Last Updated：2025/12/05
(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
音声機能が動作しない場合はこちらをご確認ください
See correct answer
direct limit
Edit Histories（0）
Source Word
direct limit
Noun
(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
in
the
indexing
poset,
then
there
exist
f_ik:A_i→A_k
and
f_jk:A_j→A_k.
If
a_i∈A_i,a_j∈A_j
such
that
f_ik(a_i)=f_jk(a_j)
then
a_i∼a_j.
If
k
=
j
then
f_jj(a_j)=a_j,f_ij(a_i)=a_j.)
Japanese Meaning
直極限：与えられたdirect systemの各構成対象の互いに交わらない和集合を、対応する射（モルフィズム）によって定められる同値関係で分割して得られる同値類全体を指す。各同値類は、射の働きにより連続的な流れが集まる「排水盆」として解釈される。また、これは代数学や圏論などで、系全体の構造を一つの対象に集約する手法として用いられる考え方である。
Sense（1）
(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
in
the
indexing
poset,
then
there
exist
f_ik:A_i→A_k
and
f_jk:A_j→A_k.
If
a_i∈A_i,a_j∈A_j
such
that
f_ik(a_i)=f_jk(a_j)
then
a_i∼a_j.
If
k
=
j
then
f_jj(a_j)=a_j,f_ij(a_i)=a_j.)
（ plural ）
