+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Category theory +-- {: .hide} [[!include category theory - contents]] =-- #### Limits and colimits +--{: .hide} [[!include infinity-limits - contents]] =-- =-- =-- # Contents * table of contents {: toc} ## Definition \begin{definition} An **initial object** in a [[category]] $\mathcal{C}$ is an [[object]] $\emptyset$ such that for all objects $x \,\in\, \mathcal{C}$, there is a unique [[morphism]] $\varnothing \xrightarrow{\exists !} x$ with [[source]] $\varnothing$. and [[target]] $x$. \end{definition} \begin{remark} An initial object, if it exists, is unique up to unique [[isomorphism]], so that we may speak of [[the]] initial object. \end{remark} \begin{remark}\label{InitialObjectIsEmptyColimit} When it exists, the initial object is the [[colimit]] over the [[empty diagram]]. \end{remark} \begin{remark} Initial objects are also called _coterminal_, and (rarely, though): _coterminators_, _universal initial_, _co-universal_, or simply _universal_. \end{remark} \begin{definition} An initial object $\varnothing$ is called a **[[strict initial object]]** if all morphisms $x \xrightarrow{\;} \varnothing$ into it are [[isomorphisms]]. \end{definition} \begin{remark} Initial objects are the [[duality|dual]] concept to [[terminal objects]]: an initial object in $C$ is the same as a terminal object in the [[opposite category]] $C^{op}$. \end{remark} \begin{remark} An object that is both initial and [[terminal object|terminal]] is called a [[zero object]]. \end{remark} ## Examples * An initial object in a [[partial order|poset]] is a [[bottom element]]. * The [[empty set]] is an initial object in [[Set]]. * Likewise, the [[empty category]] is an initial object in [[Cat]], the [[empty space]] is an initial object in [[Top]], and so on. * The [[trivial group]] is the initial object (in fact, the [[zero object]]) of [[Grp]] and [[Ab]]. * The [[ring]] of [[integers]] $\mathbb{Z}$ is the initial object of [[Ring]]. * The [[field]] of [[rational numbers]] $\mathbb{Q}$ is the initial object of $Field_0$ (category of fields with [[characteristic]] $0$) and the [[prime field]] $\mathbb{F}_p$ is the initial object of $Field_p$ (category of fields with characteristic $p$), but none are the initial object of [[Field]] (category of all fields), which actually doesn't have one at all. * The initial object of a [[under category|coslice category]] $x/C$ is the [[identity morphism]] $x \to x$. * An initial object in a category of [[central extensions]] of a given algebraic object is called a _[[universal central extension]]_. ## Properties ### Left adjoints to constant functors \begin{proposition} \label{AdjointsToConstantFunctors} Let $\mathcal{C}$ be a [[category]]. 1. The following are equivalent: 1. $\mathcal{C}$ has a [[terminal object]]; 1. 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]]. 1. Dually, the following are equivalent: 1. $\mathcal{C}$ has an [[initial object]]; 1. 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]]. \end{proposition} +-- {: .proof} ###### Proof Since the unique [[hom-set]] in the [[terminal category]] is [[generalized the|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 {#ConesOverTheIdentity} 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](#MacLane)): +--{: .num_lemma #cone} ###### 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} ###### 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. =-- +-- {: .num_theorem #LimitOverIdentityFunctorIsInitialObject} ###### Theorem An object $I$ in a [[category]] $C$ is initial iff $I$ is the [[limit]] of the [[identity functor]] $Id_C$. =-- +-- {: .proof} ###### 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 \ref{cone} we conclude $L$ is initial. =-- +-- {: .num_remark #RelevanceForAdjointFunctorTheorem} ###### Remark **(relevance for [[adjoint functor theorem]])** Theorem \ref{LimitOverIdentityFunctorIsInitialObject} is actually a key of entry into the [[adjoint functor theorem|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 functor|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.](#PointwiseExpressionOfLeftAdjoints)). 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. =-- ## Related concepts * [[terminal object]], [[bi-terminal object]] * [[bottom type]] * [[initial object in a 2-category]] * [[initial object in an (∞,1)-category]] * [[h-initial object]] * [[adjoint functor theorem]] ## References Textbook accounts: * [[Francis Borceux]], Section 2.3 in Vol. 1: *Basic Category Theory* of: *[[Handbook of Categorical Algebra]]*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) ([doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)) * {#MacLane} [[Saunders MacLane]], _[[Categories for the Working Mathematician]]_ [[!redirects initial object]] [[!redirects initial objects]] [[!redirects initial]] [[!redirects coterminal object]] [[!redirects coterminal objects]] [[!redirects coterminator]] [[!redirects coterminators]]