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\begin{document}

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\section*{free monoid}

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

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{induction}{}\paragraph*{{Induction}}\label{induction}

[[!include induction - contents]]

\hypertarget{finite_lists_and_free_monoids}{}\section*{{Finite lists and free monoids}}\label{finite_lists_and_free_monoids}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{as_functions}{As functions}\dotfill \pageref*{as_functions} \linebreak
\noindent\hyperlink{recursively}{Recursively}\dotfill \pageref*{recursively} \linebreak
\noindent\hyperlink{by_general_abstract_nonsense}{By general abstract nonsense}\dotfill \pageref*{by_general_abstract_nonsense} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{the_free_monoid_monad}{The free monoid monad}\dotfill \pageref*{the_free_monoid_monad} \linebreak
\noindent\hyperlink{foundational_relevance}{Foundational relevance}\dotfill \pageref*{foundational_relevance} \linebreak
\noindent\hyperlink{stacks_and_queues}{Stacks and queues}\dotfill \pageref*{stacks_and_queues} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Given a [[set]] $S$, the \textbf{free [[monoid]]} on $S$ is the set $S^*$ of all \textbf{lists} (finite [[sequences]]) of elements of $S$, made into a monoid using \textbf{concatenation}. The [[free functor]] from [[Set]] to [[Mon]] takes $S$ to $S^*$.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

We will give three definitions, which can all be shown equivalent.

\hypertarget{as_functions}{}\subsubsection*{{As functions}}\label{as_functions}

An element of $S^*$ consists of a [[natural number]] $n$ (possibly $n = 0$) and function from $[n]$ to $S$, where $[n]$ is the [[subset]] $\{i\colon \mathbf{N} \;|\; i \lt n\}$ of $\mathbf{N} = \{0, 1, 2, \ldots\}$. Such an element is called a \textbf{list} or (to specify $n$) \textbf{$n$-tuple} of elements of $S$. The number $n$ is called the \textbf{length} of the list.

The \textbf{[[empty list]]} is the unique list of length $0$. It may be written $()$, $*$, or $\epsilon$, perhaps with a subscript $S$ if desired.

If $n \gt 0$, then the list which assigns $0, \ldots, n - 1$ to $a_0, a_1, \ldots, a_{n-1}$ may be written $(a_0, a_1, \ldots, a_{n-1})$. For example, if $a,b,c$ are elements of $S$, then $(a,b,c)$ is an element of $S^*$.

Given two lists $x$ and $y$, the former of length $m$ and the latter of length $n$, their \textbf{concatenation} $x * y$ is a list of length $m + n$, given as follows:

\begin{displaymath}
i \mapsto \left\{ \array { x_i     & if\; i \lt m \\
                              y_{i-m} & if\; i \geq m } \right.
\end{displaymath}
One can now show that concatenation is associative with the empty list as identity; hence $S^*$ is a monoid.

\hypertarget{recursively}{}\subsubsection*{{Recursively}}\label{recursively}

The (underlying) set $S^*$ may be defined as an [[inductive type]] as follows. There are two basic constructors, one with no arguments, and one with two arguments, of which one is an element of $S$ and the other is an element of $S^*$. By the yoga of inductive types, that is a complete definition, but we spell it out in more detail while also giving terminology and notation.

So, a \textbf{list} is either the \textbf{[[empty list]]} or the \textbf{cons} (short for `constructor' and deriving from Lisp) of an element $a$ of $S$ and a (previously constructed) list $x$. The empty list may may be written $()$, $*$, or $nil$, perhaps with a subscript $S$ if desired; the cons of $a$ and $x$ may be written $a : x$, $(a) * x$, or $cons(a,x)$. We interpret the definition [[recursion|recursively]], so we can list the elements of $S^*$ in the order in which they appear:

\begin{itemize}%
\item $()$,
\item $a : ()$,
\item $a : b : ()$,
\item $a : b : c : ()$,
\item etc.

\end{itemize}
Here, $a, b, c, \ldots$ are elements of $S$. We may continue the `etc' as far as we like, but no farther; while there are lists of arbitrarily long finite length, there are no lists of infinite length. (We would get such lists, however, if we interpreted the definition [[corecursion|corecursively]], known in computer science as a [[stream]].) We normally abbreviate the lists above as follows:

\begin{itemize}%
\item $()$,
\item $(a)$,
\item $(a,b)$,
\item $(a,b,c)$,
\item etc.

