> 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.
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 stuff, structure, property. In that formalization objects in some category $D$ are objects in some category $C$ equipped with extra structure if there is a functor $p \colon D \to C$ such that
Depending on author and situation, more properties are required of this functor (Ehresmann 57, Ehresmann 65, Adamek-Rosicky-Vitale 09, remark 13.18):
$p$ is an amnestic functor ($p$-vertical isomorphisms are identities),
$p$ is an isofibration (isomorphisms can be lifted along $p$).
However, notice that these two conditions violate the principle of equivalence for categories. In the terminology of strict categories one might hence refer to these conditions as expressing “strict extra 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 $L$-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 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
is again faithful.
Thus, the English word “structure” is used in several slightly differing mathematical senses.
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 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…”
In model theory, however, the term structure is not a mass noun; it refers to a 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 language. When one adds axioms to a language to make a theory, then a structure of the language where those axioms get interpreted as properties satisfied by the structure is called a 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).
There are gazillions of examples of objects equipped with extra structure. The most familiar is maybe
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.
Evident as the notion of 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 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.
functions that preserves extra structure are called homomorphisms; relations that preserve extra structure are called logical relations_
Charles Ehresmann, Gattungen in Lokalen Strukturen, 1957
Charles Ehresmann, Catégories et Structures, Dunod, 1965
Adamek, Rosicky, Vitale, Algebraic theories pdf (2009)