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\begin{document}

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\section*{universal construction}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{yoneda_lemma}{}\paragraph*{{Yoneda lemma}}\label{yoneda_lemma}

[[!include Yoneda lemma - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{example}{Example}\dotfill \pageref*{example} \linebreak
\noindent\hyperlink{ConcreteExample}{Concrete examples}\dotfill \pageref*{ConcreteExample} \linebreak
\noindent\hyperlink{classes_of_examples}{Classes of examples}\dotfill \pageref*{classes_of_examples} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Universal properties are commonly used in mathematics, often without mentioning the term ``universal property''.

For example, if one were asked to give a map $\mathbb{R} \to \mathbb{R} \times \mathbb{C}$, they might write down something like $x \mapsto (x^2, x + i x)$. In effect, what is done is that a pair of maps $\mathbb{R} \to \mathbb{R}$ and $\mathbb{R} \to \mathbb{C}$ is given, namely $(x \mapsto x^2, x \mapsto x + i x)$. The \textbf{universal property} of the [[product]] says that giving a map to $A \times B$ is the same as giving a map to $A$ and a map to $B$, and moreover this correspondence is natural in some precise sense.

Similarly, given [[rings]] $R$ and $S$, if we want to extend a [[ring homomorphism]] $R \to S$ to a homomorphism from the [[polynomial ring]] $R[x] \to S$, all we have to do is to specify an element of $S$ that we send $x$ to. In other words, a homomorphism $R[x] \to S$ is the same as a homomorphism $R \to S$ and an element of $S$.

For it to be a universal property, just the existence of such a bijection is not sufficient. We will need some conditions to make sure the bijection is ``natural''. Abstractly, this says that the bijection is given by a [[natural isomorphism]] of certain [[functors]]. More concretely, by the [[Yoneda lemma]], this is equivalent to saying the bijection is ``mediated'' by some ``universal maps'', which is how universal properties are usually formulated. See \hyperlink{ConcreteExample}{Concrete examples} for more details.

Recall that by the [[Yoneda lemma]], specifying how we can map in or out of an object uniquely determines the object up to [[isomorphism]]. So we can use these universal properties as \emph{definitions} of the constructions! These are known as \textbf{universal constructions}. Of course, these definitions are not actually ``constructions''. We still have to do the concrete constructions the good, old way to show that there are objects satisfying the universal property (or apply general theorems such as the [[adjoint functor theorem]]).

\hypertarget{example}{}\subsection*{{Example}}\label{example}

\hypertarget{ConcreteExample}{}\subsubsection*{{Concrete examples}}\label{ConcreteExample}

We first look at a few concrete examples of universal properties. These are all special cases of the ones described below.

\begin{example}
\label{}\hypertarget{}{}
The [[product]] of two objects (eg. sets, groups, rings etc.) is specified by the property that maps $f: X \to A \times B$ biject naturally with pairs of maps $(f_1: X \to A, f_2: X \to B)$.

The naturality condition is that if $f: X \to A \times B$ corresponds to $f_1: X \to A, f_2: X \to B$, and $g: Y \to X$ is a map, then $f \circ g$ corresponds to $f_1 \circ g$ and $f_2 \circ g$, so that the bijection respects composition.

In particular, the [[identity map]] $id: A \times B \to A \times B$ corresponds to a pair of [[projection maps]] $\pi_1: A \times B \to A$ and $\pi_2: A \times B \to B$. Then if $f: X \to A \times B$ is a map, then by naturality, it corresponds to $\pi_1 \circ f: X \to A$ and $\pi_2 \circ f: X \to B$.

Suppose we are not given a bijection, but just an object $P$ with maps $\pi_1: P \to A$ and $\pi_2: P \to B$. Then as above, we obtain a function from maps $X \to P$ to pairs of maps $X \to A, X \to B$ by composition. This makes $P$ into the product of $A$ and $B$ exactly when this function is a bijection, ie. for any pair of maps $f_1 : X \to A, f_2: X \to B$, there is a unique map $f: X \to P$ whose compositions with $\pi_1, \pi_2$ are $f_1, f_2$ respectively (naturality is easy to check).

(The experienced reader will notice that this is just a special case of the [[Yoneda lemma]])

Thus, the universal property can be stated as follows: $C$ is a product of $A$ and $B$ if there exists maps $\pi_1: C \to A$ and $\pi_2: C \to B$ such that given any pair of maps $f_1: X \to A$ and $f_2: X \to B$, there is a unique map $f: X \to C$ such that the following diagram commutes:

\begin{displaymath}
\itexarray{
& & X & & \\
& ^\mathllap{f_1}\swarrow & \downarrow^\mathrlap{f} & \searrow^\mathrlap{f_2}\\
A & \underset{\pi_1}{\leftarrow} & C & \underset{\pi_2}{\rightarrow} & B
}
\end{displaymath}
In this case, we tend to write $A \times B$ for $C$.

