List of mathematical jargon in the context of "Universe (mathematics)"

In category theory, a branch of mathematics, given a morphism f: X → Y and a morphism g: Z → Y, a lift or lifting of f to Z is a morphism h: X → Z such that f = g ∘ h. We say that f factors through h.

Lifts are ubiquitous; for example, the definition of fibrations (see Homotopy lifting property) and the valuative criteria of separated and proper maps of schemes are formulated in terms of existence and (in the last case) uniqueness of certain lifts.

In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function space. Often in mathematical jargon, especially in analysis or geometry, a function could refer to a map of the form ${\displaystyle X\to \mathbb {R} }$ or ${\displaystyle X\to \mathbb {C} }$ where ${\displaystyle X}$ is the space in question. Whilst other maps of the form ${\displaystyle X\to Y}$ between any two spaces are simply referred to as maps. Example of this can be the space of compactly supported functions on a topological space. However in a larger context a function space could just consist of a set of functions (set theoretically) equipped with possibly some extra structure.