nLab
projection

Definition

General

Generally tn category theory a projection is one of the canonical morphisms $p_i$ out of a (categorical) product:

$$p_i:\prod _j{X}_j\to X_i.$$p_i \colon \prod_j X_j \to X_i \,.

or, more generally out of a limit

$$p_i:\underset{{\leftarrow }_j}{\lim }{X}_j\to X_i.$$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).

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 $V\simeq \ker (V)\oplus \mathrm{im}(V)$ and in terms of this $P$ is the composition

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

A projector is a projection followed by an inclusion.