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fibred category, prestack. The dual of a cartesian fibration is called an op-fibration; in particular, not a cocartesian fibration. A right fibration
Cartesian_fibration
Concept in mathematical category theory
construction. A key property is that π {\displaystyle \pi } is a cartesian fibration (or that C F {\displaystyle C_{F}} is a category fibered over C {\displaystyle
Category_of_elements
inclusion; hence, a Kan fibration is exactly a map that is both a left and right fibration. A right fibration is a cartesian fibration such that each fiber
Fibration_of_simplicial_sets
Concept in category theory
mapping paths to paths using φ {\displaystyle \varphi } is a fibration. Fibre bundles: Fibre products exist in the category Top {\displaystyle {\text{Top}}}
Fibred_category
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
quaternionic Hopf Fibration, ncatlab.org. https://ncatlab.org/nlab/show/quaternionic+Hopf+fibration Smith, Benjamin. "Benjamin H. Smith's Hopf fibration notes" (PDF)
Hopf_fibration
Generalization of a category
q:M\to \Delta ^{1}} that is both cartesian and cocartesian fibrations. Since q {\displaystyle q} is a cartesian fibration, by the Grothendieck construction
Quasi-category
Most general completion of a commutative square given two morphisms with same codomain
mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms
Pullback_(category_theory)
Classic science experiment demonstrating the Archimedes' principle and the ideal gas law
Dancing Cartesian Devil A Cartesian diver or Cartesian devil is a classic science experiment which demonstrates the principle of buoyancy (Archimedes'
Cartesian_diver
Topics referred to by the same term
often called product that yields the product of a sequence Direct product Cartesian product of sets Direct product of groups Semidirect product Product of
Product
Branch of mathematics
implicit. A fibration in the sense of Hurewicz is the dual notion of a cofibration: that is, a map p : X → B {\displaystyle p:X\to B} is a fibration if given
Homotopy_theory
Type theory in logic and mathematics
universal fibration was univalent: the associated fibration of pairwise homotopy equivalences between the fibers is equivalent to the paths-space fibration of
Homotopy_type_theory
Right inverse of a fiber bundle map
(category theory) Fibration Gauge theory (mathematics) Principal bundle Pullback bundle Vector bundle Husemöller, Dale (1994), Fibre Bundles, Springer
Section_(fiber_bundle)
Four-dimensional analog of the icosahedron
20 cell rings is one of 5 fibrations within the fibration of 12 of 72 decagons: a fibration of a fibration. All the fibrations have this two level structure
600-cell
structure. Fiber bundles are mathematical objects that locally look like a cartesian product of a base space and another space, the typical "fiber", but may
Bundle_map
Map in projective geometry
Segre embedding is a map used in projective geometry to consider the cartesian product of two projective spaces as a projective variety. It is named
Segre_embedding
How spheres of various dimensions can wrap around each other
_{i}(S^{15})\oplus \pi _{i-1}(S^{7}).} The three fibrations have base space Sn with n = 2m, for m = 1, 2, 3. A fibration does exist for S1 (m = 0) as mentioned
Homotopy_groups_of_spheres
Method to avoid singularities in classical mechanics
collisions. The Kustaanheimo–Stiefel map is an extension of the complex Hopf fibration h C : S 3 → S 2 {\displaystyle h_{\mathbb {C} }\colon S^{3}\rightarrow
Kustaanheimo–Stiefel regularization
Kustaanheimo–Stiefel_regularization
Equations of motion for viscous fluids
singularities comes from considering the flow along the lines of a Hopf fibration. Let r {\textstyle r} be a constant radius of the inner coil. One set
Navier–Stokes_equations
Generalized Euclidean space in mathematics
equivalently, the Riemann sphere. See Hopf fibration for details of the projectivization construction in this case. The Cartesian product of projective Hilbert spaces
Projective_Hilbert_space
4-dimensional object
geometric object embedded in 4-dimensional Euclidean space, defined as the Cartesian product of two disks of respective radii r1 and r2: D = { ( x , y , z
Duocylinder
Process in mathematics
notion of a pullback as precomposition, and the notion of a pullback as a Cartesian square. In that example, the base space of a fiber bundle is pulled back
Pullback
NEMA grade designation for glass-reinforced epoxy laminate material
fiber orientations in the XY plane of the board (in-plane). In terms of Cartesian coordinates, lengthwise is along the x-axis, crosswise is along the y-axis
FR-4
Transforming a function in such a way that it only takes a single argument
duality between the mapping cone and the mapping fiber (cofibration and fibration) can be understood as a form of currying, which in turn leads to the duality
Currying
American-Israeli designer and academic
designer Iris van Herpen to 3D-print a collection of wearable couture. Cartesian Wax, Monocoque, Raycounting (2007, MoMA) Carpal Skin (2010, Museum of
