nLab
initial object

Contents

Context

Category theory

Limits and colimits

Contents

Definition

Definition

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

Remark

An initial object, if it exists, is unique up to unique isomorphism, so that we may speak of the initial object.

Remark

When it exists, the initial object is the colimit over the empty diagram.

Remark

Initial objects are also called coterminal, and (rarely, though): coterminators, universal initial, co-universal, or simply universal.

Definition

An initial object \varnothing is called a strict initial object if all morphisms xx \xrightarrow{\;} \varnothing into it are isomorphisms.

Remark

Initial objects are the dual concept to terminal objects: an initial object in CC is the same as a terminal object in the opposite category C opC^{op}.

Remark

An object that is both initial and terminal is called a zero object.

Examples

Properties

Left adjoints to constant functors

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.

Cones over the identity

By definition, an initial object is equipped with a universal cocone under the unique functor C\emptyset\to C from the empty category. On the other hand, if II is initial, the unique morphisms !:Ix!: I \to x form a cone over the identity functor, i.e. a natural transformation ΔIId C\Delta I \to Id_C from the constant functor at the initial object to the identity functor. In fact this is almost another characterization of an initial object (e.g. MacLane, p. 229-230):

Lemma

Suppose ICI\in C is an object equipped with a natural transformation p:ΔIId Cp:\Delta I \to Id_C such that p I=1 I:IIp_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 XCX\in C, namely p Xp_X, so it suffices to show that any f:IXf:I\to X must be equal to p Xp_X. But the naturality of pp implies that Id C(f)p I=p XΔ 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 f1 I=p X1 If \circ 1_I = p_X \circ 1_I, i.e. f=p Xf=p_X as desired.

Theorem

An object II in a category CC is initial iff II is the limit of the identity functor Id CId_C.

Proof

If II is initial, then there is a cone (! X:IX) XOb(C)(!_X: I \to X)_{X \in Ob(C)} from II to Id CId_C. If (p X:AX) XOb(C)(p_X: A \to X)_{X \in Ob(C)} is any cone from AA to Id CId_C, then p X=fp Yp_X = f \circ p_Y for any f:YXf:Y\to X, and so in particular p X=! Xp Ip_X = !_X \circ p_I. Since this is true for any XX, p I:AIp_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=! Iq=1 Iq=qp_I = !_I \circ q = 1_I \circ q = q (using that ! I=1 I!_I = 1_I by initiality). Thus (! X:IX) XOb(C)(!_X: I \to X)_{X \in Ob(C)} is the limit cone.

Conversely, if (p X:LX) XOb(C)(p_X: L \to X)_{X \in Ob(C)} is a limit cone for Id CId_C, then fp Y=p Xf\circ p_Y = p_X for any f:YXf:Y\to X, and so in particular p Xp L=p Xp_X \circ p_L = p_X for all XX. This means that both p L:LLp_L: L \to L and 1 L:LL1_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.

Remark

(relevance for adjoint functor theorem)

Theorem is actually a key of entry into the general adjoint functor theorem. Showing that a functor G:CDG: C \to D has a left adjoint is tantamount to showing that each functor D(d,G)D(d, G-) is representable, i.e., that the comma category dGd \downarrow G has an initial object (c,θ:dGc)(c, \theta: d \to G c) (see at adjoint functor, this prop.). This is the limit of the identity functor, but typically this is the limit over a large diagram whose existence is not guaranteed. The point of a solution set condition is to replace this with a small diagram which is cofinal in the large diagram.

References

Textbook accounts:

Last revised on October 15, 2021 at 12:25:29. See the history of this page for a list of all contributions to it.