The notion of a Cauchy sequence goes back to work of Bolzano and Cauchy; it provides a criterion for convergence. The construction of the real numbers from the rationals via equivalence classes of Cauchy sequences is due to Cantor [Ca1872] and Méray [Me1869]. In fact, Charles Méray was apparently the first to provide a rigorous theory of irrational numbers, shortly before Georg Cantor. The achievement of Méray is well explained in a biographical note by Abraham Robinson [Ro2008]. About Méray's article "Remarques sur la nature des quantités définies par la condition de servir de limites à des variables données" from 1869 he writes: "The paper marked the first appearance in print of an 'arithmetical' theory of irrational numbers. Some years earlier Weierstrass had, in his lectures, introduced the real numbers as sums of sequences or, more precisely, indexed sets, of rational numbers; but he had not published his theory and there is no trace of any influence of Weierstrass' thinking on Méray's. Dedekind also seems to have developed his theory of irrationals at an earlier date, but he did not publish it until after the appearance of Cantor's relevant paper in 1872."
The construction of the real numbers via equivalence classes of Cauchy sequences admits an obvious generalisation in the setting of metric spaces, leading to the completion of a metric space.
For categories there is a notion of Cauchy completion that was introduced in a paper by Borceux and Dejean [BD1986]. For a given category $\cal C$, this identifies with the idempotent completion, which one obtains by taking the category of all retracts of representable functors in the category of functors from $\cal C$ to the category of sets. A more sophisticated notion of Cauchy completion for enriched categories is due to Lawvere [La1973]; it connects to the completion of metric spaces when a metric space is viewed as an enriched category.
Cauchy sequences: The notion of a Cauchy sequence can be defined in any category $\cal C$ as a sequence of morphisms $X_0 \to X_1 \to X_2 \to \cdots$ satisfying the following: for every object $C$ there exists $N_C\ge 0$ such that for all $n \ge m \ge N_C$ the morphism $X_m \to X_n$ induces a bijection $\Hom(C,X_m) \to \Hom(C,X_n)$ [Kr2018]. This leads to another notion of Cauchy completion of $\cal C$.
Completion via Cauchy sequences: For a category $\cal C$ there is a natural construction of a completion $\hat{\cal C}$ that contains $\cal C$ as a full subcategory so that each Cauchy sequence in $\cal C$ admits a colimit in $\hat{\cal C}$. A morphism between sequences $(X_n)$ and $(Y_n)$ is given by a compatible sequence of morphisms $(X_n \to Y_n)$ in $\cal C$, and one calls this eventually invertible if for any object $C$ the induced map $\Hom(C,X_n) \to \Hom(C,Y_n)$ is bijective for $n \gg 0$. The completion of $\cal C$ is now obtained from the category of all Cauchy sequences (with the obvious morphisms) by formally inverting the eventually invertible morphisms (so by attaching formal inverses). This yields a new category $\hat {\cal C}$, and $\cal C$ identifies with the full subcategory of constant sequences. The morphisms $X \to Y$ in $\hat {\cal C}$ can be identified with equivalence classes of pairs of morphisms $X \to Y' \leftarrow Y$ such that $Y \to Y'$ is eventually invertible, by taking such a pair to the composition of $X \to Y'$ with the inverse of $Y \to Y'$.
Example 1: View the set $\mathbb{Q}$ of rational numbers with the usual ordering as a category (with a unique morphism $x \to y$ if and only if $x \leq y$). Then the completion of $\mathbb{Q}$ identifies with the ordered set $\mathbb{R}$ of real numbers (plus an element $\infty$), by taking a Cauchy sequence to its limit, if the sequence is bounded, and to $\infty$ otherwise.
Example 2: Fix any ring and consider the category $\cal C$ of modules of finite composition length. Under a mild finiteness condition on $\cal C$, the completion of $\cal C$ identifies with the category of all artinian modules, by taking a Cauchy sequence to its colimit.
Example 3: Fix a right coherent ring and consider the category $\cal C$ of perfect complexes. The full subcategory of the completion of $\cal C$ given by the objects having bounded cohomology identifies with the bounded derived category of finitely presented modules, by taking a Cauchy sequence to its colimit.
Further examples will be added here as they become available. Everybody is welcome to submit such examples (by sending a message to Henning Krause).
[BD1986] | F. Borceux and D. Dejean Cauchy completion in category theory, Cahiers de Topologie et Géométrie Différentielle Catégoriques 27 (1986) no. 2, 133–146, [Link]. |
[Ca1872] | G. Cantor, Ueber die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen, Math. Ann. 5 (1872), no. 1, 123–132, doi:10.1007/BF01446327, [pdf], [Reprint]. |
[Kr2018] | H. Krause, Completing perfect complexes, with appendices by T. Barthel and B. Keller, arXiv:1805.10751. |
[La1973] | F. W. Lawvere, Metric spaces, generalized logic, and closed categories, Rend. Sem. Mat. Fis. Milano 43 (1973), 135–166, doi:10.1007/BF02924844, [Reprint]. |
[Me1869] | C. Méray, Remarques sur la nature des quantités définies par la condition de servir de limites à des variables données, Revue des sociétés savantes 4 (1869), 280–289 [Link]. |
[Ro2008] | A. Robinson, Méray, Hugues Charles Robert, in: Complete Dictionary of Scientific Biography, Vol. 9, Detroit, Charles Scribner's Sons, 2008, 307–308, [Link]. |