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\begin{document}

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\section*{structured set}

\hypertarget{structured_sets}{}\section*{{Structured sets}}\label{structured_sets}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{abstract}{Abstract}\dotfill \pageref*{abstract} \linebreak
\noindent\hyperlink{Concrete}{Concrete}\dotfill \pageref*{Concrete} \linebreak
\noindent\hyperlink{conversions}{Conversions}\dotfill \pageref*{conversions} \linebreak
\noindent\hyperlink{abstract_to_concrete}{Abstract to concrete}\dotfill \pageref*{abstract_to_concrete} \linebreak
\noindent\hyperlink{concrete_to_abstract}{Concrete to abstract}\dotfill \pageref*{concrete_to_abstract} \linebreak
\noindent\hyperlink{back_and_forth}{Back and forth}\dotfill \pageref*{back_and_forth} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{structured_objects}{Structured objects}\dotfill \pageref*{structured_objects} \linebreak
\noindent\hyperlink{see_also}{See also}\dotfill \pageref*{see_also} \linebreak


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

A structured set is, of course, a [[set]] equipped with [[extra structure]]. It is not the individual structured set that matters so much as the [[category]] of sets with a particular sort of structure.

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

\hypertarget{abstract}{}\subsubsection*{{Abstract}}\label{abstract}

Very abstractly, we may define a \textbf{structured set} as an [[object]] of any [[concrete category]], that is an object of any category $C$ equipped with a [[faithful functor]] $U\colon C \to Set$ to the [[category of sets]]. (Some authors require that $U$ be [[representable functor|representable]] for a concrete category, but we do not need that here.) Given two structured sets $X$ and $Y$ (in the same category $C$), a [[function]] $f\colon U(X) \to U(Y)$ between their underlying sets \textbf{preserves} the structure if it lies in the image of $U$, that is if there exists a (necessarily unique since $U$ is faithful) [[morphism]] $\tilde{f}\colon X \to Y$ in $C$ such that $U(\tilde{f}) = f$.

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

More concretely, we may define a \textbf{type of structure on sets} as an operation $T$ that, to any set $A$, assigns a set $T(A)$ of \textbf{$T$-structures} on $A$. At minimum, we should have $T(A) \cong T(B)$ whenever $A \cong B$ (where $\cong$ is [[isomorphism]] in $Set$), so that the concept is [[structural set theory|structural]]. But for good behaviour, we actually want something more coherent; we want an additional operation that, to any [[bijection]] $f\colon A \to B$, assigns a bijection $T(f)\colon T(A) \to T(B)$, such that:

\begin{itemize}%
\item $T(\id_A) = \id_{T(A)}$,
\item $T(f g) = T(f) T(g)$.

\end{itemize}
In other words, $T\colon Set_\cong \to Set_\cong$ (or equivalently $T\colon Set_\cong \to Set$) is a [[functor]] from the [[underlying groupoid]] of $Set$ to itself (or equivalently to all of $Set$). In particular, any [[automorphism]] of a single set $A$ defines an automorphism of the $T$-structures on $A$, giving an [[action]] of the [[symmetric group]] $S_A$ on $T(A)$.

(Compare the notion of [[structure type]] from [[combinatorics]], which is a set-valued functor on the groupoid of [[finite sets]]. Every combinatorial structure type can be interpreted as a type of structure, where only finite sets are capable of supporting the structure.)

Given a type $T$ of structure on sets, we define a \textbf{$T$-structured set} to be a set $A$ equipped with an [[element]] of $T(A)$. Given $T$-structured sets $X = (A,\sigma)$ and $Y = (B,\tau)$, a bijection $f\colon A \to B$ \textbf{preserves} the $T$-structure on $X$ and $Y$ if $T(f)(\sigma) = \tau$.

In general, there is no notion of whether an arbitrary function $f\colon A \to B$ preserves $T$-structure, although such a notion may be defined in many cases. So to get a concrete construction of a [[concrete category]], we specify whatever morphisms we like, subject to the restriction that they form a category and have the correct [[core]] given above.

[[Bourbaki]]'s theory of structure, while not described in category-theoretic terms, is essentially the above.

\hypertarget{conversions}{}\subsection*{{Conversions}}\label{conversions}

Morally, either of the abstract and concrete versions can be converted into the other. Technically, there are some restrictions.

\hypertarget{abstract_to_concrete}{}\subsubsection*{{Abstract to concrete}}\label{abstract_to_concrete}

Given a category $C$ and a faithful functor $U\colon C \to Set$, we may define a type $T$ of structure on sets as follows:

For each set $A$, consider the [[essential fiber|essential fibre]] of $U$ over $A$, the collection of pairs $(X,f)$ where $X$ is an object of $C$ and $f\colon U(X) \to A$ is a bijection. We consider two such pairs $(X,\sigma)$ and $(Y,\tau)$ to be [[equivalent]] if there is an [[isomorphism]] $h\colon X \to Y$ in $C$ such that $U(h) ; \tau = \sigma$. (Because $U$ is faithful, any such $h$ must be unique.) Define $T(A)$ to be the [[quotient set]] of the essential fibre modulo this [[equivalence relation]]. That is, a $T$-structure on $A$ is an equivalence class $[(X,\sigma)]$.

Given a bijection $f\colon A \to B$, we must define $T(f)\colon T(A) \to T(B)$. So given $(X,\sigma)$ as above, let $T(f)$ map $[(X,\sigma)]$ to $[(X,\sigma;f)]$ in $T(B)$. It's easy to check that this is well defined as a function from $T(A)$ to $T(B)$; we can also check that this makes $T$ into a functor and that the abstract and concrete definitions of whether a bijection preserves $T$-structure agree.

