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\begin{document}

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\section*{structure}

\begin{quote}%
This entry is about a general concepts of ``mathematical structure'' in [[category theory]]. It subsumes but is more general than the concept of [[structure in model theory]].


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

\hypertarget{mathematics}{}\paragraph*{{Mathematics}}\label{mathematics}

[[!include mathematicscontents]]

\hypertarget{mathematics_2}{}\paragraph*{{Mathematics}}\label{mathematics_2}

[[!include mathematicscontents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{notions_of_structure}{Notions of structure}\dotfill \pageref*{notions_of_structure} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

It is common in informal language to speak of mathematical objects ``equipped with extra structure'' of some sort. The archetypical examples are [[algebras over a Lawvere theory]] in [[Set]]: these are [[sets]] equipped with the structure of certain algebraic operations. For instance a [[group]] $(G, e, {\cdot})$ is a [[set]] $G$ equipped with a binary operation ${\cdot} : G \times G \to G$, etc.

One may formalize the notion of structure using the language of [[category theory]]. This is discussed at \emph{[[stuff, structure, property]]}. In that formalization [[objects]] in some [[category]] $D$ are objects in some category $C$ \emph{equipped with extra structure} if there is a [[functor]] $p \colon D \to C$ such that

\begin{itemize}%
\item $p$ is a [[faithful functor]].

\end{itemize}
 Depending on author and situation, more properties are required of this functor (\hyperlink{Ehresmann57}{Ehresmann 57}, \hyperlink{Ehresmann65}{Ehresmann 65}, \hyperlink{AdamekRosickyVitale}{Adamek-Rosicky-Vitale 09, remark 13.18}):

\begin{itemize}%
\item $p$ is an [[amnestic functor]] ($p$-vertical [[isomorphisms]] are [[identities]]),


\item $p$ is an [[isofibration]] (isomorphisms can be lifted along $p$).



\end{itemize}
 However, notice that these two conditions violate the [[principle of equivalence]] for [[categories]]. In the terminology of \emph{[[strict categories]]} one might hence refer to these conditions as expressing ``strict extra structure''.

\hypertarget{notions_of_structure}{}\subsection*{{Notions of structure}}\label{notions_of_structure}

A special class of examples of this is the notion of [[structure in model theory]]. In this case one defines a ``language'' $L$ that describes the constants, functions (say operations) and relations with which we want to equip sets, and then sets equipped with those operations and relations are called $L$-[[structure in model theory|structures]] for that language. (Equivalently one might say ``sets with $L$-structure''. Or one might generally say ``$X$-structure'' for ``set with $X$-structure''.) In this case there is a [[faithful functor]] from $L$-structures to their underlying sets, and so this is a special case of the general definition.

We instead say [[model]] of a [[theory]] when we restrict to those structures which satisfy the axioms of a theory (in other words, satisfy \emph{properties} specified by the axioms). In this case there is a full and faithful functor from the category of models of a theory $T$ to the category of structures of the underlying language $L(T)$, while the composition of forgetful functors

\begin{displaymath}
Mod_T \to Struct_{L(T)} \to Set
\end{displaymath}
is again faithful.

\begin{remark}
\label{}\hypertarget{}{}
Thus, the English word ``structure'' is used in several slightly differing mathematical senses.

\begin{enumerate}%
\item Within category theory itself, ``structure'' can function as a kind of mass noun, as in a phrase like ``forgetting structure''. Here it refers to data comprising operations, relations, constants, and \emph{also properties} borne by models of a theory or relative theory, considered abstractly (for example, the functor $Grp \to Set$ which forgets group structure, or the functor $Ring \to Ab$ which forgets multiplicative structure). On the other hand, it can also operate in the singular, where one says for example ``a topological group is a topological space equipped with a group structure, such that\ldots{}''


\item In model theory, however, the term \emph{structure} is not a mass noun; it refers to a \emph{particular} set (or ``structures'' for a family of sets) together with functions, relations, and elements that interpret the symbols of operations, predicates, and constants of a \emph{language}. When one adds axioms to a language to make a \emph{theory}, then a structure of the language where those axioms get interpreted as properties \emph{satisfied} by the structure is called a \emph{model} of the theory. Thus, in summary, the category theorist might refer to ``the structure of a group'' as consisting of a multiplication, a unit, etc., satisfying group axioms, while the model theorist would say that each particular group (like $\mathbb{Z}$) is a model of a theory of groups. For a model theorist, being a model does entail being a structure for the language of groups, but she would also say that a structure for the language of groups need not satisfy any of the axioms of a group (like associativity or unitality).



\end{enumerate}
\end{remark}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

There are gazillions of examples of objects equipped with extra structure. The most familiar is maybe

\begin{itemize}%
\item [[algebraic structure]].

\end{itemize}
Generally the [[forgetful functor]] from a category of algebras over an [[algebraic theory]] down to the base category exhibits the equipment with the corresponding algebraic structure.

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item Evident as the notion of \emph{mathematical structure} may seem these days, it was at least not made explicit until the middle of the 20th century. Then it was the influence of the \emph{[[Bourbaki]]}-project (see there for more) and then later the development of [[category theory]] which made the notion explicit and finally led to the above formalization.


\item [[functions]] that preserves extra structure are called \emph{[[homomorphisms]]}; [[relations]] that preserve extra structure are called [[logical relations]]\_


\item [[Birkhoff's HSP theorem]]


\item [[exceptional structure]]



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

\begin{itemize}%
\item [[Charles Ehresmann]], \emph{Gattungen in Lokalen Strukturen}, 1957


\item [[Charles Ehresmann]], \emph{Cat\'e{}gories et Structures}, Dunod, 1965


\item Adamek, Rosicky, Vitale, \emph{Algebraic theories} \href{http://www.iti.cs.tu-bs.de/~adamek/algebraic_theories.pdf}{pdf} (2009)



\end{itemize}
[[!redirects structure]] [[!redirects structures]]



\end{document}