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A stack may be implemented as, for example, a singly linked list with a pointer to the top element. A stack may be implemented to have a bounded capacity. If the stack is full and does not contain enough space to accept another element, the stack is in a state of stack overflow. A stack is needed to implement depth-first search.
For example, if is a scheme in (/), then it determines the contravariant functor = (,) and the corresponding fibered category is the stack associated to X. Stacks (or prestacks) can be constructed as a variant of this construction.
In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory.Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves, and the moduli stack of elliptic curves.
For example, if denotes the moduli stack of rank-n vector bundles, then there is a presentation given by the trivial bundle over (). A quasi-affine morphism between algebraic stacks is a morphism that factorizes as a quasi-compact open immersion followed by an affine morphism .
An effective quotient orbifold, e.g., [/] where the action has only finite stabilizers on the smooth space , is an example of a quotient stack. [2]If = with trivial action of (often is a point), then [/] is called the classifying stack of (in analogy with the classifying space of ) and is usually denoted by .
Deligne–Mumford stacks are typically constructed by taking the stack quotient of some variety where the stabilizers are finite groups. For example, consider the action of the cyclic group C n = a ∣ a n = 1 {\displaystyle C_{n}=\langle a\mid a^{n}=1\rangle } on C 2 {\displaystyle \mathbb {C} ^{2}} given by a ⋅ : ( x , y ) ↦ ( ζ n x , ζ ...
For example, a stack may have operations push(x) and pop(), that operate on the only existing stack. ADT definitions in this style can be easily rewritten to admit multiple coexisting instances of the ADT, by adding an explicit instance parameter (like S in the stack example below) to every operation that uses or modifies the implicit instance.
A differentiable stack is a stack : together with a special kind of representable submersion (every submersion described above is asked to be surjective), for some manifold . The map F X → C {\displaystyle F_{X}\to {\mathcal {C}}} is called atlas, presentation or cover of the stack X {\displaystyle X} .