QUOTIENT STACK Meaning and
Definition
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Quotient stack, also known as a stack quotient, is a concept in mathematics and geometry that refers to a type of geometric object obtained by taking a quotient of a given space or stack by a group or an equivalence relation.
In general, a stack is a mathematical structure that extends the notion of a space. It is a collection of objects (or spaces) together with transformation maps between them, which satisfy certain conditions. A quotient stack is consequently defined as the result of dividing this stack by a group or equivalence relation.
To be more specific, let's consider a stack defined on a category of spaces. The quotient stack is then constructed by taking the category of objects or spaces that are invariant under the group or relation, and considering these objects as the new points in the quotient stack. The morphisms or transformation maps between the new points are given by the original maps restricted to these invariant objects.
The quotient stack is a powerful tool in algebraic geometry and generalizes the concept of a quotient space. It allows for the study of geometric objects that are obtained by dividing a space by a group or relations that have nontrivial stabilizers. These objects often appear in various geometric constructions, such as moduli spaces, where the quotient stack provides a more comprehensive representation of the geometry involved.
Etymology of QUOTIENT STACK
The word "quotient" comes from the Latin word "quotiens", meaning "how many times" or "as often as". It originally referred to the result of dividing one quantity by another.
The term "stack" in mathematics typically refers to a collection of mathematical objects arranged one on top of another. In algebraic geometry, a "stack" is a generalization of the concept of a space, where the points are replaced by more complicated geometric objects.
The term "quotient stack" combines these two concepts. It refers to a geometric object formed by "stacking" or putting together certain geometric objects in a way that reflects the division or quotient in some sense. It is often used in the context of algebraic geometry and category theory to study the geometric properties of objects obtained by taking a quotient or division.
