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Concept in homological algebra
especially homological algebra, a differential graded category, often shortened to dg-category or DG category, is a category whose morphism sets are endowed
Differential_graded_category
Index of articles associated with the same name
may refer to the category of augmented differential graded commutative algebras. A graded Lie algebra is a Lie algebra that is graded as a vector space
Graded_structure
Algebraic structure in homological algebra
about a topological or geometric space. Explicitly, a differential graded algebra is a graded associative algebra with a chain complex structure that
Differential_graded_algebra
G} -graded category. Differential graded category Graded (mathematics) Graded algebra Slice category Zhang, James J. (1 March 1996). "Twisted graded algebras
Graded_category
Mathematical concept
differential graded module, or dg-module, is a Z {\displaystyle \mathbb {Z} } -graded module together with a differential; i.e., a square-zero graded
Differential_graded_module
particular abstract algebra and topology, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a graded vector space with added Lie algebra and
Differential graded Lie algebra
Differential_graded_Lie_algebra
Additive category in homological algebra
references for details. More generally, the homotopy category Ho(C) of a differential graded category C is defined to have the same objects as C, but morphisms
Homotopy category of chain complexes
Homotopy_category_of_chain_complexes
resolution of the differential graded algebra ( B ∙ , 0 ) {\displaystyle (B_{\bullet },0)} where B ∙ {\displaystyle B_{\bullet }} is the graded algebra with
Derived_scheme
Swiss mathematician
Congress of Mathematicians in Madrid in 2006, with a talk On differential graded categories. Keller is a fellow of the American Mathematical Society. with
Bernhard_Keller
Manifold with supersymmetry structure
geometry, graded manifolds are extensions of the concept of manifolds based on ideas coming from supersymmetry and supercommutative algebra. Both graded manifolds
Graded_manifold
differential graded category is a category whose Hom sets are equipped with structures of differential graded modules. In particular, if the category has only
Glossary_of_category_theory
Mathematical concept in category theory
R-algebra. the category of graded modules is a graded R-algebra. the category of chain complexes of R-modules is a differential graded algebra. A monoid
Monoid_(category_theory)
categories Ho ( dga ) ≃ Ho ( A ∞ -alg ) {\displaystyle {\text{Ho}}({\text{dga}})\simeq {\text{Ho}}(A_{\infty }{\text{-alg}})} of differential graded algebras
Homotopy_associative_algebra
Type of algebraic structure
Filtered algebra, a generalization Graded (mathematics) Graded category Graded vector space Tensor algebra Differential graded module Sakarovitch, Jacques (2009)
Graded_ring
mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra
Graded_Lie_algebra
Various mathematical dualites
graded rings, differential graded rings). Algebras over the so-called commutative operad are commutative algebras, i.e., commutative (possibly graded
Koszul_duality
of a differential graded Lie algebra. To be a little more specific, the Jacobi identity only holds up to homotopy. Therefore, a differential graded Lie
Homotopy_Lie_algebra
Algebraic study of differential equations
+ 1 ∘ d m = 0 {\displaystyle d_{m+1}\circ d_{m}=0} . A differential graded algebra is a graded algebra A {\textstyle A} with a linear derivation d : A
Differential_algebra
Mathematical theory of topological spaces
rational homotopy category is equivalent to the homotopy category of connected differential graded Lie algebras. (The associated graded Lie algebra ker
Rational_homotopy_theory
Behrend (2002). Behrend, Kai (2002). "Differential Graded Schemes II: The 2-category of Differential Graded Schemes". arXiv:math/0212226. Lurie, Jacob
Étale_spectrum
Branch of mathematics
with a sheaf of commutative differential graded algebras. Sometimes authors take the convention that these are negatively graded, so O X n = 0 {\displaystyle
Derived_algebraic_geometry
Tool in homological algebra
complex Differential graded algebra Differential graded Lie algebra Dold–Kan correspondence says there is an equivalence between the category of chain
Chain_complex
the category of Lie algebras. In particular, it is a simplicial abelian group, and thus is subject to the Dold–Kan correspondence. Differential graded Lie
Simplicial_Lie_algebra
Example of the Koszul duality
{\displaystyle A^{!}} is coherent). J.-W. He and Q.-S. Wu. “Koszul differential graded algebras and BGG correspondence”. In: J. Algebra 320.7 (2008), pp
BGG_correspondence
Branch of mathematics
Differential geometry over a noncommutative algebra requires a replacement for differential forms. One common starting point is a differential graded
Noncommutative_geometry
\mathbb {Q} } of rational numbers is Quillen equivalent to the category of differential graded algebras over Q {\displaystyle \mathbb {Q} } . Example: The
Commutative_ring_spectrum
