nLab initial object

Contents

category theory

Applications

Limits and colimits

limits and colimits

Contents

Definition

An initial object in a category $C$ is an object $\emptyset$ such that for any object $x$ of $C$, there is a unique morphism $!:\emptyset\to x$. 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, coterminator, universal initial, co-universal, or simply universal.

Initial objects are the dual concept to terminal objects: an initial object in $C$ is the same as a terminal object in $C^{op}$. 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 $x\to \emptyset$ must be an isomorphism. The initial objects of a poset, of $Set$, $Cat$, $Top$, 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).

Properties

Proposition

Let $\mathcal{C}$ be a category.

1. The following are equivalent:

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

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

$\ast \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longleftarrow}} {\bot} \mathcal{C}$

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

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

$\mathcal{C} \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longleftarrow}} {\bot} \ast$

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 $\mathcal{C}$

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

or of a terminal object

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

respectively.

Cones over the identity

By definition, an initial object is equipped with a universal cocone under the unique functor $\emptyset\to C$ from the empty category. On the other hand, if $I$ is initial, the unique morphisms $!: I \to x$ form a cone over the identity functor, i.e. a natural transformation $\Delta I \to Id_C$ 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 $I\in C$ is an object equipped with a natural transformation $p:\Delta I \to Id_C$ such that $p_I = 1_I : I\to I$. Then $I$ is an initial object of $C$.

Proof

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

Theorem

An object $I$ in a category $C$ is initial iff $I$ is the limit of the identity functor $Id_C$.

Proof

If $I$ is initial, then there is a cone $(!_X: I \to X)_{X \in Ob(C)}$ from $I$ to $Id_C$. If $(p_X: A \to X)_{X \in Ob(C)}$ is any cone from $A$ to $Id_C$, then $p_X = f \circ p_Y$ for any $f:Y\to X$, and so in particular $p_X = !_X \circ p_I$. Since this is true for any $X$, $p_I: A \to I$ defines a morphism of cones, and it is the unique morphism of cones since if $q$ is any morphism of cones, then $p_I = !_I \circ q = 1_I \circ q = q$ (using that $!_I = 1_I$ by initiality). Thus $(!_X: I \to X)_{X \in Ob(C)}$ is the limit cone.

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

Remark

Theorem is actually a key of entry into the general adjoint functor theorem. Showing that a functor $G: C \to D$ has a left adjoint is tantamount to showing that each functor $D(d, G-)$ is representable, i.e., that the comma category $d \downarrow G$ has an initial object $(c, \theta: d \to G c)$ (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.