direct limit
Noun
Meaning
(algebra) A set of equivalence classes which partition the disjoint union of the members of a direct system; each equivalence class being a sort of “drainage basin” of the mappings (of the morphisms) of the direct system, if these are analogically considered as “rivers”. (If
i
≤
k
,
j
≤
k
{\displaystyle i\leq k,\ j\leq k}
in the indexing poset, then there exist
f
i
k
:
A
i
→
A
k
{\displaystyle f_{ik}:A_{i}\rightarrow A_{k}}
and
f
j
k
:
A
j
→
A
k
{\displaystyle f_{jk}:A_{j}\rightarrow A_{k}}
. If
a
i
∈
A
i
,
a
j
∈
A
j
{\displaystyle a_{i}\in A_{i},\ a_{j}\in A_{j}}
such that
f
i
k
(
a
i
)
=
f
j
k
(
a
j
)
{\displaystyle f_{ik}(a_{i})=f_{jk}(a_{j})}
then
a
i
∼
a
j
{\displaystyle a_{i}\sim a_{j}}
. If k = j then
f
j
j
(
a
j
)
=
a
j
,
f
i
j
(
a
i
)
=
a
j
{\displaystyle f_{jj}(a_{j})=a_{j},\ f_{ij}(a_{i})=a_{j}}
.)
A direct limit has “canonical functions” which map each element of the disjoint union to its equivalence class.
Direct limits in the algebraic sense are models of category-theoretic colimits.
(category theory) a colimit
Source: en.wiktionary.org