Branch of mathematics
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It
Differential_geometry
Algebraic structure used in theoretical physics
theoretical physics, a superalgebra is a Z 2 {\displaystyle \mathbb {Z} _{2}} -graded algebra. That is, it is an algebra over a commutative ring or field with
Superalgebra
Algebraic structure used in topology
advantage that the product on differential forms is graded-commutative, whereas the product on singular cochains is only graded-commutative up to chain homotopy
Cohomology
Branch of mathematics
of Jean-Pierre Serre, quasi-coherent sheaves on Proj of a graded ring are the same as graded modules over the ring up to finite dimensional factors. The
Noncommutative algebraic geometry
Noncommutative_algebraic_geometry
Module over a sheaf of differential operators
K-linear combinations of differential operators xα∂β with |α| + |β| ≤ p (using multiindex notation). The associated graded ring is seen to be isomorphic
D-module
In algebra, given a differential graded algebra A over a commutative ring R, the derived tensor product functor is − ⊗ A L − : D ( M A ) × D ( A M ) →
Derived_tensor_product
Commutative monoid in simplicial abelian groups
{\displaystyle \pi _{*}A=\oplus _{i\geq 0}\pi _{i}A} the structure of a graded-commutative graded ring as follows. By the Dold–Kan correspondence, π ∗ A {\displaystyle
Simplicial_commutative_ring
Theory in algebraic geometry
over k } ⟶ { graded K -algebras } {\displaystyle H^{*}:\{{\text{smooth projective varieties over }}k\}\longrightarrow \{{\text{graded }}K{\text{-algebras}}\}}
Weil_cohomology_theory
Exact sequence used to describe the structure of an object
which the free modules Ei may be graded in such a way that the di and ε are graded linear maps. Among these graded free resolutions, the minimal free
Resolution_(algebra)
Algebra associated to any vector space
\alpha .} In addition to studying the graded structure on the exterior algebra, Bourbaki (1989) studies additional graded structures on exterior algebras,
Exterior_algebra
Algebra based on a vector space with a quadratic form
Elements that are pure in this Z2-grading are simply said to be even or odd. Remark. The Clifford algebra is not a Z-graded algebra, but is Z-filtered, where
Clifford_algebra
Ukrainian mathematician
Roiter, A. V.; Kleiner, M. M. (1975). "Representations of differential graded categories". Representations of algebras (Proc. Internat. Conf., Carleton
Andrei_Roiter
Branch of mathematics
called boundary maps or differentials, Isomorphisms of Er+1 with H(Er), the homology of Er with respect to dr. A doubly graded spectral sequence has a
Homological_algebra
Algebraic structure used in theoretical physics
have the structure of a Z2-graded Hopf algebra. Likewise the representations of this Hopf algebra turn out to be Z2-graded comodules. This Hopf algebra
Supergroup_(physics)
naturally to the setting of graded commutative algebra, allowing for a natural foundation of calculus on supermanifolds, graded manifolds and associated
Differential calculus over commutative algebras
Differential_calculus_over_commutative_algebras
Blood test
A white blood cell differential is a medical laboratory test that provides information about the types and amounts of white blood cells in a person's blood
White_blood_cell_differential
forgetful functor. The free Lie algebra on a set X is naturally graded. The 1-graded component of the free Lie algebra is just the free vector space on
Free_Lie_algebra
Differential geometry of supermanifolds
Supergeometry is differential geometry of modules over graded commutative algebras, supermanifolds and graded manifolds. Supergeometry is part and parcel
Supergeometry
Canadian mathematician (born 1942)
tome 9–10, 1983, 1–261 with Yves Félix and Jean-Claude Thomas: Differential graded algebras in topology, in I.M. James (editor) Handbook in Algebraic
Stephen_Halperin
Tool in homological algebra
reality spectral sequences mostly occur in the category of doubly graded modules over a ring R (or doubly graded sheaves of modules over a sheaf of rings)
Spectral_sequence
Theory for associative algebras over rings
a resolution of F p {\displaystyle \mathbb {F} _{p}} as the free differential graded algebras Z → ⋅ p Z {\displaystyle \mathbb {Z} \xrightarrow {\cdot
Hochschild_homology
Conjecture in symplectic geometry
category of the Calabi–Yau. Conjecture (Thomas–Yau–Joyce): An oriented, graded, almost-calibrated Lagrangian L {\displaystyle L} splits as a graded Lagrangian
Thomas–Yau_conjecture
Sheaf consisting of modules on a ringed space; generalizing vector bundles
associated sheaf) in a natural way. Similarly, if R is a graded ring and X is the Proj of R, then any graded module defines an OX-module in a natural way. O-modules
Sheaf_of_modules
Lubricant used in vehicles and machinery
lubricant made specifically for transmissions, transfer cases, and differentials in automobiles, trucks, and other machinery. It has high viscosity and
Gear_oil
Ring that is also a vector space or a module
Temperley-Lieb algebra. A differential graded algebra is an associative algebra together with a grading and a differential. For example, the de Rham algebra
