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Simplicial Commutative Ring
In algebra, a simplicial commutative ring is a commutative monoid in the category of simplicial abelian groups, or, equivalently, a simplicial object in the category of commutative rings. If ''A'' is a simplicial commutative ring, then it can be shown that \pi_0 A is a ring and \pi_i A are modules over that ring (in fact, \pi_* A is a graded ring over \pi_0 A.) A topology-counterpart of this notion is a commutative ring spectrum. Examples *The ring of polynomial differential forms on simplexes. Graded ring structure Let ''A'' be a simplicial commutative ring. Then the ring structure of ''A'' gives \pi_* A = \oplus_ \pi_i A the structure of a graded-commutative graded ring as follows. By the Dold–Kan correspondence, \pi_* A is the homology of the chain complex corresponding to ''A''; in particular, it is a graded abelian group. Next, to multiply two elements, writing S^1 for the simplicial circle, let x:(S^1)^ \to A, \, \, y:(S^1)^ \to A be two maps. Then the composition :(S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. The word ''alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dold–Kan Correspondence
In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states that there is an equivalence between the category of (nonnegatively graded) chain complexes and the category of simplicial abelian groups. Moreover, under the equivalence, the nth homology group of a chain complex is the nth homotopy group of the corresponding simplicial abelian group, and a chain homotopy corresponds to a simplicial homotopy. (In fact, the correspondence preserves the respective standard model structures.) Example: Let ''C'' be a chain complex that has an abelian group ''A'' in degree ''n'' and zero in all other degrees. Then the corresponding simplicial group is the Eilenberg–MacLane space K(A, n). There is also an ∞-category-version of the Dold–Kan correspondence. The book "Nonabelian Algebraic Topology" cited below has a Section 14.8 on cubical versions of the Dold–Kan theorem, and relates them to a pre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers \mathbb; and ''p''-adic integers. Commutative algebra is the main technical tool in the local study of schemes. The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras. Overview Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings. Considerations related to modular arithmetic have le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E N-ring
In mathematics, an \mathcal_n-algebra in a symmetric monoidal infinity category ''C'' consists of the following data: *An object A(U) for any open subset ''U'' of Rn homeomorphic to an ''n''-disk. *A multiplication map: *:\mu: A(U_1) \otimes \cdots \otimes A(U_m) \to A(V) :for any disjoint open disks U_j contained in some open disk ''V'' subject to the requirements that the multiplication maps are compatible with composition, and that \mu is an equivalence if m=1. An equivalent definition is that ''A'' is an algebra in ''C'' over the little ''n''-disks operad. Examples * An \mathcal_n-algebra in vector spaces over a field is a unital associative algebra if ''n'' = 1, and a unital commutative associative algebra if ''n'' ≥ 2. * An \mathcal_n-algebra in categories is a monoidal category if ''n'' = 1, a braided monoidal category if ''n'' = 2, and a symmetric monoidal category if ''n'' ≥ 3. * If Λ is a commutative ring, then X \mapsto C_*(\Omega^n X; \Lambda) define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Opposite Category
In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself. In symbols, (C^)^ = C. Examples * An example comes from reversing the direction of inequalities in a partial order. So if ''X'' is a set and ≤ a partial order relation, we can define a new partial order relation ≤op by :: ''x'' ≤op ''y'' if and only if ''y'' ≤ ''x''. : The new order is commonly called dual order of ≤, and is mostly denoted by ≥. Therefore, duality plays an important role in order theory and every purely order theoretic concept has a dual. For example, there are opposite pairs child/parent, descendant/ancestor, infimum/supremum, down-set/up-set, ideal/ filter etc. This order theoretic duality is in turn a special ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Scheme
In algebraic geometry, a derived scheme is a pair (X, \mathcal) consisting of a topological space ''X'' and a sheaf \mathcal either of simplicial commutative rings or of commutative ring spectra on ''X'' such that (1) the pair (X, \pi_0 \mathcal) is a scheme and (2) \pi_k \mathcal is a quasi-coherent \pi_0 \mathcal-module. The notion gives a homotopy-theoretic generalization of a scheme. A derived stack is a stacky generalization of a derived scheme. Differential graded scheme Over a field of characteristic zero, the theory is closely related to that of a differential graded scheme. By definition, a differential graded scheme is obtained by gluing affine differential graded schemes, with respect to étale topology. It was introduced by Maxim Kontsevich "as the first approach to derived algebraic geometry." and was developed further by Mikhail Kapranov and Ionut Ciocan-Fontanine. Connection with differential graded rings and examples Just as affine algebraic geometry is equiva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module Spectrum
