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

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...

{\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...

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

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...

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

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 of learning tasks and understanding. The NTL Model outlines how specific neural structures of the human brain shape the nature of thought and language...

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)...

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...

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...

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...

"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...

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...

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

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

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...

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

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

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...

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

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...

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...

objects are all projective. The category SSet {\displaystyle {\textbf {SSet}}} of simplicial setspg 1.3 there is a model category structure where the fibrations...

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...

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...

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