Neri_Oxman
op-fibration A functor π:C → D is an op-fibration if, for each object x in C and each morphism g : π(x) → y in D, there is at least one π-coCartesian morphism
Glossary_of_category_theory
Fiber bundle whose fibers are group torsors
mathematical object that formalizes some of the essential features of the Cartesian product X × G {\displaystyle X\times G} of a topological space X {\displaystyle
Principal_bundle
Four-dimensional analog of the dodecahedron
disjoint irregular great hexagons (a discrete fibration of the 120-cell) in four different ways. Each fibration has its distinct left (and right) isoclinic
120-cell
Indian defense projects under DRDO
across different projects. These include "Pythagoras processor" to convert cartesian to polar coordinates, ANUCO, a floating point coprocessor and several
Projects_of_DRDO
Method to avoid singularities in classical mechanics
introduced them in 1920. The Levi-Civita map is an extension of the real Hopf fibration h R : S 1 → S 1 {\displaystyle h_{\mathbb {R} }\colon S^{1}\rightarrow
Levi-Civita_regularization
Mathematical construction used in homotopy theory
point of the theory is that the geometric realization of a Kan fibration is a Serre fibration of spaces. With the model structure in place, a homotopy theory
Simplicial_set
Generalisation of a sheaf; a fibered category that admits effective descent
0)\\\downarrow &&\downarrow \\(U,u)&\to &(N_{x}//G_{x},0)\end{matrix}}} is cartesian, and there exists an etale morphism f : ( [ W / G x ] , w ) → ( X , x
Stack_(mathematics)
Representation of a quantum mechanical system
C 2 {\displaystyle \mathbb {C} ^{2}} to the Bloch sphere is the Hopf fibration, with each ray of spinors mapping to one point on the Bloch sphere. Given
Bloch_sphere
Non-orientable mathematical surface
3-manifolds, it is known that a solid Klein bottle is homeomorphic to the Cartesian product of a Möbius strip and a closed interval. The solid Klein bottle
Klein_bottle
)\cdot Y\right)^{2}=e^{i\delta }I} a SU(2) double cover. See also Hopf fibration. The matrix shown here is from openQASM 3.0, which differs from U ( θ
List_of_quantum_logic_gates
Construct allowing differentiation of tangent vector fields of manifolds
isomorphism of T(FM) with the trivial bundle FM × aff(n), where aff(n) is the Cartesian product of Rn and gl(n) (viewed as the Lie algebra of the affine group
Affine_connection
p_{2}:M\times M\to M} given by projection the respective factors of the Cartesian product, which restrict to give projections p 1 , p 2 : M ( 2 ) → M .
Grothendieck_connection
Science of using a material's refractive index for optical effects
where n is the refractive index and S is the arc length of the curve. If Cartesian coordinates are used, this equation is modified to incorporate the change
Gradient-index_optics
Category where every morphism is invertible; generalization of a group
morphism p : E → B {\displaystyle p:E\to B} of groupoids is called a fibration if for each object x {\displaystyle x} of E {\displaystyle E} and each
Groupoid
Generalization of the concept of vector bundle
the zero section. Given any topological space B {\displaystyle B} , the cartesian product B × R n {\displaystyle B\times \mathbb {R} ^{n}} (together with
Microbundle
Branch of logic using category theory to study mathematical structures
theories of βη-equational logic over simply typed lambda calculus and Cartesian closed categories. Categories arising from theories via term model constructions
Categorical_logic
Mathematics concept
tangent space to the fiber. A simple example of a smooth fiber bundle is a Cartesian product of two manifolds. Consider the bundle B1 := (M × N, pr1) with
Vertical and horizontal bundles
Vertical_and_horizontal_bundles
Metric on a complex projective space endowed with Hermitian form
of rotations. This quotient is realized explicitly by the famous Hopf fibration S1 → S2n+1 → CPn, the fibers of which are among the great circles of S
Fubini–Study_metric
Mapping of mathematical formulas to a particular meaning
categorically corresponds to one ("total") category, capturing the logic, being fibred over another ("base") category, capturing the type theory. Both universal
Structure (mathematical logic)
Structure_(mathematical_logic)
Regular object in four dimensional geometry
representative not only of its own fibration of Clifford parallel planes but also of the other congruent fibrations. For example, rotation class [ 4 ]
24-cell
Concept in differential geometry
decomposition theorem, a principle for splitting a Riemannian manifold into a Cartesian product of Riemannian manifolds by splitting the tangent bundle into irreducible
Holonomy
Generalization of definite integrals to functions of multiple variables
between the surface defined by the function (on the three-dimensional Cartesian plane where z = f(x, y)) and the plane which contains its domain. If there
Multiple_integral
Swedish pioneering physician (1907–1986)
