nLab projection

Definition

General

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

$$p_i \colon \prod_j X_j \to X_i \,.$$

or, more generally out of a limit

$$p_i \colon \underset{\leftarrow_j}{\lim} X_j \to X_i \,.$$

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 $P:V\to V$ 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 $P \colon V \to V$ canonically induces a decomposition of $V$ as a direct sum $V \simeq ker(V) \oplus im(V)$ and in terms of this $P$ is the composition

$$P \colon V \simeq im(V) \oplus ker(V)\to im(V) \hookrightarrow V$$

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

A projector is a projection followed by an inclusion.

Related concepts

A different concept of a similar name is projection formula.