nLab
projection
Contents
Context
Limits and colimits
limits and colimits

1-Categorical
limit and colimit

limits and colimits by example

commutativity of limits and colimits

small limit

filtered colimit

sifted colimit

connected limit , wide pullback

preserved limit , reflected limit , created limit

product , fiber product , base change , coproduct , pullback , pushout , cobase change , equalizer , coequalizer , join , meet , terminal object , initial object , direct product , direct sum

finite limit

Kan extension

weighted limit

end and coend

fibered limit

2-Categorical
(∞,1)-Categorical
Model-categorical
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 coprojection s .

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.
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