Limits and colimits

Category theory




Generally in category theory a projection is one of the canonical morphisms p ip_i out of a (categorical) product:

p i: jX jX i. p_i \colon \prod_j X_j \to X_i \,.

or, more generally out of a limit

p i:lim jX jX i. p_i \colon \underset{\leftarrow_j}{\lim} X_j \to X_i \,.

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 coprojections.

In linear algebra

In linear algebra an idempotent linear operator P:VVP:V\to V 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 P:VVP \colon V \to V canonically induces a decomposition of VV as a direct sum Vker(V)im(V)V \simeq ker(V) \oplus im(V) and in terms of this PP is the composition

P:Vim(v)ker(V)im(V)V P \colon V \simeq im(v) \oplus ker(V)\to im(V) \hookrightarrow V

of the projection (in the above sense of maps out of products) out of the direct sum im(V)ker(V)im(V)×ker(V)im(V) \oplus ker(V) \simeq im(V) \times ker(V) followed by the subobject inclusion of im(V)im(V). Hence:

A projector is a projection followed by an inclusion.

A different concept of a similar name is projection formula.

Revised on April 6, 2017 00:37:03 by Toby Bartels (