initial object


Category theory

Limits and colimits



An initial object in a category CC is an object \emptyset such that for any object xx of CC, there is a unique morphism !:x!:\emptyset\to x. An initial object, if it exists, is unique up to unique isomorphism, so we speak of the initial object.

An initial object may also be called coterminal, universal initial, co-universal, or simply universal.

Initial objects are the dual concept to terminal objects: an initial object in CC is the same as a terminal object in C opC^{op}. An object that is both initial and terminal is called a zero object.

Hence the initial object may also be viewed as the colimit over the empty diagram.


Strict initial objects

An initial object \emptyset is called a strict initial object if any morphism xx\to \emptyset must be an isomorphism. The initial objects of a poset, of SetSet, CatCat, TopTop, and of any topos (in fact, any extensive category, or any cartesian closed category) are strict. At the other extreme, a zero object is only a strict initial object if the category is trivial (equivalent to the terminal category).


Left adjoints to constant functors


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.


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 \,,


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:


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.


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 If=p_I as desired.


An object II is initial in CC iff II is the limit of Id CId_C.


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, and 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=! Xp Ip_X = !_X \circ p_I. Hence p I:AIp_I: A \to I defines a morphism of cones, and the unique morphism of cones since if qq any morphism of cones, then p I=! Iq=1 Iq=qp_I = !_I \circ q = 1_I \circ q = q. 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 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 Theorem 1 we conclude LL is initial.

This corollary 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). 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.

Last revised on June 12, 2018 at 12:39:27. See the history of this page for a list of all contributions to it.