# nLab projection

Contents

### Context

#### Limits and colimits

limits and colimits

category theory

# Contents

## 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.

A different concept of a similar name is projection formula.

Last revised on October 20, 2022 at 11:23:05. See the history of this page for a list of all contributions to it.