parallel pair of morphisms

Unknown

For these categories the database has no info if they satisfy this property or not.

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