Exact completion

In category theory, a branch of mathematics, the exact completion constructs a Barr-exact category from any finitely complete category. It is used to from the effective topos and other realizability toposes.

Construction

Let C be a category with finite limits. Then the exact completion of C (denoted C_ex) has for its objects pseudo-equivalence relations in C.^[1] A pseudo-equivalence relation is like an equivalence relation except that it need not be jointly monic. An object in C_ex thus consists of two objects X₀ and X₁ and two parallel morphisms x₀ and x₁ from X₁ to X₀ such that there exist a reflexivity morphism r from X₀ to X₁ such that x₀r = x₁r = 1_X0; a symmetry morphism s from X₁ to itself such x₀s = x₁ and x₁s = x₀; and a transitivity morphism t from X₁ × _{x1, X0, x0} X₁ to X₁ such that x₀t = x₀p and x₁t = x₁q, where p and q are the two projections of the aforementioned pullback. A morphism from (X₀, X₁, x₀, x₁) to (Y₀, Y₁, y₀, y₁) in C_ex is given by an equivalence class of morphisms f₀ from X₀ to Y₀ such that there exists a morphism f₁ from X₁ to Y₁ such that y₀f₁ = f₀x₀ and y₁f₁ = f₀x₁, with two such morphisms f₀ and g₀ being equivalent if there exists a morphism e from X₀ to Y₁ such that y₀e = f₀ and y₁e = g₀.

Examples

• If the axiom of choice holds, then Set_ex is equivalent to Set.
• More generally, let C be a small category with finite limits. Then the category of presheaves Set^Cop is equivalent to the exact completion of the coproduct completion of C.^[2]
• The effective topos is the exact completion of the category of assemblies.^[2]

Properties

• If C is an additive category, then C_ex is an abelian category.^[3]
• If C is cartesian closed or locally cartesian closed, then so is C_ex.^[4]

References

1. Menni, Matias (2000). "Exact completion and toposes" (PDF). Retrieved 18 September 2016.
2. Carboni, A. (15 September 1995). "Some free constructions in realizability and proof theory". Journal of Pure and Applied Algebra. 103 (2): 117–148. Retrieved 18 September 2016.
3. Carboni, A.; Magno, R. Celia (December 1982). "The free exact category on a left exact one". Journal of the Australian Mathematical Society. 33 (3): 295–301. Retrieved 18 September 2016.
4. Carboni, A.; Rosolini, G. (1 December 2000). "Locally cartesian closed exact completions". Journal of Pure and Applied Algebra. 154 (1–3): 103–116. Retrieved 18 September 2016.

External links

• Exact completion in nLab