Two Preservation Thorems in Infinitary Logic

J. Rosický

A regular sentence in L_{λλ} is defined as (∀x)(φ(x) → (∃y)ψ(x, y)) where φ and ψ are (infinitary) conjunctions of atomic formulas (and x and y are strings of variables). Regular sentences correspond to the algebraic concept of injectivity. In a case when ∃ is the unique existence quantifier, one speaks about limit sentences; they are related to the concept of orthogonality.

Theorem 1 (see [4]) Let λ be a regular cardinal. A class of structures is axiomatized by regular sentences in L_{λλ} iff it is closed under products, λ-directed colimits and λ-pure substructures.

Theorem 2 (see [2]) Let λ be an uncountable regular cardinal. A class of structures is axiomatized by limit sentences in L_{λλ} iff it is closed under limits and λ-directed colimits.

Recall that λ-pure means elementary w.r.t. positive-primitive formulas in L_{λλ} (see [1]). Any class closed under limits and λ-directed colimits is closed under products and λ-pure substructures. For λ = ω, Theorem 1 is an easy consequence of the Compactness Theorem. Theorem 2 does not hold in this case (see [5]).

A geometric sentence in L_{∞λ} is defined as (∀x)(φ(x) → ⋁_{i∈I}(∃y_{i})ψ(x, y_{i})) (where φ, ψ_{i}, x and y_{i} are

as above). Classes of structures axiomatized by geometric sentences are closed under λ-directed colimits and λ-pure substructures. The converse is not true even for λ = ω (see [4]).

The just stated results are motivated by the theory of accessible categories (see [3] and [1]) which is closely related to infinitary model theory and is useful in algebra, homotopy theory or theoretical computer science.

[1] J. Adámek and J. Rosický: Locally presentable and accessible categories Cambridge Univ. Press 1994.

[2] M. Hébert and J. Rosický: Uncountable orthogonality is a closure property, submitted.

[3] M. Makkai and R. Paré: Accessible categories: the foundations of categorical model theory Contemporary Math. 104, AMS, Providence 1989

[4] J. Rosický, J. Adámek and F. Borceux: More on injectivity in locally presentable categories, submitted

[5] H. Volger: Preservation theorems for limits of structures and global sections of sheaves of structures Math. Z. 166 (1979), 27–53

Address: Masaryk University, Janáčkovo nám. 2a, 662 95 Brno, Czech Republic

E-mail: rosickymath.muni.cz