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mathematics, injective and projective model structures are special model structures on functor categories into a model category. Both model structures do not
Injective and projective model structure
Injective_and_projective_model_structure
_{k}^{n})\rightarrow \operatorname {RFib} (\Delta ^{n}).} Injective and projective model structure, induced model structures on functor categories Lurie, Jacob (2009)
Co- and contravariant model structure
Co-_and_contravariant_model_structure
Function that preserves distinctness
{\displaystyle s\in S} to itself) is injective. In particular, the identity function X → X {\displaystyle X\to X} is always injective (and in fact bijective). If the
Injective_function
Homological construction
projectives, i.e. for every object X there is an epimorphism from a projective object P to X, one can use projective resolutions instead of injective
Derived_category
Mathematical category with weak equivalences, fibrations and cofibrations
third model structure lying in between the projective and injective. The process of forcing certain maps to become weak equivalences in a new model category
Model_category
Inclusion of one mathematical structure in another, preserving properties of interest
is given by some injective and structure-preserving map f : X → Y {\displaystyle f:X\rightarrow Y} . The precise meaning of "structure-preserving" depends
Embedding
Completion of the usual space with "points at infinity"
concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus
Projective_space
Type of machine learning model
model of learning tasks and understanding. The NTL model outlines how specific neural structures of the human brain shape the nature of thought and language
Large_language_model
Protocol for communicating between LLMs and applications
official extension to the Model Context Protocol built on mcp-ui. While the base MCP specification is restricted to text and structured data, MCP Apps standardizes
Model_Context_Protocol
Mathematical set with some added structure
full-fledged, no structure is lost. In other words, the latter transition is injective (one-to-one), while the former transition is not injective (many-to-one)
Space_(mathematics)
is endowed with (at least) three model structures, namely the projective, the Reedy, and the injective model structure. The weak equivalences / fibrations
Simplicial_presheaf
Simplicial object in the category of simplicial sets
transported over to the category of bisimplicial sets using the injective and projective model structure. But it is more useful to instead take the analog replacements
Bisimplicial_set
Generalization of vector spaces from fields to rings
spaces. Projective Projective modules are direct summands of free modules and share many of their desirable properties. Injective Injective modules are
Module_(mathematics)
Type of mathematical space
mean a projective homogeneous variety, that is, a smooth projective variety X over a field F with a transitive action of a reductive group G (and smooth
Generalized_flag_variety
Map in projective geometry
embedding is a map used in projective geometry to consider the cartesian product of two projective spaces as a projective variety. It is named after Corrado
Segre_embedding
Homological construction in category theory
where the I i are all injective (this is known as an injective resolution of X). Applying the functor F to this sequence, and chopping off the first
Derived_functor
Mathematical system
with projective determinacy are hard to find. ZFC + {there are n Woodin cardinals: n is a natural number} is conservative over Z2 with projective determinacy[citation
Second-order_arithmetic
Overview of and topical guide to algebraic structures
modules, projective modules, injective modules and flat modules are studied in abstract algebra. These structures are defined over two sets, a ring R and an
Outline of algebraic structures
Outline_of_algebraic_structures
Concept in algebraic geometry
the Kodaira dimension κ(X) measures the size of the canonical model of a projective variety X. Soviet mathematician Igor Shafarevich in a seminar introduced
Kodaira_dimension
Type of mathematical model
the mapping is injective), it is said to be identifiable. In some cases, the model can be more complex. In Bayesian statistics, the model is extended by
Statistical_model
Two-dimensional manifold
"sides". For example, the sphere and torus are orientable, while the real projective plane is not (because the real projective plane with one point removed
Surface_(topology)
space Gauss–Bolyai–Lobachevsky space Grassmannian Complex projective space Real projective space Euclidean space Stiefel manifold Upper half-plane Sphere
List of differential geometry topics
List_of_differential_geometry_topics
Invariant in projective geometry
is essentially the only projective invariant of a quadruple of collinear points; this underlies its importance for projective geometry. The cross-ratio
Cross-ratio
One-to-one correspondence
complex plane. An injective non-surjective function (injection, not a bijection) An injective surjective function (bijection) A non-injective surjective function
Bijection
is of finite type and has finite fibers. quasi-projective A quasi-projective variety is a locally closed subvariety of a projective space. quasi-separated