in Philadelphia (Spiegel et al. 1947). Their original frame, using a Cartesian coordinate systems and similar in design and operation to the Clarke-Horsley
Lars_Leksell
Mathematician, natural philosopher and astronomer (1664–1753)
came under the influence of the rector, Jean-Robert Chouet, a prominent Cartesian. Before he was eighteen, Fatio wrote to the director of the Paris Observatory
Nicolas_Fatio_de_Duillier
Tools for studying groups based on techniques from algebraic topology
H\to 1} Using the associated Eilenberg-Maclane spaces there is a Serre fibration K ( N , 1 ) → K ( G , 1 ) → K ( H , 1 ) {\displaystyle K(N,1)\to K(G,1)\to
Group_cohomology
List of terms created from a person's name
(lifting device) René Descartes, French philosopher – Cartesian coordinate system, Cartesianism David Deutsch, Israeli-British physicist – Church–Turing–Deutsch
List_of_eponyms_(A–K)
Special orthogonal group
Zamboj, Michal (8 January 2021). "Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space". Journal of Computational
Rotations in 4-dimensional Euclidean space
Rotations_in_4-dimensional_Euclidean_space
Concept in mathematics
Y = [0, T] × H1(Ω), which as a Cartesian product also has the structure of a Banach bundle over the manifold [0, T] with fibre H1(Ω), in which case elements/solutions
Banach_bundle
}(G)<\theta (G)} ? Graham's pebbling conjecture on the pebbling number of Cartesian products of graphs Meyniel's conjecture that cop number is O ( n ) {\displaystyle
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
History of maths
categories fibrations and cofibrations are independent and that for an ABC model category MD is an ABC model category. To an ABC (co)fibration category
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Description of the orientation of a rigid body
(2010). "Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration". The International Journal of Robotics Research. 29 (7). Section 8 —
Euler_angles
Concept in geometric topology
{\displaystyle h_{*}} is homotopy-invariant. It preserves homotopy co-cartesian squares. This reflects the fact that h ∗ {\displaystyle h_{*}} has Mayer-Vietoris
Assembly_map
Philosophical and political conception of society as a living organism
organicism and vitalism, were born from the quest for getting rid of the Cartesian picture of reality, a view that has been claimed to be the most destructive
Organicism
Type of mathematical functions
complex coordinate space C n {\displaystyle \mathbb {C} ^{n}} is the Cartesian product of n copies of C {\displaystyle \mathbb {C} } , and when C n {\displaystyle
Function of several complex variables
Function_of_several_complex_variables
Descartes (1596–1650) allowed those orbits to be plotted on a graph, in Cartesian coordinates. Building on earlier work by many predecessors, Isaac Newton
History_of_mathematics
Type of angular momentum in light
using SI units. The i {\displaystyle i} -superscripted symbols denote the cartesian components of the corresponding vectors. For a monochromatic wave this
Orbital angular momentum of light
Orbital_angular_momentum_of_light
Photographic printing process that produces a blue print
or their unspooling. Kate Cordsen applies Japanese aesthetics and non-Cartesian perspective in her mural-scale cyanotype landscapes. Betty Hahn was early
Cyanotype
On the homotopy groups of the infinite symmetric product of a connected CW complex
satisfied as p*−1([e]) ≅ SP(A) holds. One may wonder whether p* is not even a fibration. However, that turns out not to be the case: Take an arbitrary path xt
Dold–Thom_theorem
Mathematical series
with respect to a weight which is combined multiplicatively when taking Cartesian products. Suppose that A is a set with a function w : A → N assigning
Dirichlet_series
Algebraic structure in linear algebra
above reduces to this example if an arrow is represented by a pair of Cartesian coordinates of its endpoint. The simplest example of a vector space over
Vector_space
Generalization of category
V={\textbf {Set}}} is the category of sets with ⊗ {\displaystyle \otimes } cartesian product, then a category enriched over it is an ordinary category. If
2-category
Theorem in measure theory
disintegration theorem applies. When Y {\displaystyle Y} is written as a Cartesian product Y = X 1 × X 2 {\displaystyle Y=X_{1}\times X_{2}} and π i : Y
Disintegration_theorem
Four-dimensional analog of the octahedron
entire fibration of two completely orthogonal great squares. The 5-cell and the 16-cell are the only regular 4-polytopes where each discrete fibration has
16-cell
Subject area in mathematics
for relative K-groups arises as the long exact homotopy sequence of a fibration K(R,I) → K(R) → K(R/I). Quillen gave two constructions, the "plus-construction"
Algebraic_K-theory
Type of non-magnetic compass based on the rotation of the Earth
the Earth can be approximated as being an inertial frame. We establish cartesian coordinates ( X 1 , Y 1 , Z 1 ) {\displaystyle (X_{1},Y_{1},Z_{1})} for
Gyrocompass