\end{itemize}
We still must define the monoidal structure on $S^*$; we define the \textbf{concatenation} $x * y$ of $x$ and $y$ recursively in $x$. To be explicit:

\begin{itemize}%
\item $() * y = y$;
\item $(a : x) * y = a : (x * y)$ (with parentheses for grouping, but the parentheses can be dropped now that have this definition).

\end{itemize}
One can now show that concatenation is associative with the empty list as identity; hence $S^*$ is a monoid.

\hypertarget{by_general_abstract_nonsense}{}\subsubsection*{{By general abstract nonsense}}\label{by_general_abstract_nonsense}

To prove that the category [[Mon]] of monoids is a [[complete category]], one normally shows that the [[forgetful functor]] $U$ (from $Mon$ to the category [[Set]] of sets) preserves all [[limits]]. Then, the [[adjoint functor theorem]] defines a [[left adjoint]] to $U$ if a size condition is met; this adjoint is the functor $*$ that takes a set to its free monoid $S^*$.

To be sure, meeting the solution set condition basically requires starting the constructions in one of the other definitions above; but the proofs may all be thrown onto the adjoint functor theorem.

Another abstract approach is given in the following general theorem, which applies to more general [[monoid in a monoidal category|monoids in a monoidal category]]:

\begin{theorem}
\label{monoid_in_monoidal}\hypertarget{monoid_in_monoidal}{}
Suppose $C$ is a monoidal category with countable coproducts for which the tensor product distributes over countable coproducts (for example, a [[cocomplete category|cocomplete]] [[monoidal biclosed category]]). Then a left adjoint to the forgetful functor $Mon(C) \to C$ exists, taking an object $c$ to

\begin{displaymath}
\sum_{n \geq 0} c^{\otimes n},
\end{displaymath}
which thereby becomes the free monoid on $c$.

\end{theorem}
This applies immediately to $C = Set$, as this is a cocomplete [[cartesian closed category]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item If $S$ is the [[empty set]], then $S^*$ consists only of the empty list; it is the trivial monoid, one manifestation of the [[point]].
\item If $S$ is the point, then $S^*$ is $\mathbf{N}$; the only information in a list of indistinguishable points is the length of the list. The monoid operation on $\mathbf{N}$ is addition.
\item If $S$ is $\mathbf{N}$, then $\mathbf{N}^*$ is still a [[denumerable set]]. But note that $\mathbf{N} \cong \mathbf{N}^*$ only as sets (that is, ${|\mathbf{N}^*|} = {|\mathbf{N}|} = \aleph_0$ as [[cardinal numbers]]); they are quite different as monoids.
\item Generalising the above, ${|S^*|} = {|S|}$ is $S$ is an [[infinite set]], or more generally ${|S^*|} = max(\aleph_0,S)$ if $S$ is an [[inhabited set]]. (These theorems probably require the [[axiom of choice]], but I haven't checked thoroughly.)

\end{itemize}
\hypertarget{the_free_monoid_monad}{}\subsection*{{The free monoid monad}}\label{the_free_monoid_monad}

If the free monoid functor $F\colon Set \to Mon$ is followed by the forgetful functor $U\colon Mon \to Set$, then we get a [[monad]] on $Set$. This monad is very important in [[computer science]], where it is known as the \textbf{list monad}.

The list monad bears the same relation to [[multicategories]] as the [[identity monad]] on $Set$ bears to ordinary [[categories]]. This relation should be explained at [[generalized multicategory]].

\hypertarget{foundational_relevance}{}\subsection*{{Foundational relevance}}\label{foundational_relevance}

Every definition of free monoid makes use of some form of [[axiom of infinity]], either $\mathbf{N}$ directly or the ability to form general inductive types. Indeed, as $\mathbf{N} = pt^*$, the axiom of infinity follows from the existence of free monoids.

In [[topos theory]] the equivalent of the above theorem \ref{monoid_in_monoidal} is due to C. J. Mikkelsen:

\begin{prop}
\label{free_monoids_in_topos}\hypertarget{free_monoids_in_topos}{}
Let $\mathcal{S}$ be a topos and $\mathbf{mon}(\mathcal{S})$ its category of internal monoids. Then $\mathcal{S}$ has a NNO precisely if the forgetful functor $U:\mathbf{mon}(\mathcal{S})\to \mathcal{S}$ has a left adjoint.

