Very abstractly, we may define a structured set as an object of any concrete category, that is an object of any category equipped with a faithful functor to the category of sets. (Some authors require that be representable for a concrete category, but we do not need that here.) Given two structured sets and (in the same category ), a function between their underlying sets preserves the structure if it lies in the image of , that is if there exists a (necessarily unique since is faithful) morphism in such that .
More concretely, we may define a type of structure on sets as an operation that, to any set , assigns a set of -structures on . At minimum, we should have whenever (where is isomorphism in ), so that the concept is structural. But for good behaviour, we actually want something more coherent; we want an additional operation that, to any bijection , assigns a bijection , such that:
In other words, (or equivalently ) is a functor from the underlying groupoid of to itself (or equivalently to all of ). In particular, any automorphism of a single set defines an automorphism of the -structures on , giving an action of the symmetric group on .
(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 of structure on sets, we define a -structured set to be a set equipped with an element of . Given -structured sets and , a bijection preserves the -structure on and if .
In general, there is no notion of whether an arbitrary function preserves -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.
Morally, either of the abstract and concrete versions can be converted into the other. Technically, there are some restrictions.
Given a category and a faithful functor , we may define a type of structure on sets as follows:
For each set , consider the essential fibre of over , the collection of pairs where is an object of and is a bijection. We consider two such pairs and to be equivalent if there is an isomorphism in such that . (Because is faithful, any such must be unique.) Define to be the quotient set of the essential fibre modulo this equivalence relation. That is, a -structure on is an equivalence class .
Given a bijection , we must define . So given as above, let map to in . It's easy to check that this is well defined as a function from to ; we can also check that this makes into a functor and that the abstract and concrete definitions of whether a bijection preserves -structure agree.
Technicality: If is a large category, then might be a proper class instead of a set. In this case, we can pass to a larger universe; it is not essential for that both copies of be the same size. But for the above description to make sense as it is, we must require that have essentially small fibres.
Given a type of structure on sets, we cannot quite reconstruct the category , but we can reconstruct its core . That is, we can say what the objects and isomorphisms of are, if not the morphisms of in general.
An object of is simply a pair consisting of a set and an element of . Given two such objects, an isomorphism from to is simply a structure-preserving map from to , that is a bijection such that . Then it is straightforward to check that this defines a groupoid . This groupoid, the groupoid of -structured sets, is naturally equipped with a faithful forgetful functor , given by .
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 (not the!) category of -structured sets by picking, for each and , a collection of functions from to , such that:
whenever and , then ;
the identity map on belongs to ; and
a bijection from to is an isomorphism (as defined above) if and only if both and .
The last condition states precisely that the underlying groupoid of any concrete category of -structured sets is the groupoid of -structured sets.
Any category of -structured sets is still (like the groupoid of such sets) a concrete category.
If we start with a type of structures on sets, construct from this a groupoid and a faithful functor and then construct from this another type of structures, then will be equivalent to in the sense that there is a natural isomorphism between them as functors from to .
If we start with a category and a faithful functor , construct from this a type of structures , and then construct from this a groupoid with a faithful functor , then will in fact be the core of , with to restriction of 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.
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).
Given any category whatsoever, we may define a type of structure on objects of as a functor to from the underyling groupoid of . Then any faithful functor whatsoever defines a type of structure on objects of (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 presents the objects of as objects of with extra structure.