# nLab structure

Contents

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.

# Contents

## 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 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”.

## 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 $L$-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

$Mod_T \to Struct_{L(T)} \to Set$

is again faithful.

###### Remarks

Thus, the English word “structure” is used in several slightly differing mathematical senses.

1. 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…”

2. 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).

## Examples

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.

## References

Last revised on February 3, 2020 at 23:40:05. See the history of this page for a list of all contributions to it.