Associative_algebra
Manifold upon which it is possible to perform calculus
The exterior derivative is a linear operator on the graded vector space of all smooth differential forms on a smooth manifold M {\displaystyle M} . It
Differentiable_manifold
Polish mathematician
Sciences, Poland, Katarzyna Grabowska (11 November 2015). "Graded Bundles in the Category of Lie Groupoids". Symmetry, Integrability and Geometry: Methods
Janusz_Grabowski
an internal operation is the exterior algebra of V; it is a graded algebra, with the graded piece of weight k being called the k-th exterior power of V
Glossary_of_tensor_theory
Invariant of mathematical knots
a singly graded knot homology theory using Lagrangian intersection Floer homology, which they conjecture to be isomorphic to a singly graded version of
Khovanov_homology
Concept in algebraic topology
they form a differential graded algebra as follows: the graded set of groups form a graded R-module; this can be given the structure of a graded R-algebra
Singular_homology
Associative algebra together with a Lie bracket that satisfies Leibniz's law
The difference between the two is in the grading of the product itself. For the Poisson superalgebra, the grading is given by | { a , b } | = | a | + | b
Poisson_algebra
Mathematical concept
two differentials, the horizontal differential d h : C p , q → C p + 1 , q {\displaystyle d^{h}:C_{p,q}\to C_{p+1,q}} and the vertical differential d v
Double_complex
Generalization of vector bundles
{N} } -graded ring, be a projective scheme over a Noetherian ring R 0 {\displaystyle R_{0}} . Then each Z {\displaystyle \mathbb {Z} } -graded R {\displaystyle
Coherent_sheaf
Concept in mathematics
operators into other areas of mathematics, specifically, those that have a differential algebra. They also play a central role in some recent developments in
Universal_enveloping_algebra
Concept in homological algebra
triangulated category may be endowed with several distinct t-structures. A non-standard example of a t-structure on the derived category of (graded) modules
T-structure
Branch of functional analysis
functional analysis, it has direct applications to representation theory, differential geometry, quantum statistical mechanics, quantum information, and quantum
Operator_algebra
equivalent to the property d 2 = 0 {\displaystyle d^{2}=0} . In other words, the graded vector space Ω X ∗ ( E ) {\displaystyle \Omega _{X}^{*}(E)} is a cochain
Flat_vector_bundle
Cohomology theory for Lie algebras
of differential forms on G {\displaystyle G} . Using an averaging process, this complex can be replaced by the complex of left-invariant differential forms
Lie_algebra_cohomology
History of maths
This is a timeline of category theory and related mathematics. Its scope ("related mathematics") is taken as: Categories of abstract algebraic structures
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Infinitesimal calculus on functions defined on a geometric algebra
other mathematical theories including vector calculus, differential geometry, and differential forms. With a geometric algebra given, let a {\displaystyle
Geometric_calculus
Generalization of vector spaces from fields to rings
submodules becomes stationary after finitely many steps. Graded A graded module is a module over a graded ring R = ⨁x Rx together with a direct sum decomposition
Module_(mathematics)
Mathematical sequence
p = the p th graded part of H d R n ( C ∙ ( f − 1 U , Ω X ∙ ) ) {\displaystyle E_{\infty }^{n-p,p}={\text{the }}p{\text{th graded part of }}H_{dR}^{n}(C^{\bullet
Leray_spectral_sequence
Equivalence between the categories of chain complexes and simplicial abelian groups
that there is an equivalence between the category of (nonnegatively graded) chain complexes and the category of simplicial abelian groups. Moreover, under
Dold–Kan_correspondence
Algebraic structure
which guarantee commutativity of a ring are also known. A graded ring R = ⨁i∊Z Ri is called graded-commutative if, for all homogeneous elements a and b, ab
Commutative_ring
Analysis of tissue to identify colorectal cancer characteristics
adenoma with "high-grade dysplasia", because prognosis and management are essentially the same. Conventional adenocarcinoma may be graded as follows Moderately
Histopathology of colorectal adenocarcinoma
Histopathology_of_colorectal_adenocarcinoma
Cohomology with real coefficients computed using differential forms
Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth
De_Rham_cohomology
Mathematician
and Arne Wessel, demonstrates that the presheaf of graded manifolds obtained via the differential-geometric fat point is representable by the tangent
Chenchang_Zhu
Australian and American mathematician (born 1975)
was awarded the Fields Medal in 2006 for his contributions to partial differential equations, combinatorics, harmonic analysis, and additive number theory
Terence_Tao
Algebraic structure
regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have been introduced for generalizing some properties of polynomial
Polynomial_ring
Mathematics study in geometry