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Category whose objects are sets and whose morphisms are functions
additive nor preadditive. Every non-empty set is an injective object in Set. Every set is a projective object in Set (assuming the axiom of choice). The
Category_of_sets
together with the metric graph structure L on Γ such that L(ei) = xi for i = 1, ..., k. One can show that j is in fact an injective map, that is, distinct points
Outer_space_(mathematics)
Programming technique of receiving dependencies
dependencies by external code (an "injector"), which it is not aware of. Dependency injection makes implicit dependencies explicit and helps solve the following
Dependency_injection
Differentiable function whose derivative is everywhere injective
everywhere injective. Explicitly, f : M → N is an immersion if D p f : T p M → T f ( p ) N {\displaystyle D_{p}f:T_{p}M\to T_{f(p)}N\,} is an injective function
Immersion_(mathematics)
Curve defined as zeros of polynomials
zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three
Algebraic_curve
Concept in homotopy theory
objects are all projective. The category SSet {\displaystyle {\textbf {SSet}}} of simplicial setspg 1.3 there is a model category structure where the fibrations
Cofibration
Two closely related mathematical subjects
are applied to analytic spaces and analytic techniques to algebraic varieties. Let X {\displaystyle X} be a projective complex algebraic variety. Because
Algebraic geometry and analytic geometry
Algebraic_geometry_and_analytic_geometry
Statistical Markov model
dependency structure enables identifiability of the model and the learnability limits are still under exploration. Hidden Markov models are generative models, in
Hidden_Markov_model
General concept and operation in mathematics
categorical duality are projective and injective modules in homological algebra, fibrations and cofibrations in topology and more generally model categories. Two
Duality_(mathematics)
Generalized manifold
s^{-1})} is injective for every x ∈ X {\displaystyle x\in X} . Two different orbifold atlases give rise to the same orbifold structure if and only if their
Orbifold
Topological space that locally resembles Euclidean space
sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical
Manifold
Barrier that stops or slows the flow of water
dam is a structure that impounds water or restricts its flow. Dams are classified into four basic types: gravity dams are massive structures made of concrete
Dam
Euclidean space without distance and angles
from any projective plane by removing one line and all the points on it, and conversely any affine plane can be used to construct a projective plane as
Affine_space
Organisms used to study biology across species
been very useful for the study of gene structure and gene regulation (e.g. phages Lambda and T4). Disease models are divided into three categories: homologous
Model_organism
Technique for the generative modeling of a continuous probability distribution
diffusion model consists of two major components: the forward diffusion process, and the reverse sampling process. The goal of diffusion models is to learn
Diffusion_model
Association of one output to each input
most one element. An empty function is always injective. If X is not the empty set, then f is injective if and only if there exists a function g : Y → X {\displaystyle
Function_(mathematics)
Structure at the rear of the vertebrate brain, beneath the cerebrum
equilibrium, posture, and motor learning in humans. Anatomically, the human cerebellum has the appearance of a separate structure attached to the bottom
Cerebellum
Mathematical set that can be enumerated
countable there is an injective function h : T → N {\displaystyle h:T\to \mathbb {N} } . Then if f : S → T {\displaystyle f:S\to T} is injective the composition
Countable_set
Algebraic structure with an associative operation and an identity element
the monoid into its Grothendieck group is not injective. More precisely, if a • b = a • c, then b and c have the same image in the Grothendieck group
Monoid
Algorithm for modelling sequential data
to be amplified and less important tokens to be diminished. Because self-attention alone is permutation-invariant, transformers inject positional information
Transformer_(deep_learning)
Statistical model used in machine learning
A flow-based generative model is a generative model used in machine learning that explicitly models a probability distribution by leveraging normalizing
Flow-based_generative_model
Supergeometric generalization of a manifold
view that the structure of a supermanifold M is contained in its sheaf OM of "smooth functions". In the dual point of view, an injective map corresponds
Supermanifold
Generalization of affine connections
P = stabilizer of a point on the projective null hyperquadric. Contact projective connections: Here G = SP(2n+2) and P is the stabilizer of the ray generated
Cartan_connection
Organ central to the nervous system
models of neurons and neural networks, which can be simulated using computers. Some useful models are abstract, focusing on the conceptual structure of
Brain
category that has enough injectives and enough projectives and such that the class of injective objects coincides with that of projective objects. Fukaya category
Glossary_of_category_theory