Vector bundle of cotangent spaces at every point in a manifold
M} be a smooth manifold and let M × M {\displaystyle M\times M} be the Cartesian product of M {\displaystyle M} with itself. The diagonal mapping Δ {\displaystyle
Cotangent_bundle
product of two objects X {\displaystyle X} and Y {\displaystyle Y} is the Cartesian product of sets X × Y {\displaystyle X\times Y} where a realizer of (
Effective_topos
Australian neurophysiologist (1903–1997)
with his concept of three worlds. I was a dualist, now I am a trialist! Cartesian dualism has become unfashionable with many people. They embrace monism
John Eccles (neurophysiologist)
John_Eccles_(neurophysiologist)
Algebraic object with geometric applications
Wiktionary Array data type, for tensor storage and manipulation Bitensor Cartesian tensor Fibre bundle Glossary of tensor theory Multilinear projection One-form
Tensor
Assignment of a tensor continuously varying across a region of space
v i A k i . {\displaystyle v_{k}\mapsto v_{i}A_{k}^{i}.} The list of Cartesian coordinate basis vectors e k {\displaystyle \mathbf {e} _{k}} transforms
Tensor_field
Type of derivative in differential geometry
spherical coordinates differs from the naive derivative of the components in Cartesian coordinates. On an abstract manifold such a definition is meaningless
Lie_derivative
Mathematical parametrization of vector spaces by another space
bundles, while those of rank 2 are less commonly called plane bundles. The Cartesian product X × R k {\displaystyle X\times \mathbb {R} ^{k}} , equipped with
Vector_bundle
Term in theoretical linguistics
whatever—chosen to transmit the corresponding signals. Note that the Cartesian assumption of mind's independence of matter implied—in the human case
Digital_infinity
General concept and operation in mathematics
correspond to each other while considering the opposite category. For example, Cartesian products Y1 × Y2 and disjoint unions Y1 ⊔ Y2 of sets are dual to each
Duality_(mathematics)
Process of monitoring and controlling the movement of a craft or vehicle
etc is defined as a position using a reference point/coordinates (see Cartesian coordinate system). Positions can either be referenced as latitude/longitude
Navigation
Using 3D printing to construct buildings
manipulator mounted onto an overhead to locate the print nozzle in XYZ cartesian coordinates while robotic arms offer additional degrees of freedom to
Construction_3D_printing
equivalence relation. Thom computed, as a ring under disjoint union and cartesian product, the cobordism ring N ∗ {\displaystyle {\mathfrak {N}}_{*}} of
Timeline_of_bordism
Refractive property of materials
Finding the allowed values of k for a given ω is easiest done by using Cartesian coordinates with the x, y and z axes chosen in the directions of the symmetry
Birefringence
Generalization of an ordered basis of a vector space
with an ordered basis of vectors in the associated difference space. A Cartesian frame of an affine space is a choice of origin along with an orthonormal
Moving_frame
Subset of lambda calculus
}} expression. In such circumstances the × type operator is not a true cartesian product, and is generally written ⊗ to make this clear. Hasegawa, Masahito
Kappa_calculus
of objects like simplicial sets as well. Moreover, if the category is cartesian closed, the distributive law X × (Y ∐ Z) ≅ X × Y ∐ X × Z holds and therefore
Symmetric_product_(topology)
Mathematical structure in differential geometry
{\displaystyle f\in {\mathcal {C}}^{\infty }(M)} , is automatically Poisson. The Cartesian product ( M 0 × M 1 , π 0 × π 1 ) {\displaystyle (M_{0}\times M_{1},\pi
Poisson_manifold
British quantum physicist (1935–2025)
everything can be made explicit at a given time" and adding: 'Within the Cartesian order, complementarity seems totally mysterious. There exists no structural
Basil_Hiley
Concept in differential geometry
{Dflg}}} is closed under many categorical operations: for instance, it is Cartesian closed, complete and cocomplete, and more generally it is a quasitopos
Diffeology
Subclass of liquid crystal
distance along the director twist axis (usually defined as the z-axis in Cartesian coordinates) is: n x = cos ( q z ) {\displaystyle n_{x}=\cos(qz)} n
Cholesteric_liquid_crystal
Philosophy done in Romania or by Romanians
brain. He thought that sensorial input is transmitted through the nervous fibres under the form of "shakings", or vibrations, which provoke the apparition
Romanian_philosophy
Robotic arm installed on the ISS Russian Segment
degree of freedom at a time in a joint space, or one degree of freedom in Cartesian Space is able to be used. Also on the console is the emergency stop button
European_Robotic_Arm
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CARTESIAN FIBRATION
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CARTESIAN FIBRATION
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CARTESIAN FIBRATION
CARTESIAN FIBRATION
CARTESIAN FIBRATION
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