categories which can be associated to any point w 0 ∈ A 1 {\displaystyle w_{0}\in \mathbb {A} ^{1}} , a Z / 2 {\displaystyle \mathbb {Z} /2} -graded category
Derived noncommutative algebraic geometry
Derived_noncommutative_algebraic_geometry
is the rank of the module. differential A differential graded module or dg-module is a graded module with a differential. direct sum A direct sum of
Glossary_of_module_theory
Social group defined by shared traits
are characterized by a high degree of ethnocentrism, competition, and differential power. Ethnocentrism is the tendency to look at the world primarily from
Ethnicity
Yang–Mills coupled to a Higgs field
mathematics, the Yang–Mills–Higgs equations are a set of non-linear partial differential equations for a Yang–Mills field, given by a connection, and a Higgs
Yang–Mills–Higgs_equations
Differential variety
forms a graded Lie-Rinehart algebra together with H ¯ ∙ ( O ) {\displaystyle {\overline {H}}^{\bullet }({\mathcal {O}})} ; secondary differential p {\displaystyle
Diffiety
Algebraic object with geometric applications
as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in
Tensor
Software used for psychometric analysis
Issayeva. There is also an R Shiny tool for reproducible Rasch analysis, differential item functioning, equating, and examination of group effects. Additionally
Psychometric_software
Intrusion of water causing destruction and attack
special methods, longer drying times, or substantial water vapor pressure differentials. Preventing water damage is far more cost-effective than restoration
Water_damage
Symplectic topology tool
to show that the counts of flow lines defining the differential are finite, so that the differential is well-defined and squares to zero. Thus the Floer
Floer_homology
American mathematician
studied generalizations of these structures for ring spectra and differential graded algebras. She has more recently used algebraic topology to understand
Kathryn_Hess
Swelling due to a compromised lymphatic system
Tiwari A, Cheng KS, Button M, Myint F, Hamilton G (February 2003). "Differential diagnosis, investigation, and current treatment of lower limb lymphedema"
Lymphedema
Algebraic structure
are fundamental models for linear time-invariant systems. In partial differential equations, a semigroup is associated to any equation whose spatial evolution
Semigroup
Association of cohomology classes to principal bundles
They are one of the unifying geometric concepts in algebraic topology, differential geometry, and algebraic geometry. The notion of characteristic class
Characteristic_class
Mathematical theory
Rham cohomology (not just its associated graded), Fontaine constructed a filtered ring BdR whose associated graded is BHT and conjectured the following (called
P-adic_Hodge_theory
Mathematical operation on vector spaces
product operation form an algebra, called the tensor algebra, which is graded by the order of a tensor. For tensors of type (1, 1) there is a canonical
Tensor_product
Algebraic structure with addition and multiplication
representation theory, operator algebras in functional analysis, rings of differential operators, and cohomology rings in topology. The conceptualization of
Ring_(mathematics)
American football player and coach (born 1987)
scored, with Smith-Njigba leading the league in receiving yards and Darnold graded as the league's best quarterback by Pro Football Focus while leading in
Klint_Kubiak
Branch of mathematics
rings with an infinity category of differential graded commutative algebras, or of simplicial commutative rings or a similar category with an appropriate
Algebraic_geometry
Compact heavy equipment with differential steering
straight alignment on the body of the machine. Turning is accomplished by differential steering, in which the left and right wheel pairs are operated at different
Skid-steer_loader
institutions) is that a study average at least grade 9.5, final thesis graded 10, not a single grade below 8 and completing the curriculum without delays
Academic_grading_in_Serbia
High-gain voltage amplifier with a differential input
op amp, op-amp, or opamp) is a DC-coupled electronic amplifier with a differential input, a (usually) single-ended output voltage, and an extremely high
Operational_amplifier
Raised body temperature caused by disease
diminutive form of the Latin word for fever), was once used to refer to a low-grade fever lasting only a few days. This term fell out of use in the early 20th
Fever
Branch of mathematics that studies abstract algebraic structures
methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics and topology. The
Representation_theory
Bacterial infection of the inner layers of the skin called the dermis
superficial wound usually creates less lameness (grade 1–2 of 5) than that caused by septic arthritis (grade 4–5). The horse exhibits inflammatory edema,
Cellulitis
Surveying technique
to a reference. Optical levelling, also known as spirit levelling and differential levelling, employs an optical level, which consists of a precision telescope
Levelling
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