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 co-projectioons.

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 June 8, 2015 13:24:08 by Mike Shulman (