Complexessequences

Complexessequences is a term used in some branches of mathematics to denote a directed system of chain complexes. Let I be a directed index set (often the natural numbers with the usual order). To each i ∈ I one assigns a chain complex C_i with differential d_i, and for i ≤ j a chain map f_{i→j}: C_i → C_j such that f_{i→k} = f_{j→k} ∘ f_{i→j} for all i ≤ j ≤ k. The data (I, {C_i}, {f_{i→j}}) is called a complexessequence. A morphism between two complexessequences consists of a family of chain maps compatible with the connecting maps.

One studies the direct limit (colimit) of a complexessequence, denoted lim_i C_i, which is again a chain complex with a differential induced by the d_i. Under suitable conditions—for example, finiteness hypotheses or Mittag-Leffler conditions on the induced homology systems—one can relate the homology of the limit to the limit of the homologies: H_*(lim_i C_i) ≈ lim_i H_*(C_i). In general, one must account for potential higher derived limits (lim^1 terms) via spectral sequences.

The mapping telescope provides a concrete model for the colimit of a directed system of chain complexes, encoding the limit as a single complex built from the C_i and the connecting maps.

Applications occur in algebraic topology, algebraic geometry, and homological algebra, particularly when analyzing stabilized or filtered objects, pro-objects, or successive approximations.

See also chain complex, directed system, mapping telescope, homology, spectral sequence.