pullbacki

Pullbacki is a term occasionally used in mathematical literature to denote a generalized or indexed form of the pullback construction, typically arising in category theory and its applications. It is not a universally standardized term, but it describes a family of fiber-product-like objects that generalize the ordinary pullback to a collection of maps indexed by a set I.

Definition and idea

In a category C with enough limits, take a base object Z and a family of morphisms f_i: X_i → Z for i in I. A pullbacki consists of an object P equipped with projection morphisms p_i: P → X_i for each i in I, such that all composites f_i ∘ p_i are equal to a common morphism c: P → Z. The object P is universal with this property: for any object Q with maps q_i: Q → X_i making all f_i ∘ q_i equal to the same map to Z, there exists a unique factorization Q → P commuting with the p_i and f_i.

Relation to ordinary pullbacks

If I has two elements and the maps f_1, f_2: X_1, X_2 → Z are given, the pullbacki reduces to the ordinary pullback X_1 ×_Z X_2. For larger index sets, the pullbacki is a limit over the diagram formed by the X_i together with their maps to Z.

Examples and applications

In Sets, P consists of tuples (x_i) in the product ∏ X_i such that all f_i(x_i) agree in Z. In topology, one obtains fibered products over Z; in algebraic geometry, fibered products of schemes over a base scheme are a familiar special case. The concept supports constructions that require coordinating multiple maps to a common base.

Usage and notes

The term pullbacki appears in only some texts and discussions; other authors describe the same construction using limits over a multi-map diagram. As with pullbacks in general, existence depends on the category having the appropriate limits.