# nLab initial object

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, 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).

## 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:

###### Theorem

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. $f=p_I$ as desired.

###### Corollary

An object $I$ is initial in $C$ iff $I$ is the limit of $Id_C$.

###### Proof

If $I$ is initial, then there is a cone $(!_X: I \to X)_{X \in Ob(C)}$ from $I$ to $Id_C$, and if $(p_X: A \to X)_{X \in Ob(C)}$ is any cone from $A$ to $Id_C$, then $p_X = !_X \circ p_I$. Hence $p_I: A \to I$ defines a morphism of cones, and the unique morphism of cones since if $q$ any morphism of cones, then $p_I = !_I \circ q = 1_I \circ q = q$. 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 $p_X \circ p_L = p_X$ 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 Theorem 1 we conclude $L$ is initial.

This corollary 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 $(c, \theta: d \to G c)$. 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 March 8, 2018 at 04:25:09. See the history of this page for a list of all contributions to it.