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Many structures in mathematics are what are called "essentially algebraic"algebraic". This includes all algebraic structures (which model theories given by a functional signature and universally quantified equations): groups, rings, Lie algebras, etc. It also includes algebra-like structures which don't quite fit in this mold, for example categories, where some of the operations may only be partially defined (but the domains of the operations are given by equations involving other operations). There are various nice ways of expressing the syntax of an essentially algebraic theory: one is via limit sketches, another is just to use finitely complete categories, in much the same way that (following Lawvere) classical algebraic theories can be expressed as certain categories with products.

which preserve all limits (have left adjoints) and all filtered colimits. This is essentially Gabriel-Ulmer duality, giving a perfect dual correspondence between theories or syntax of finitary essentially algebraic type (small finitely complete categories) and semantic categories of models of such (which can be described in pure category terms as locally finitely accessible presentable categories).

This perfect duality between syntax and semantics is part of a long story in categorical model theory, culminating the theory of accessible and locally presentable categories. Spiritually, it is reminiscent of many other dualities such as the Tannaka-Krein duality mentioned by Finn Lawler earlier. (

Edit: This is in response to Martin's comment. I'll add give a sample application and some links nontrivial consequences (which, to be sure, can be arrived at via other avenues). The moral for me is that Gabriel-Ulmer duality is a relatively simple statement that can become, after a while, a useful part of one's daily thinking.

My example is the category $\text{Cocomm}$ of cocommutative coalgebras over (let's say) a field $k$. What can we say about it? Well, it's easy to see it is cocomplete: one constructs colimits (coproducts, coequalizers, etc.) as one would on the underlying vector spaces, lifting the vector-space colimit to a coalgebra structure in a canonical way. Much more significantly, there is a fundamental theorem about (cocommutative) coalgebras: each is the union (in particular a filtered colimit) of its finite-dimensional subcoalgebras. Meanwhile, a finite-dimensional coalgebra $C$ has the property that $\hom(C, -): \text{Cocomm} \to Set$ preserves filtered colimits.

People who have taken Gabriel-Ulmer duality into their hearts will immediately recognize the import of these results: $Cocomm$ is locally finitely presentable. This implies something at first unexpected: not only are coalgebras coalgebraic over vector spaces: they are models of an essentially algebraic theory! Indeed, Gabriel-Ulmer duality implies that $Cocomm$ is equivalent to the category of finitely continuous (left exact) functors

$$\text{Cocomm}_{\text{fin.dim.}}^{op} \to Set$$

or if you prefer, to the category of finitely continuous functors $\text{CommAlg}_{fd} \to Set$ (so in this later.case, the relevant "theory" is the category of finite-dimensional commutative algebras).

This unexpected recognition has a host of useful consequences. For example, the category $Cocomm$ is complete. Those who think this is obvious are invited to construct infinite products and equalizers explicitly with their bare hands -- it is not trivial. Also, the category is cartesian closed. Indeed $C$ and $D$ are viewed as cocommutative coalgebras, then the exponential $D^C$ is identified with the left exact functor that takes a finite-dimensional algebra $A$ to $\hom(A^\ast \otimes_k C, D)$, a very pretty and explicit formula.

Similar considerations apply to (not necessarily cocommutative) coalgebras, and to differential graded coalgebras (cocommutative or not).

I think of Gabriel-Ulmer duality as one of the early key results in categorical model theory.

Many structures in mathematics are what are called "essentially algebraic". This includes all algebraic structures (which model theories given by a functional signature and universally quantified equations): groups, rings, Lie algebras, etc. It also includes algebra-like structures which don't quite fit in this mold, for example categories, where some of the operations may only be partially defined (but the domains of the operations are given by equations involving other operations). There are various nice ways of expressing the syntax of an essentially algebraic theory: one is via limit sketches, another is just to use finitely complete categories, in much the same way that (following Lawvere) classical algebraic theories can be expressed as certain categories with products.

If we think of a small finitely complete category as a "theory" $T$, then a model of $T$ in this way of thinking is a finitely continuous functor $T \to Set$. The category of models would then be the category $Cont(T, Set)$ of such functors and transformations between them.

What is amazing and very nice is that the theory $T$ can be recovered from the category of models $M$, by considering functors

$$M \to Set$$

which preserve all limits (have left adjoints) and all filtered colimits. This is essentially Gabriel-Ulmer duality, giving a perfect dual correspondence between theories or syntax of finitary essentially algebraic type (small finitely complete categories) and semantic categories of models of such (which can be described in pure category terms as locally finitely accessible categories).

This perfect duality between syntax and semantics is part of a long story in categorical model theory, culminating the theory of accessible and locally presentable categories. Spiritually, it is reminiscent of many other dualities such as the Tannaka-Krein duality mentioned by Finn Lawler earlier. (I'll add some links to this later.)