Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
Generally in category theory a projection is one of the canonical morphisms out of a (categorical) product:
or, more generally out of a limit
Hence a projection is a component of a limiting cone over a given diagram.
In fact, in older literature the filtered diagrams of spaces or algebraic systems (usually in fact indexed by a codirected set) were called projective systems (or inverse systems).
Dually, for colimits the corresponding maps in the opposite direction are sometimes caled co-projectioons.
In linear algebra
In linear algebra an idempotent linear operator is called a projection onto its image. See at projector.
In functional analysis, one sometimes requires additionally that this idempotent is in fact self-adjoint; or one can use the slightly different terminology projection operator.
This relates to the previous notion as follows: the existence of the projector canonically induces a decomposition of as a direct sum and in terms of this is the composition
of the projection (in the above sense of maps out of products) out of the direct sum followed by the subobject inclusion of . Hence:
A projector is a projection followed by an inclusion.
A different concept of a similar name is projection formula.
Revised on June 8, 2015 13:24:08
by Mike Shulman