Generalizations of codimension-1 subvarieties of algebraic varieties
If X is a projective curve over k, then the divisor of a nonzero rational function f on X has degree zero. As a result, for a projective curve X, the
Divisor_(algebraic_geometry)
History of maths
category theory and related mathematics. Its scope ("related mathematics") is taken as: Categories of abstract algebraic structures including representation
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Diode that emits light from an organic compound
which are based on a p–n diode crystalline solid structure. In LEDs, doping is used to create p- and n-regions by changing the conductivity of the host
OLED
Database normalization
same candidate key as the table T is excluded. Attribute–value system Injective function First normal form (1NF) Second normal form (2NF) Third normal
Fifth_normal_form
Motor vehicle
1963 through 1967 model years. After 10 years of tinkering with the basic C1 package, albeit much improved, Corvette's designers and engineers knew it
Chevrolet_Corvette_(C2)
Mathematical descriptions of the properties of certain cells in the nervous system
temporal structure characterized by adaptation, bursting, or initial bursting followed by regular spiking. Generalized integrate-and-fire models such as
Biological_neuron_model
Hungarian and American mathematician and physicist (1903–1957)
result in projective geometry to the continuous dimensional case. This coordinatization theorem stimulated considerable work in abstract projective geometry
John_von_Neumann
Self-intersecting, highly symmetrical mapping of the real projective plane into 3D space
real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane;
Roman_surface
American sports car by the Chevrolet division of General Motors (GM)
saw fuel management system upgrades for the 2022 model year which featured a new fuel pump and injectors. The base price was also increased by $1200. A
Chevrolet_Corvette
Size of a set in mathematics
it is called injective. If a function covers every member in the output set, it is called surjective. If a function is both injective and surjective, it
Cardinality
Fundamental space of geometry
as defining a projective space as the set of the vector lines in a vector space of dimension one more. As for affine spaces, projective spaces are defined
Euclidean_space
Relationship between elements of two sets
the green and blue binary relations in the diagram are injective, but the red one is not (as it relates both − 1 {\displaystyle -1} and 1 {\displaystyle
Binary_relation
Infectious agent that replicates in cells
protein tail fibres. This tail structure acts like a molecular syringe, attaching to the bacterial host and then injecting the viral genome into the cell
Virus
Pin or bolt used as part of a pivot or hinge
describes the attachment point between the landing gear structure and the aircraft structure. The pintle is the bolt around which the landing gear rotates
Pintle
Shading language
shader assembly language, and went on to become the required shading language for the unified shader model of Direct3D 10 and higher. It was developed
High-Level_Shader_Language
National airline of Pakistan
aircraft skidded and eventually came to rest in sand 50 meters from the runway. The aircraft sustained damage to its right wing structure and right engine
Pakistan International Airlines
Pakistan_International_Airlines
Semiconductor light source
and balanced charge injection. In other words, they still used flat emissive layer, but tried to optimize the balance of electrons and holes injected
Light-emitting_diode
Category with direct sums and certain types of kernels and cokernels
{A} _{3}} identifying the simple projective, simple injective and indecomposable projective-injective modules. The essential image of I is a full, additive
Abelian_category
Deaths involving use of large language models
There have been multiple incidents where interaction with a large language model (LLM) chatbot has been cited as a direct or contributing factor in a person's
Deaths_linked_to_chatbots
Hypothetical climatic effect of nuclear war
short- and long-term atmospheric effects of a nuclear war using a three-dimensional model: First one to three months: 10–25% of soot injected is immediately
Nuclear_winter
efficient enough to effect a reduction in the rate of maternal and infant mortality The Abiye model has been recognized by the World Bank as a viable template
Abiye (Safe Motherhood) Project
Abiye_(Safe_Motherhood)_Project
Algebraic structure associated with a topological space
^{n}} and f : U → R n {\displaystyle f:U\to \mathbb {R} ^{n}} is an injective continuous map, then V = f ( U ) {\displaystyle V=f(U)} is open and f is
Homology_(mathematics)
Monitoring and Storage Project), was conducted there. Injection continued even after the study concluded. Between 2000 and 2018, over 30 Mt CO2 was injected. As
List of carbon capture and storage projects
List_of_carbon_capture_and_storage_projects
Prehistoric impact crater in Mexico
alongside the much older Sudbury and Vredefort impact structures, and the only one whose peak ring is intact and directly accessible for scientific
Chicxulub_crater
Property that assigns truth values to k-tuples of individuals
relation, when R is unique on {X1}, it is also said to be left-unique or injective, and when R is unique on {X2}, it is also said to be univalent or right-unique