\end{prop}
For a proof see Johnstone (\hyperlink{JT77}{1977},p.190).

Furthermore then it is a theorem due to [[Andreas Blass]] (\hyperlink{Blass}{1989}) that $\mathcal{S}$ has a NNO precisely if $\mathcal{S}$ has an [[classifying topos for the theory of objects|object classifier]] $\mathcal{S}[\mathbb{O}]$.

A consequence of this, discussed in sec. B4.2 of (Johnstone 2002,I p.431), is that [[classifying topos|classifying toposes]] for [[geometric theories]] over $\mathcal{S}$ exist precisely if $\mathcal{S}$ has a NNO.

From a different perspective then, in a topos the existence of free objects over various gadgets like e.g. [[algebraic theory|algebraic theories]] or [[geometric theory|geometric theories]] often hinge on the existence of free monoids, the intuition being that the free monoids permit to construct a free model \emph{syntactically} by providing for the (syntactic) building blocks needed for this process.

Notice that algebraic theories can nevertheless have free algebras even if the ambient topos lacks a NNO. This may happen for algebraic theories that have the property that the free algebra on a finite set of generators has a finite carrier e.g. in the topos $FinSet$ of finite sets [[graphic category|free graphic monoids]] exist.

\hypertarget{stacks_and_queues}{}\subsection*{{Stacks and queues}}\label{stacks_and_queues}

In [[computer science]], lists often appear as \emph{stacks} (not to be confused with the [[stacks]] from higher sheaf theory) and \emph{queues}.

Fix a [[monoidal category]] that has [[coproducts]] with the [[unit object]] $I$. Given an [[object]] $A$, an object of \textbf{stacks} on $A$ is an object $S_A$ equipped with [[morphisms]] $push_A\colon S_A \otimes A \to S_A$ and $pop_A\colon S_A \to S_A \otimes A + I$ such that these diagrams commute:

\begin{displaymath}
\array {
   S_A \otimes A &                     & \overset{\iota_{S_A \otimes A,I}}\to &                  & S_A \otimes A + I \\
                 & {}_{push_A}\searrow &                                      & \nearrow_{pop_A} &                                       & \searrow^{push_A + id_I} \\
                 &                     & S_A                                  &                  & \underset{\iota_{S_A \otimes A,I}}\to &                          & S_A + I
   }
\end{displaymath}
The idea is that $push_A$ and $pop_A$ are as close to [[inverse morphism|inverses]] as reasonably possible, but $pop_A$ takes us to $S_A \otimes A + I$ rather than to $S_A \otimes A$, because of the empty stack.

Queues are a little more complicated. An object of \textbf{queues} on $A$ is an object $Q_A$ equipped with morphisms $ins_A\colon A \otimes Q_A \to Q_A$ (`insert') and $rem_A\colon Q_A \to Q_A \otimes A + I$ (`remove'). These operations are far from inverses; whereas popping a stack returns the last item to be pushed onto it, removing an item from a queue returns the \emph{first} item to have been inserted into it.

What are the diagrams for this? I seem to recall that we need a [[distributive category]]; in particular, we need a [[cartesian monoidal category]], so that $\otimes$ is $\times$. But perhaps a [[2-rig]] will be sufficient?

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[classifying topos for the theory of objects]]


\item [[natural numbers object]]


\item [[arithmetic pretopos]]


\item [[tensor algebra]]


\item [[cofree coalgebra]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Jean Bénabou]], \emph{Some Remarks on Free Monoids in a Topos} , pp.20-29 in LNM \textbf{1488} Springer Heidelberg 1991.


\item [[Andreas Blass]], \emph{Classifying topoi and the axiom of infinity} , Algebra Universalis \textbf{26} (1989) pp.341-345.


\item [[Peter Johnstone]], \emph{Topos Theory} , Academic Press New York 1977. (Dover reprint Minneola 2014, chap. 6)



\end{itemize}
[[!redirects list]] [[!redirects lists]] [[!redirects finite list]] [[!redirects finite lists]] [[!redirects finite sequence]] [[!redirects finite sequences]]

[[!redirects concatenation]] [[!redirects cons]]

[[!redirects free monoid]] [[!redirects free monoid functor]] [[!redirects free monoid monad]] [[!redirects list monad]]

[[!redirects queue]] [[!redirects queues]]

[[!redirects list object]] [[!redirects list objects]]



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