Flat morphismIn mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e., is a flat map for all P in X. A map of rings is called flat if it is a homomorphism that makes B a flat A-module. A morphism of schemes is called faithfully flat if it is both surjective and flat. Two basic intuitions regarding flat morphisms are: flatness is a generic property; and the failure of flatness occurs on the jumping set of the morphism.
Étale morphismIn algebraic geometry, an étale morphism (etal) is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology.
A¹ homotopy theoryIn algebraic geometry and algebraic topology, branches of mathematics, A1 homotopy theory or motivic homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. The theory is due to Fabien Morel and Vladimir Voevodsky. The underlying idea is that it should be possible to develop a purely algebraic approach to homotopy theory by replacing the unit interval [0, 1], which is not an algebraic variety, with the affine line A1, which is.
Subobject classifierIn , a subobject classifier is a special object Ω of a category such that, intuitively, the subobjects of any object X in the category correspond to the morphisms from X to Ω. In typical examples, that morphism assigns "true" to the elements of the subobject and "false" to the other elements of X. Therefore, a subobject classifier is also known as a "truth value object" and the concept is widely used in the categorical description of logic. Note however that subobject classifiers are often much more complicated than the simple binary logic truth values {true, false}.
Kan fibrationIn mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard structure on simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category. The name is in honor of Daniel Kan. For each n ≥ 0, recall that the , , is the representable simplicial set Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological standard -simplex: the convex subspace of Rn+1 consisting of all points such that the coordinates are non-negative and sum to 1.
Homotopy category of chain complexesIn homological algebra in mathematics, the homotopy category K(A) of chain complexes in an A is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain complexes Kom(A) of A and the D(A) of A when A is ; unlike the former it is a , and unlike the latter its formation does not require that A is abelian. Philosophically, while D(A) turns into isomorphisms any maps of complexes that are quasi-isomorphisms in Kom(A), K(A) does so only for those that are quasi-isomorphisms for a "good reason", namely actually having an inverse up to homotopy equivalence.
Zero morphismIn , a branch of mathematics, a zero morphism is a special kind of morphism exhibiting properties like the morphisms to and from a zero object. Suppose C is a , and f : X → Y is a morphism in C. The morphism f is called a constant morphism (or sometimes left zero morphism) if for any W in C and any g, h : W → X, fg = fh. Dually, f is called a coconstant morphism (or sometimes right zero morphism) if for any object Z in C and any g, h : Y → Z, gf = hf. A zero morphism is one that is both a constant morphism and a coconstant morphism.
Homotopy groups of spheresIn the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.
ToposIn mathematics, a topos (USˈtɒpɒs, UKˈtoʊpoʊs,_ˈtoʊpɒs; plural topoi ˈtɒpɔɪ or ˈtoʊpɔɪ, or toposes) is a that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the and possess a notion of localization; they are a direct generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic. The mathematical field that studies topoi is called topos theory.
Finitely generated groupIn algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of S and of inverses of such elements. By definition, every finite group is finitely generated, since S can be taken to be G itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated.