Finitary_relation
Series of rotary-powered sports cars
GSL-SE had a fuel injected 1,308 cc (1.3 L) 13B RE-EGI engine rated at 135 hp (101 kW; 137 PS) and 133 lb⋅ft (180 N⋅m). GSL-SE models had much the same
Mazda_RX-7
American digital asset and data center infrastructure company
through Clearstream and later planned integration across Deutsche Börse Group's service portfolio. AllUnity also partnered with Projective Group to integrate
Galaxy_Digital_(company)
Algebraic structure
active research. Projective modules can be defined to be the direct summands of free modules. If R is local, any finitely generated projective module is actually
Commutative_ring
US military fighter aircraft in service 1930-1941
Boeing F2B and F3B with the United States Navy, the Boeing Model 99 first flew on 25 June 1928. The new aircraft was smaller, lighter and more agile than
Boeing_P-12
Compact luxury crossover SUV
2007 model. The facelifted model went on sale in August 2009 for the 2010 model year, which featured the addition of Acura's "power plenum" grille and a
Acura_RDX
Motor vehicle
with both its brand name and it's model name. The first generation Smart ForTwo (a joint venture project between Mercedes-Benz and Swatch) was internally
Smart_Fortwo
Japanese video game designer (born 1963)
releases, and occasionally drawing comparisons to his own works. In 2022, Kojima began a podcast for Spotify called Hideo Kojima presents Brain Structure. The
Hideo_Kojima
2024 video game
"adventure-first" video game, and the team was particularly inspired by Indiana Jones and the Fate of Atlantis (1992). Its gameplay structure was inspired by Metro
Indiana Jones and the Great Circle
Indiana_Jones_and_the_Great_Circle
Motor vehicle from 1965
It was renamed as the M 130 for both carburetted and fuel-injected versions creating the 280 model series. The former on the 280 S, made 140 PS (103 kW;
Mercedes-Benz_W108/W109
Technique invented by Paul Cohen for proving consistency and independence results
B^{*}} is injective. For any set B ⊆ [ 0 , 1 ] {\displaystyle B\subseteq [0,1]} such that B ∈ M {\displaystyle B\in M} and M ⊨ {\displaystyle M\models } " B
Forcing_(mathematics)
Automated software testing technique
used to test programs that take structured inputs. This structure is specified, such as in a file format or protocol and distinguishes valid from invalid
Fuzzing
Defunct American particle accelerator at Fermilab in Illinois (1983–2011)
fundamental fermion predicted by the Standard Model of particle physics. On July 2, 2012, scientists of the CDF and DØ collider experiment teams at Fermilab
Tevatron
Relativistic quantum mechanical wave equation
complex phase, so particles belong to projective representations rather than regular representations, with the projective representations of the Lorentz group
Dirac_equation
Branch of mathematics
finite-dimensional, and this allows identifying V with its bidual. (In the infinite-dimensional case, the canonical map is injective, but not surjective
Linear_algebra
American experimental spaceplane
The main structure is of a graphite/epoxy composite material. From front to back, it contains the crew cabin, oxidizer tank, fuel casing, and rocket nozzle
SpaceShipOne
Group of unitary complex matrices with determinant of 1
the motions of the Poincaré disk model of hyperbolic plane geometry. Indeed, for a point [z, 1] in the complex projective line, the action of SU(1,1) is
Special_unitary_group
Motor vehicle
African-made model was exported to the United Kingdom between 1991 and 1993 as the "Sao Penza" and fitted with a 1.3-litre fuel-injected engine. The importer
Mazda_Familia
current, and thus injectors, are offset by 90 degrees with two injectors, and by 60 degrees with three injectors. A spheromak plasma structure forms naturally
Dynomak
Micronation
model and trade system would have been hyper-capitalistic, modeled after the writings of Ayn Rand, Napoleon Hill, Robert Heinlein, Dale Carnegie, and
New_Utopia
Peoples Temple cult settlement in Guyana, led by Jim Jones
two from apparent cyanide poisoning, many of whom were injected against their will. Jones and some Peoples Temple members referred to the act as a "revolutionary
Jonestown
Motor vehicle
carburetted models (Rover O8) 820e – 4-cylinder 16-valve single point injected models (Rover M16e) 820i – 4-cylinder 16-valve multi point injected models (Rover
Rover_800_series
American luxury car
of mid-sized and full-sized luxury cars produced between 1939 and 2020 by Lincoln, a division of the American automaker Ford. The model line was introduced
Lincoln_Continental
Compact crossover SUV
model tested by the Insurance Institute for Highway Safety (IIFHS) was the LX model with standard driver and passenger airbags. The car's structure received
Honda_CR-V
Software engineering agent developed by OpenAI
integration let Codex inspect a project's structure, consult current Apple developer documentation, build projects, run tests, and revert changes through automatically
OpenAI_Codex_(AI_agent)
U.S. presidential administration since 2025
hostages, dismantling Hamas's military capabilities, and establishing a transitional governance structure in the Gaza Strip. In October 2025, President Trump
Second presidency of Donald Trump
Second_presidency_of_Donald_Trump
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