In algebra, a module spectrum is a spectrum with an action of a ring spectrum; it generalizes a module in abstract algebra. The ∞-category of (say right) module spectra is stable A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...; hence, it can be considered as either analog or generalization of the derived category of modules over a ring. K-theory Lurie defines the K-theory of a ring spectrum ''R'' to be the K-theory of the ∞-category of perfect modules over ''R'' (a perfect module being defined as a compact object in the ∞-category of module spectra.) See also * G-spectrum References *J. LurieLecture 19: Algebraic K-theory of Ring Spectra {{algebra-stub Homotopy theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations: \begin (2 + 3) + 4 &= 2 + (3 + 4) = 9 \,\\ 2 \times (3 \times 4) &= (2 \times 3) \times 4 = 24 . \end Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any rea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Sphere
In geometry and combinatorics, a simplicial (or combinatorial) ''d''-sphere is a simplicial complex homeomorphic to the ''d''-dimensional sphere. Some simplicial spheres arise as the boundaries of convex polytopes, however, in higher dimensions most simplicial spheres cannot be obtained in this way. One important open problem in the field was the g-conjecture, formulated by Peter McMullen, which asks about possible numbers of faces of different dimensions of a simplicial sphere. In December 2018, the g-conjecture was proven by Karim Adiprasito in the more general context of rational homology spheres. Examples * For any ''n'' ≥ 3, the simple ''n''-cycle ''C''''n'' is a simplicial circle, i.e. a simplicial sphere of dimension 1. This construction produces all simplicial circles. * The boundary of a convex polyhedron in R3 with triangular faces, such as an octahedron or icosahedron, is a simplicial 2-sphere. * More generally, the boundary of any (''d''+1)-dimensional compac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chain Complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels. A cochain complex is similar to a chain complex, except that its homomorphisms are in the opposite direction. The homology of a cochain complex is called its cohomology. In algebraic topology, the singular chain complex of a topological space X is constructed using continuous maps from a simplex to X, and the homomorphisms of the chain complex capture how these maps restrict to the boundary of the simplex. The homology of this chain complex is called the singular homology of X, and is a commonly used invariant of a topological space. Chain complexes are studied in homological algebra, but are used in several areas of mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Differential Form
In algebra, the ring of polynomial differential forms on the standard ''n''-simplex is the differential graded algebra: :\Omega^*_( = \mathbb _0, ..., t_n, dt_0, ..., dt_n(\sum t_i - 1, \sum dt_i). Varying ''n'', it determines the simplicial commutative dg algebra: :\Omega^*_ (each u: \to /math> induces the map \Omega^*_( \to \Omega^*_( , t_i \mapsto \sum_ t_j). References * Aldridge Bousfield and V. K. A. M. Gugenheim, §1 and §2 of: On PL De Rham Theory and Rational Homotopy Type In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by and . This simplification of homo ..., Memoirs of the A. M. S., vol. 179, 1976. * External links * https://ncatlab.org/nlab/show/differential+forms+on+simplices * https://mathoverflow.net/questions/220532/polynomial-differential-forms-on-bg Differential algebra Ring theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoid Object
In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) in a monoidal category is an object ''M'' together with two morphisms * ''μ'': ''M'' ⊗ ''M'' → ''M'' called ''multiplication'', * ''η'': ''I'' → ''M'' called ''unit'', such that the pentagon diagram : and the unitor diagram : commute. In the above notation, is the identity morphism of , is the unit element and α, λ and ρ are respectively the associativity, the left identity and the right identity of the monoidal category C. Dually, a comonoid in a monoidal category C is a monoid in the dual category Cop. Suppose that the monoidal category C has a symmetry ''γ''. A monoid ''M'' in C is commutative when . Examples * A monoid object in Set, the category of sets (with the monoidal structure induced by the Cartesian product), is a monoid in the usual sense. * A monoid object in Top, the category of topological spaces (with the monoidal structure induced by t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