Technicality: If $C$ is a [[large category]], then $T(A)$ might be a [[proper class]] instead of a set. In this case, we can pass to a larger [[universe]]; it is not essential for $T\colon Set_\cong \to Set$ that both copies of $Set$ be the same size. But for the above description to make sense as it is, we must require that $U$ have essentially small fibres.

\hypertarget{concrete_to_abstract}{}\subsubsection*{{Concrete to abstract}}\label{concrete_to_abstract}

Given a type $T$ of structure on sets, we cannot quite reconstruct the category $C$, but we can reconstruct its [[core]] $C_\cong$. That is, we can say what the [[objects]] and [[isomorphisms]] of $C$ are, if not the [[morphisms]] of $C$ in general.

An object of $C$ is simply a pair $(A,\sigma)$ consisting of a set $A$ and an element $\sigma$ of $T(A)$. Given two such objects, an isomorphism from $(A,\sigma)$ to $(B,\tau)$ is simply a structure-preserving map from $A$ to $B$, that is a bijection $f\colon A \to B$ such that $T(f)(\sigma) = \tau$. Then it is straightforward to check that this defines a [[groupoid]] $C_\cong$. This groupoid, the \textbf{groupoid of $T$-structured sets}, is naturally equipped with a faithful [[forgetful functor]] $U\colon C_\cong \to Set$, given by $U(A,\sigma) \coloneqq A$.

While defining isomorphisms of structured sets is an exact science, choosing more general morphisms of structured sets is something of an art. In principle, we may define a (\emph{not} the!) \textbf{category of $T$-structured sets} by picking, for each $(A,\sigma)$ and $(B,\tau)$, a collection $Hom_{\sigma,\tau}(A,B)$ of [[functions]] from $A$ to $B$, such that:

\begin{itemize}%
\item whenever $f \in Hom_{\sigma,\tau}(A,B)$ and $g \in Hom_{\tau,\upsilon}(B,C)$, then $f ; g \in Hom_{\sigma,\upsilon}(A,C)$;


\item the [[identity map]] on $A$ belongs to $Hom_{\sigma,\sigma}(A,A)$; and


\item a [[bijection]] $f$ from $A$ to $B$ is an isomorphism (as defined above) if and only if both $f \in Hom_{\sigma,\tau}(A,B)$ and $f^{-1} \in Hom_{\tau,\sigma}(B,A)$.



\end{itemize}
The last condition states precisely that the [[underlying groupoid]] of any concrete category of $T$-structured sets is the groupoid of $T$-structured sets.

Any category of $T$-structured sets is still (like the groupoid of such sets) a concrete category.

\hypertarget{back_and_forth}{}\subsubsection*{{Back and forth}}\label{back_and_forth}

If we start with a type $T$ of structures on sets, construct from this a groupoid $C_\cong$ and a faithful functor $U\colon C_\cong \to Set$ and then construct from this another type $T'$ of structures, then $T'$ will be equivalent to $T$ in the sense that there is a [[natural isomorphism]] between them as functors from $Set_\cong$ to $Set$.

If we start with a category $C$ and a faithful functor $U\colon C \to Set$, construct from this a type of structures $T\colon Set_\cong \to Set$, and then construct from this a groupoid $C_\cong$ with a faithful functor $U\colon C_\cong \to Set$, then $C_\cong$ will in fact be the [[core]] of $C$, with $U\colon C_\cong \to Set$ to restriction of $U\colon C \to Set$ to this core, up to [[equivalence of categories]].

(Do we need proofs?)

Thus the abstract and concrete approaches to structured sets are equivalent, except that the concrete approach does not include a specification of what are the noninvertible morphisms between structured sets.

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

Almost everything in contemporary [[mathematics]] is an example of a structured set; here we list only a few representative ones (and perhaps also some exceptions).

\begin{itemize}%
\item [[sets]] themselves (trivially);
\item [[groups]], [[rings]], and other objects of [[universal algebra]];
\item [[topological spaces]] and other concrete (based on sets of [[points]]) notions of [[spaces]];
\item some other notions of [[spaces]] such as [[locales]], as long as we only consider isomorphisms, but \emph{not} using functions as the other morphisms;
\item \emph{not} some other notions of spaces such as [[homotopy types]] (at least if defined via topological spaces);
\item [[strict categories]] (where the underlying set is the set of [[morphisms]]) but \emph{not} [[categories]] treated up to [[equivalence of categories|equivalence]].

\end{itemize}
\hypertarget{structured_objects}{}\subsection*{{Structured objects}}\label{structured_objects}

Given any category $S$ whatsoever, we may define a \textbf{type of structure} on objects of $S$ as a functor $T\colon S_\cong \to Set$ to $Set$ from the underyling groupoid of $S$. Then any [[faithful functor]] $U\colon C \to S$ whatsoever defines a type of structure on objects of $S$ (at least if its fibres are essentially small), in the same way as a concrete category defines a type of structure on sets. Indeed, we say that $U$ presents the objects of $C$ as objects of $S$ with \textbf{[[extra structure]]}.

\hypertarget{see_also}{}\subsection*{{See also}}\label{see_also}

\begin{itemize}%
\item [[category of elements]]
\item [[Grothendieck construction]]

\end{itemize}
[[!redirects structured set]] [[!redirects structured sets]]



\end{document}