Question

Let me try to explain the situation in greater detail. I guess the correspondences are better explained in terms of sketches. (This nLab page needs expansion; Adámek and Rosický give a nice account of sketches; another account can be found in Barr and Wells.) A sketch asserts the existence of certain limits and colimits, or just learned something absolutely fascinating regarding categories limits in the case of a limit sketch, taken together these assertions can be formulated as a sentence in L_∞,∞ (sketchy details below). Like such sentences, every sketch S has a category Mod(S) of modelsunder homomorphisms. These are not always Sketches and accessible as witnessed by the theory with categories go hand in hand.

If S is a single axiom ∃x,y(x≠y) which sketch, then Mod(S) is not completean accessible category, and every accessible category is equivalent to the category of models of a sketch.However

If S is a limit sketch, Adamek then Mod(S) is a locally presentable category, and Rosicky have shown that very locally presentable category is equivalent to the statement that if category of models of some infinitary a limit sketch.

When translated into L_∞,∞, a limit sketch becomes a theory with all homomorphisms axioms of the form

$\forall\bar{x}(\phi(\bar{x})\to\exists!\bar{y}\psi(\bar{x},\bar{y})),$

where $\phi$ and $\psi$ are conjunction of atomic formulas (and the variable lists $\bar{x}$ and $\bar{y}$ can be infinite). When the category is locally finitely presentable, then these axioms can be stated in L_ω,ω. Theories with axioms of this type are essentially characterized by the fact that Mod(T) has equalizers finite limits.

If T is a theory in L_ω,ω and Mod(T) which is closed under finite limits (computed in Mod(∅)), then it Mod(T) is locally finitely presentable category (and hence finitely admissible).

Every locally finitely presentable category is equivalent to a category Mod(T) where T is a limit theory in L_ω,ω (i.e. with axioms as described above).

It is natural to conjecture that this equivalence continues when ω is replaced by ∞. Adámek and Rosický have shown in A remark on accessible and axiomatizable theories (Comment. Math. Univ. Carolin. 37, 1996) is that for a complete category being equivalent equivalent to a (complete) category of models of a sentence in L_∞,∞ and being accessible are equivalent provided that Vopenka's Principle holds. In fact, they proved it twice!

On preaccessible categoriesthis equivalence is itself equivalent to Vopenka's Principle. (It is apparently unknown whether accessible can be strengthened to locally presentable.)

Now, Jif T is a sentence in L_∞,∞, then the category Elem(T) (models of T under elementary embeddings) is always an accessible category. Pure ApplThe category Mod(T) is unfortunately not necessarily accessible. Algebra 105 (1995), 225-232When translated into L_∞,∞ sketches become sentences of a special form. A remark on formula in L_∞,∞ is positive existential if it has the form

$\bigvee_{i \in I} \exists\bar{y}_i \phi_i(\bar{x},\bar{y}_i)$

where each $\phi_i$ is a conjunction of atomic formulas. A basic sentence in L_∞,∞ is conjunction of sentences of the form

$\forall\bar{x}(\phi(\bar{x})\to\psi(\bar{x}))$

where $\phi$ and $\psi$ are positive existential formulas.

A category is accessible if and axiomatizable categories, Commentonly if it is equivalent to a category Mod(T) where T is a basic sentence in L_∞,∞.Math

It would be great if one could simply replace accessible by finitely accessible and sentence in L_∞,∞ by theory in L_ω,ω, as in the locally presentable case above. UnivUnfortunately, this is simply not true. CarolinThe category of models of the basic sentence $\forall x\exists y(x \mathrel{E} y)$ in the language of graphs is accessible but not finitely accessible. 37 (1996), 411-414A counterexample in the other direction is the category of models of $\bigvee_{n<\omega} f^{n+1}(a) = f^n(a)$, which is finitely accessible but not axiomatizable in L_ω,ω.