\end{example}
Note that if we are talking about sets, then an element of a set $X$ is equivalent to a map $1 \to X$ from the singleton set $1$. Thus in particular, the above definition says an element of $A \times B$ is the same as a pair of elements $(a, b)$, where $a \in A$ and $b \in B$.

\begin{example}
\label{}\hypertarget{}{}
The [[free group|free]] [[group]] on $n$ generators is a group $F_n$ such that [[group homomorphisms]] $F_n \to G$ bijects (naturally) with $n$ elements of $G$ (not necessarily distinct).

Similar to the above, the naturality condition says if $f: F_n \to G$ corresponds to $g_1, ..., g_n \in G$, and $h: G \to H$ is a map, then $h \circ f$ corresponds to the elements $h(g_1), ..., h(g_n)$. In particular, suppose the identity map $id: F_n \to F_n$ corresponds to $n$ elements $x_1, ..., x_n\in F_n$. Then any homomorphism $f: F_n \to G$ corresponds to the elements $f(x_1), ..., f(x_n)$ of $G$.

Thus, given the specified elements $x_1, ..., x_n \in F_n$, the universal property says given any $n$ elements of $G$, we can find a unique homomorphism $f: F_n \to G$ that sends $x_1, ..., x_n$ to the $n$ elements.

Diagrammatically, picking $n$ elements out of a set $X$ is the same as a [[function]] (of sets) $n \to X$. If we write $U(G)$ for the underlying set of the group $G$ (ie. $U$ is the [[forgetful functor]] to $Set$), the universal property of the free group says that there is a specified function $\phi: n \to U(F_n)$, such that for every function $f: n \to U(G)$, we can find a unique group homomorphism $\tilde{f}: F_n \to G$ such that the following diagram commutes:

\begin{displaymath}
\itexarray{
  U(F_n) & \overset{U(\tilde{f})}{\to} & U(G)\\
  ^\mathllap{\phi}\uparrow & \nearrow_{\mathrlap{f}}\\
  n
}
\end{displaymath}
In other words, every map $f: n \to U(G)$ factors through the universal map $\phi: n \to U(F_n)$ uniquely.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
The [[tensor product]] of [[vector spaces]] has the universal property that a [[bilinear map]] $V \times W \to U$ bijects naturally with linear maps $V \otimes W \to U$. The naturality condition is given by the existence of a universal bilinear map $\phi: V \times W \to V \otimes W$ such that every bilinear map $V \times W \to U$ factors through $\phi$ uniquely.

\end{example}
We have more degenerate examples such as the terminal object:

\begin{example}
\label{}\hypertarget{}{}
In the [[category of sets]], the singleton $1$ satisfies the property that there is always a unique map from any object to $1$. So we can say that the maps $A \to 1$ biject (necessarily naturally) with the set $1$. More generally, in any category, if an object $X$ is such that there is always a unique map from any object to $X$, then $X$ is called the [[terminal object]].

Dually, an initial object is an object $0$ such that there is a unique map from $0$ to any object $X$.

\end{example}
\hypertarget{classes_of_examples}{}\subsubsection*{{Classes of examples}}\label{classes_of_examples}

In general, the \textbf{universal constructions} in [[category theory]] include

\begin{itemize}%
\item [[representable functor]]


\item [[adjoint functor]]


\item [[limit]]/[[colimit]]


\item [[end]]/[[coend]]


\item [[Kan extension]]


\item [[dependent sum]]/[[dependent product]]



\end{itemize}
Each of these may be defined by requiring it to satisfy a \textbf{universal property}. A universal property is a property of some construction which boils down to (is manifestly equivalent to) the property that an associated object is a universal initial object of some (auxiliary) category.

In good cases, every single one of these is a special case of every other, so somehow one single concept here comes to us with many different faces.

Some or all of these have analogs in [[higher category theory]], notably in [[2-category]] theory and [[(∞,1)-category theory]]:

\begin{itemize}%
\item [[2-limit]]


\item [[2-adjunction]]


\item [[limit in a quasi-category]]


\item [[adjoint (∞,1)-functor]]



\end{itemize}
[[!redirects universal construction]] [[!redirects universal constructions]] [[!redirects universal mapping]] [[!redirects universal mappings]]

[[!redirects universal property]] [[!redirects universal properties]] [[!redirects universal mapping property]] [[!redirects universal mapping properties]] [[!redirects universal]]



\end{document}