problem 1.2.b
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main.typ
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main.typ
@@ -8,4 +8,6 @@
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#pagebreak()
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#include "problems/problem12a.typ"
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#pagebreak()
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#include "problems/problem12bprime.typ"
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#pagebreak()
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#include "problems/problem12b.typ"
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pdfs/main.pdf
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#import "@preview/fletcher:0.5.8" as fletcher: diagram, node, edge
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#problembox[
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б') Пусть даны категории $cal(I), cal(J), cal(C)$. Пусть даны функторы $phi: cal(J) -> cal(I)$, $alpha: cal(I) -> cal(C)$, $beta: cal(I)^"op" -> cal(C)$. Определите естественные морфизмы $colim(alpha compose phi) -> colim(alpha)$ и $lim(beta) -> lim(beta compose phi^"op")$ в предположении, что все пределы существуют. (Это будет на след. лекции, но попробуйте догадаться самостоятельно).
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б) Определим предпучок $f^+ cal(G)$ на $X$ как $f^+ cal(G)(U) = colim_(f(U) subset W) cal(G)(W)$, где копредел берётся по диаграмме, состоящей из множеств $cal(G)(W)$, стрелки между которыми --- отображения ограничения. Определите отображение ограничения для $f^+ cal(G)$ (используйте пункт б')). Этот предпучок называется *обратным образом предпучка* $cal(G)$.
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]
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#addinfobox[
|
||||
Пояснение: для любого $U subset X$ определим полную poset-подкатегорию $C(U)$ в $Open(Y)^"op"$, состоящую из таких открытых $W subset Y$, что $f(U) subset W$. Тогда имеем функтор $alpha(U, cal(G)): C(U) -> Set$, который получается из функтора $cal(G): Open(Y)^"op" -> Set$ ограничением его на $C(U)$. Тогда $f^+ cal(G)(U)$ --- это копредел по $alpha(U, cal(G))$ для открытого $U subset X$.
|
||||
]
|
||||
|
||||
== Решение
|
||||
|
||||
=== Для копределов
|
||||
Будем обозначать $alpha^U_cal(G) := alpha(U, cal(G))$.
|
||||
|
||||
Обозначим $iota: alpha ==> colim alpha$, $tau: (alpha compose phi) ==> colim (alpha compose phi)$ --- коконусы.
|
||||
Хотим привести это всё к виду $colim(alpha compose phi) -> colim alpha$
|
||||
|
||||
Хотим: коконус $psi: (alpha compose phi) ==> colim alpha$.
|
||||
Построим его с помощью переиндексации
|
||||
$ psi_j := iota_(phi(j)) $
|
||||
Очевидно, это коконус.
|
||||
Функтор вложения $iota_U: C(U) arrow.hook Open(Y)^op$ просто оставляет объекты и морфизмы на месте. Тогда $alpha^U_cal(G) = cal(G) compose iota_U$. В таком случае ограничения --- это отображения
|
||||
$ colim (cal(G) compose iota_U) -> colim (cal(G) compose iota_V), space.quad V subset.eq U $
|
||||
|
||||
По универсальному свойству, $forall X in cal(C)$ и коконуса $mu: (alpha compose phi) ==> X$ существует единственное $theta: colim(alpha compose phi) -> X$ т.ч
|
||||
$ theta compose tau_(j) = mu_(j) $
|
||||
Существует вложение категорий $phi^U_V: C(U) arrow.hook C(V)$.
|
||||
$ op("Ob")(C(U)) = {W in tau_Y | f(U) subset.eq W} $
|
||||
$ op("Ob")(C(V)) = {Z in tau_Y | f(V) subset.eq Z} $
|
||||
Если $f(V) subset.eq f(U)$, а $f(U) subset.eq Z$ для $Z in op("Ob")(C(U))$, то, очевидно, $f(V) subset.eq Z$. Тогда категории вложены.
|
||||
|
||||
Пололжим $X = colim alpha, mu = psi$.
