nLab projection

Context

Limits and colimits

limits and colimits

category theory

Contents

Definition

General

Generally tn category theory a projection is one of the canonical morphisms ${p}_{i}$ out of a (categorical) product:

${p}_{i}:\prod _{j}{X}_{j}\to {X}_{i}\phantom{\rule{thinmathspace}{0ex}}.$p_i \colon \prod_j X_j \to X_i \,.

or, more generally out of a limit

${p}_{i}:\underset{{←}_{j}}{\mathrm{lim}}{X}_{j}\to {X}_{i}\phantom{\rule{thinmathspace}{0ex}}.$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).

In linear algebra

In linear algebra an idempotent linear operator $P:V\to V$ 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 $P:V\to V$ canonically induces a decomposition of $V$ as a direct sum $V\simeq \mathrm{ker}\left(V\right)\oplus \mathrm{im}\left(V\right)$ and in terms of this $P$ is the composition

$P:V\simeq \mathrm{im}\left(v\right)\oplus \mathrm{ker}\left(V\right)\to \mathrm{im}\left(V\right)↪V$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 $\mathrm{im}\left(V\right)\oplus \mathrm{ker}\left(V\right)\simeq \mathrm{im}\left(V\right)×\mathrm{ker}\left(V\right)$ followed by the subobject inclusion of $\mathrm{im}\left(V\right)$. Hence:

A projector is a projection followed by an inclusion.

Revised on January 7, 2013 17:14:19 by Urs Schreiber (89.204.154.29)