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\begin{document}

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\section*{algebraic category}

\hypertarget{algebraic_categories}{}\section*{{Algebraic categories}}\label{algebraic_categories}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

An algebraic category is a [[concrete category]] which behaves very much like the categories familiar from [[algebra]], such as [[Grp]], [[Ring]], and [[Vect]], but characterised in category-theoretic terms. But many other categories are also algebraic, most famously [[compact Hausdorff space|CompHausTop]]; one can describe these in purely algebraic terms, but only using infinitary (perhaps even largely many) operations.

There are several definitions of `algebraic' in the literature. Here, we will follow AHS (see references) in using a generous interpretation, but other authors follow Johnstone in using `algebraic' to mean monadic (a stricter requirement), while some authors add finiteness conditions that remove examples such as $Comp Haus Top$. However, all of these notions are related, and we will discuss them here.

The definitions in AHS also include an [[evil]] requirement of unique strict lifts of isomorphisms, which serves to fix algebraic categories up to [[isomorphism of categories|isomorphism]] (instead of mere [[equivalence of categories|equivalence]]), which we omit.

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

Let $A$ be a [[concrete category]]; that is, $A$ is equipped with a [[forgetful functor]] $U\colon A \to Set$ to the [[Set|category of sets]]. For some authors, such a category is called `concrete' only if $U$ is [[representable functor|representable]], but that follows in all the cases considered below; in particular, if $A$ has free objects (that is, if $U$ has a [[left adjoint]] $F$), then $U$ is representable by $F(1)$, where $1$ is a [[singleton]].

\begin{udefn}
The concrete category $A$ is \textbf{algebraic} if the following conditions hold:

\begin{itemize}%
\item $A$ has [[free objects]].
\item The category $A$ has all binary [[coequalizers]].
\item The forgetful functor $U$ preserves and reflects [[extremal epimorphisms]].

\end{itemize}
\end{udefn}
\begin{udefn}
The concrete category $A$ is \textbf{monadic} if the following conditions hold:

\begin{itemize}%
\item $A$ has [[free objects]].
\item The [[adjunction]] $F \dashv U$ is [[monadic adjunction|monadic]].

\end{itemize}
\end{udefn}
\begin{udefn}
An algebraic (or monadic) category is \textbf{bounded} if the following condition holds:

\begin{itemize}%
\item For some [[cardinal number]] $\kappa$ and every $\kappa$-[[directed colimit]] in $A$, the universal [[cocone]] is jointly [[surjection|surjective]] in $Set$.

\end{itemize}
\end{udefn}
\begin{udefn}
An algebraic (or monadic) category is \textbf{finitary} if the following condition holds:

\begin{itemize}%
\item For every finitely [[directed colimit]] in $A$, the universal [[cocone]] is jointly [[surjection|surjective]] in $Set$.

\end{itemize}
\end{udefn}
Note that this is a weakening of the condition that the forgetful functor $U$ is [[finitary functor|finitary]] (that is, that $U$ preserves directed colimits); every universal cocone in $Set$ is jointly surjective, but not conversely.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

Every monadic category is algebraic; an algebraic category is monadic if and only if the forgetful functor $U$ preserves [[congruences]]. (AHS 23.41)

A category is algebraic if and only if it is a [[reflective subcategory]] of a monadic category with [[regular epimorphism|regular epic]] reflector; given an algebraic category, this monadic category is the [[Eilenberg–Moore category]] of the monad $U \circ F$. (AHS 24.3)

Every monadic category is the category of algebras for some [[variety of algebras]], although we must allow potentially a [[proper class]] of infinitary axioms; that is, every monadic category is [[equationally presentable category|equationally presentable]]. Similarly, every algebraic category is the category of algebras for some [[quasivariety of algebras]]; that is, we allow [[conditional statements]] of equations among the axioms. (AHS 24.11)

As special cases of the last item:

\begin{itemize}%
\item A concrete category is bounded monadic if and only if it is equationally presentable (presented by a variety) with a small set of operations (and hence equations).
\item A concrete category is bounded algebraic if and only if it is presented by a quasivariety with a small set of operations.
\item A concrete category is finitary monadic if and only if it is the category of algebras for some finitary variety; that is, we have only a small set of finitary operations.
\item A concrete category is finitary algebraic if and only if it is the category of algebras for some finitary quasivariety.

\end{itemize}
Also, every algebraic category whose [[forgetful functor]] preserves [[filtered colimits]] is the category of [[models]] for some [[first-order theory]]. The converse is false.

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

The typical categories studied in [[algebra]], such as [[Grp]], [[Ring]], [[Vect]], etc, are all finitary monadic categories. The [[monad]] $U \circ F$ may be thought of as mapping a set $x$ to the set of words with alphabet taken from $x$ and the connections between letters taken from the appropriate algebraic operations, with two words identified if they can be proved equal by the appropriate algebraic axioms.

The category of cancellative [[monoids]] is finitary algebraic but not monadic. The category of [[fields]] is not even algebraic.

Assuming the [[ultrafilter principle]], the category of [[compact Hausdorff spaces]] is monadic, but not bounded algebraic. The monad in question takes a set $x$ to the set of [[ultrafilters]] on $x$. (Without the ultrafilter principle, this monad still exists, but it may be quite small, possibly even the [[identity monad]]; passing to [[locales]] does not help.)

Similarly, the category of [[Stone space]]s is algebraic, but not monadic or bounded algebraic.

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

Our definitions are taken from

\begin{itemize}%
\item \textbf{AHS}: [[Jiri Adamek|Jiří Adámek]], [[Horst Herrlich]], [[George Strecker]]; \emph{Abstract and Concrete Categories: [[The Joy of Cats]]}, Sections 23 \&{} 24; \href{http://katmat.math.uni-bremen.de/acc}{web}.

\end{itemize}
Actually, AHS discusses the more general concept of algebraic (etc) \emph{functors}, generalising from $U\colon A \to Set$ to arbitrary functors (not necessarily faithful, not necessarily to $Set$). We actually take our definitions from AHS's characterisation theorems in the case of faithful functors to $Set$. We probably should discuss the more general concept, perhaps at [[algebraic functor]]; we already have [[monadic functor]].

\begin{itemize}%
\item [[Peter Johnstone]]; \emph{[[Stone Spaces]]}, Section 3.8

\end{itemize}
For Johnstone, a concrete category is `algebraic' if and only if it is monadic. However, Johnstone also discusses [[equationally presentable category|equationally presentable categories]].

[[!redirects algebraic category]] [[!redirects algebraic categories]] [[!redirects bounded algebraic category]] [[!redirects bounded algebraic categories]] [[!redirects finitary algebraic category]] [[!redirects finitary algebraic categories]]

[[!redirects monadic category]] [[!redirects monadic categories]] [[!redirects bounded monadic category]] [[!redirects bounded monadic categories]] [[!redirects finitary monadic category]] [[!redirects finitary monadic categories]]



\end{document}