|
||||
#cdiagram({
|
||||
node((-1, -1), [$alpha(phi(j))$])
|
||||
node((1, 1), [$op("colim") (alpha compose phi)$])
|
||||
node((3, -1), [$alpha (phi(j'))$])
|
||||
node((1, 3), [$op("colim") alpha$])
|
||||
edge((-1, -1), (3, -1), [$alpha(phi(f))$], label-side: left, "->")
|
||||
edge((-1, -1), (1, 1), [$tau_j$], label-side: right, "->")
|
||||
edge((3, -1), (1, 1), [$tau_j'$], label-side: left, "->")
|
||||
edge((-1, -1), (1, 3), [$psi_j$], label-side: left, "->", bend: -30deg)
|
||||
edge((3, -1), (1, 3), [$psi_j'$], label-side: left, "->", bend: 30deg)
|
||||
edge((1, 1), (1, 3), [$theta$], label-side: left, "-->")
|
||||
})
|
||||
Применим б'), подставляя
|
||||
- $cal(J) = C(U)$
|
||||
- $cal(I) = C(V)$
|
||||
- $cal(C) = Set$
|
||||
- $phi = phi^U_V$
|
||||
- $alpha = alpha^V_cal(G)$
|
||||
|
||||
Тогда существует единственное
|
||||
$ colim (alpha^V_cal(G) compose phi^U_V) ->^theta colim(alpha^V_cal(G)) $
|
||||
Это равно
|
||||
$ colim (alpha^U_cal(G)) ->^theta colim(alpha^V_cal(G)) $
|
||||
|
||||
#answerbox[
|
||||
#align(center)[
|
||||
Определили $theta: colim (alpha compose phi) -> colim alpha$
|
||||
]
|
||||
]
|
||||
|
||||
=== Для пределов
|
||||
|
||||
Имеем пределы с их каноническими конусами:
|
||||
- $pi: lim beta ==> beta$ для функтора $beta: cal(I)^"op" -> cal(C)$,
|
||||
- $q: lim (beta compose phi^"op") ==> (beta compose phi^"op")$ для функтора $beta compose phi^"op": cal(J)^"op" -> cal(C)$.
|
||||
|
||||
Построим конус $p: lim beta ==> (beta compose phi^op)$ с помощью переиндексации:
|
||||
$ p_j := pi_phi(j) $
|
||||
|
||||
Тогда по универсальному свойству предела $lim(beta compose phi^"op")$ существует единственный морфизм $eta: lim beta -> lim (beta compose phi^"op")$, делающий диаграмму коммутативной:
|
||||
|
||||
#cdiagram({
|
||||
node((0, 4), [$beta(phi(j))$])
|
||||
node((4, 4), [$beta(phi(j'))$])
|
||||
node((2, 2), [$lim (beta compose phi^"op")$])
|
||||
node((2, 0), [$lim beta$])
|
||||
edge((4, 4), (0, 4), [$beta(phi(f))$], label-side: left, "->")
|
||||
edge((2, 2), (0, 4), [$q_j$], label-side: right, "->")
|
||||
edge((2, 2), (4, 4), [$q_(j')$], label-side: left, "->")
|
||||
edge((2, 0), (2, 2), [$eta$], label-side: right, "-->")
|
||||
edge((2, 0), (0, 4), [$p_j$], label-side: right, "->", bend: -30deg)
|
||||
edge((2, 0), (4, 4), [$p_j'$], label-side: left, "->", bend: 30deg)
|
||||
})
|
||||
|
||||
#answerbox[
|
||||
#align(center)[
|
||||
Определили $eta: lim beta -> lim (beta compose phi^op)$
|
||||
Ограничение определяется как
|
||||
$ theta: f^+ cal(G)(U) -> f^+ cal(G)(V) $
|
||||
]
|
||||
]
|
||||
|
||||
71
problems/problem12bprime.typ
Normal file
71
problems/problem12bprime.typ
Normal file
@@ -0,0 +1,71 @@
|
||||
#import "../utils/boxes.typ": *
|
||||
#import "../utils/macros.typ": *
|
||||
#import "@preview/fletcher:0.5.8" as fletcher: diagram, node, edge
|
||||
|
||||
#problembox[
|
||||
б') Пусть даны категории $cal(I), cal(J), cal(C)$. Пусть даны функторы $phi: cal(J) -> cal(I)$, $alpha: cal(I) -> cal(C)$, $beta: cal(I)^"op" -> cal(C)$. Определите естественные морфизмы $colim(alpha compose phi) -> colim(alpha)$ и $lim(beta) -> lim(beta compose phi^"op")$ в предположении, что все пределы существуют. (Это будет на след. лекции, но попробуйте догадаться самостоятельно).
