initial object in nLab

Context

Category theory

Limits and colimits

• Definition
• Examples
• Properties
• Left adjoints to constant functors
• Cones over the identity
• Related concepts
• References

Definition

An initial object in a category 𝒞\mathcal{C} is an object ∅\emptyset such that for all objects x∈𝒞x \,\in\, \mathcal{C}, there is a unique morphism ∅→∃!x\varnothing \xrightarrow{\exists !} x with source ∅\varnothing. and target xx.

Proposition

Let 𝒞\mathcal{C} be a category.

1. The following are equivalent:

   1. 𝒞\mathcal{C} has a terminal object;

   2. the unique functor 𝒞→*\mathcal{C} \to \ast to the terminal category has a right adjoint

      *⊥⟶⟵𝒞 \ast \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longleftarrow}} {\bot} \mathcal{C}

   Under this equivalence, the terminal object is identified with the image under the right adjoint of the unique object of the terminal category.

2. Dually, the following are equivalent:

   1. 𝒞\mathcal{C} has an initial object;

   2. the unique functor 𝒞→*\mathcal{C} \to \ast to the terminal category has a left adjoint

      𝒞⊥⟶⟵* \mathcal{C} \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longleftarrow}} {\bot} \ast

   Under this equivalence, the initial object is identified with the image under the left adjoint of the unique object of the terminal category.

Proof

Since the unique hom-set in the terminal category is the singleton, the hom-isomorphism characterizing the adjoint functors is directly the universal property of an initial object in 𝒞\mathcal{C}

Hom 𝒞(L(*),X)≃Hom *(*,R(X))=* Hom_{\mathcal{C}}( L(\ast) , X ) \;\simeq\; Hom_{\ast}( \ast, R(X) ) = \ast

or of a terminal object

Hom 𝒞(X,R(*))≃Hom *(L(X),*)=*, Hom_{\mathcal{C}}( X , R(\ast) ) \;\simeq\; Hom_{\ast}( L(X), \ast ) = \ast \,,

respectively.

Lemma

Suppose I∈CI\in C is an object equipped with a natural transformation p:ΔI→Id Cp:\Delta I \to Id_C such that p I=1 I:I→Ip_I = 1_I : I\to I. Then II is an initial object of CC.

Proof

Obviously II has at least one morphism to every other object X∈CX\in C, namely p Xp_X, so it suffices to show that any f:I→Xf:I\to X must be equal to p Xp_X. But the naturality of pp implies that Id C(f)∘p I=p X∘Δ I(f)\Id_C(f) \circ p_I = p_X \circ \Delta_I(f), and since p I=1 Ip_I = 1_I this is to say f∘1 I=p X∘1 If \circ 1_I = p_X \circ 1_I, i.e. f=p Xf=p_X as desired.

Proof

If II is initial, then there is a cone (! X:I→X) X∈Ob(C)(!_X: I \to X)_{X \in Ob(C)} from II to Id CId_C. If (p X:A→X) X∈Ob(C)(p_X: A \to X)_{X \in Ob(C)} is any cone from AA to Id CId_C, then p X=f∘p Yp_X = f \circ p_Y for any f:Y→Xf:Y\to X, and so in particular p X=! X∘p Ip_X = !_X \circ p_I. Since this is true for any XX, p I:A→Ip_I: A \to I defines a morphism of cones, and it is the unique morphism of cones since if qq is any morphism of cones, then p I=! I∘q=1 I∘q=qp_I = !_I \circ q = 1_I \circ q = q (using that ! I=1 I!_I = 1_I by initiality). Thus (! X:I→X) X∈Ob(C)(!_X: I \to X)_{X \in Ob(C)} is the limit cone.

Conversely, if (p X:L→X) X∈Ob(C)(p_X: L \to X)_{X \in Ob(C)} is a limit cone for Id CId_C, then f∘p Y=p Xf\circ p_Y = p_X for any f:Y→Xf:Y\to X, and so in particular p X∘p L=p Xp_X \circ p_L = p_X for all XX. This means that both p L:L→Lp_L: L \to L and 1 L:L→L1_L: L \to L define morphisms of cones; since the limit cone is the terminal cone, we infer p L=1 Lp_L = 1_L. Then by Lemma we conclude LL is initial.