mathematics, injective and projective model structures are special model structures on functor categories into a model category. Both model structures do not...

{\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...

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...

_{k}^{n})\rightarrow \operatorname {RFib} (\Delta ^{n}).} Injective and projective model structure, induced model structures on functor categories Lurie, Jacob (2009)...

is given by some injective and structure-preserving map f : X → Y {\displaystyle f:X\rightarrow Y} . The precise meaning of "structure-preserving" depends...

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...

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...

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...

such as structure prediction and mutational outcome prediction, a small model using an embedding as input can approach or exceed much larger models using...

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)...

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...

spaces. Projective Projective modules are direct summands of free modules and share many of their desirable properties. Injective Injective modules are...

"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...

is endowed with (at least) three model structures, namely the projective, the Reedy, and the injective model structure. The weak equivalences / fibrations...

Segre embedding is used in projective geometry to consider the cartesian product (of sets) of two projective spaces as a projective variety. It is named after...

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...

is essentially the only projective invariant of a quadruple of collinear points; this underlies its importance for projective geometry. The cross-ratio...

of smooth projective varieties X. That is, this vector space is canonically identified with the corresponding space for any smooth projective variety which...

space Gauss–Bolyai–Lobachevsky space Grassmannian Complex projective space Real projective space Euclidean space Stiefel manifold Upper half-plane Sphere...

lifting property, co- and contravariant as well as injective and projective model structures, and the general theory of presheaves of sets, which in particular...

complex plane. An injective non-surjective function (injection, not a bijection) An injective surjective function (bijection) A non-injective surjective function...

categorical duality are projective and injective modules in homological algebra, fibrations and cofibrations in topology and more generally model categories. Two...

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...

are applied to analytic spaces and analytic techniques to algebraic varieties. Let X {\displaystyle X} be a projective complex algebraic variety. Because...

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...

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...

dependencies by external code (an "injector"), which it is not aware of. Dependency injection makes implicit dependencies explicit and helps solve the following...

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...

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...

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...