|
||||
]
|
||||
|
||||
== Решение
|
||||
|
||||
=== Для копределов
|
||||
|
||||
Обозначим $iota: alpha ==> colim alpha$, $tau: (alpha compose phi) ==> colim (alpha compose phi)$ --- коконусы.
|
||||
|
||||
Хотим: коконус $psi: (alpha compose phi) ==> colim alpha$.
|
||||
Построим его с помощью переиндексации
|
||||
$ psi_j := iota_(phi(j)) $
|
||||
Очевидно, это коконус.
|
||||
|
||||
По универсальному свойству, $forall X in cal(C)$ и коконуса $mu: (alpha compose phi) ==> X$ существует единственное $theta: colim(alpha compose phi) -> X$ т.ч
|
||||
$ theta compose tau_(j) = mu_(j) $
|
||||
|
||||
Пололжим $X = colim alpha, mu = psi$.
|
||||
#cdiagram({
|
||||
node((-1, -1), [$alpha(phi(j))$])
|
||||
node((1, 1), [$op("colim") (alpha compose phi)$])
|
||||
node((3, -1), [$alpha (phi(j'))$])
|
||||
node((1, 3), [$op("colim") alpha$])
|
||||
edge((-1, -1), (3, -1), [$alpha(phi(f))$], label-side: left, "->")
|
||||
edge((-1, -1), (1, 1), [$tau_j$], label-side: right, "->")
|
||||
edge((3, -1), (1, 1), [$tau_j'$], label-side: left, "->")
|
||||
edge((-1, -1), (1, 3), [$psi_j$], label-side: left, "->", bend: -30deg)
|
||||
edge((3, -1), (1, 3), [$psi_j'$], label-side: left, "->", bend: 30deg)
|
||||
edge((1, 1), (1, 3), [$theta$], label-side: left, "-->")
|
||||
})
|
||||
|
||||
#answerbox[
|
||||
#align(center)[
|
||||
Определили $theta: colim (alpha compose phi) -> colim alpha$
|
||||
]
|
||||
]
|
||||
|
||||
=== Для пределов
|
||||
|
||||
Имеем пределы с их каноническими конусами:
|
||||
- $pi: lim beta ==> beta$ для функтора $beta: cal(I)^"op" -> cal(C)$,
|
||||
- $q: lim (beta compose phi^"op") ==> (beta compose phi^"op")$ для функтора $beta compose phi^"op": cal(J)^"op" -> cal(C)$.
|
||||
|
||||
Построим конус $p: lim beta ==> (beta compose phi^op)$ с помощью переиндексации:
|
||||
$ p_j := pi_phi(j) $
|
||||
|
||||
Тогда по универсальному свойству предела $lim(beta compose phi^"op")$ существует единственный морфизм $eta: lim beta -> lim (beta compose phi^"op")$, делающий диаграмму коммутативной:
|
||||
|
||||
#cdiagram({
|
||||
node((0, 4), [$beta(phi(j))$])
|
||||
node((4, 4), [$beta(phi(j'))$])
|
||||
node((2, 2), [$lim (beta compose phi^"op")$])
|
||||
node((2, 0), [$lim beta$])
|
||||
edge((4, 4), (0, 4), [$beta(phi(f))$], label-side: left, "->")
|
||||
edge((2, 2), (0, 4), [$q_j$], label-side: right, "->")
|
||||
edge((2, 2), (4, 4), [$q_(j')$], label-side: left, "->")
|
||||
edge((2, 0), (2, 2), [$eta$], label-side: right, "-->")
|
||||
edge((2, 0), (0, 4), [$p_j$], label-side: right, "->", bend: -30deg)
|
||||
edge((2, 0), (4, 4), [$p_j'$], label-side: left, "->", bend: 30deg)
|
||||
})
|
||||
|
||||
#answerbox[
|
||||
#align(center)[
|
||||
Определили $eta: lim beta -> lim (beta compose phi^op)$
|
||||
]
|
||||
]
|
||||
Reference in New Issue
Block a user