Question

Proving Proposition I.5.1 of Mac Lane and Moerdijk.

score 126 · Answer 1 · 2026-03-25 22:25:56

126

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Proving Proposition I.5.1 of Mac Lane and Moerdijk.

Published on 25 Mar 2026 - 22:25

score 116 · Answer 2 · 2020-02-18 04:14:02

116

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Can the absolute Picard functor be trivial?

Published on 18 Feb 2020 - 4:14

score 298 · Answer 3 · 2026-03-29 13:29:52

298

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A sieve $S$ on $U$ in the category $\mathcal{O}(X)$ is principal iff the corresponding subfunctor $S\subset 1_U\cong{\rm Hom}(-,U)$ is a sheaf.

Published on 29 Mar 2026 - 13:29

score 232 · Answer 4 · 2026-03-25 23:58:41

232

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A presheaf $P$ on $X$ is a sheaf iff for every covering sieve $S$ on an open set $U$ of $X$ one has $PU=\varprojlim_{V\in S}PV.$

Published on 25 Mar 2026 - 23:58

score 122 · Answer 5 · 2020-02-24 21:16:07

122

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Castelnuovo-Mumford Regularity of Product of Ideal Sheaves

Published on 24 Feb 2020 - 21:16

score 47 · Answer 6 · 2020-02-25 15:52:17

47

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Why $ (\operatorname{Im}\varphi)_x=\operatorname{Im}(\varphi_x) $?

Published on 25 Feb 2020 - 15:52

score 276 · Answer 7 · 2026-03-26 06:30:50

276

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different definitions of fine sheaves / partitions of unity

Published on 26 Mar 2026 - 6:30

score 116 · Answer 8 · 2026-03-25 14:02:27

116

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Wedge product and right derived functors

Published on 25 Mar 2026 - 14:02

score 265 · Answer 9 · 2026-03-28 15:21:18

265

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For basis $\mathcal{B}$ of the topology on $X$, the restriction functor $\mathbf{r}:{\rm Sh}(X)\to{\rm Sh}(\mathcal{B})$ is an equivalence.

Published on 28 Mar 2026 - 15:21

score 313 · Answer 10 · 2026-03-28 15:21:31

313

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Prove that $F$ is locally constant iff the associated étale space over $X$ is a covering.

Published on 28 Mar 2026 - 15:21

score 181 · Answer 11 · 2026-03-28 15:26:54

181

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Just what is Mac Lane & Moerdijk's $\Lambda$ from $\S II.5$?

Published on 28 Mar 2026 - 15:26

score 73 · Answer 12 · 2020-02-28 12:41:33

73

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How to show that these presheaves are sheaves

Published on 28 Feb 2020 - 12:41

score 96 · Answer 13 · 2026-03-25 22:30:45

96

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. . . and what about $\Gamma$ in $\S II.5$ of Mac Lane and Moerdijk?

Published on 25 Mar 2026 - 22:30

score 93 · Answer 14 · 2026-03-28 15:19:53

93

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Two equivalent notions of compactness?

Published on 28 Mar 2026 - 15:19

score 327 · Answer 15 · 2026-03-25 22:23:36

327

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Show that $\Gamma$, $\Lambda$, and the associated sheaf functor are all left exact.

Published on 25 Mar 2026 - 22:23

List Question

Proving Proposition I.5.1 of Mac Lane and Moerdijk.

Can the absolute Picard functor be trivial?

A sieve $S$ on $U$ in the category $\mathcal{O}(X)$ is principal iff the corresponding subfunctor $S\subset 1_U\cong{\rm Hom}(-,U)$ is a sheaf.

A presheaf $P$ on $X$ is a sheaf iff for every covering sieve $S$ on an open set $U$ of $X$ one has $PU=\varprojlim_{V\in S}PV.$

Castelnuovo-Mumford Regularity of Product of Ideal Sheaves

Why $ (\operatorname{Im}\varphi)_x=\operatorname{Im}(\varphi_x) $?

different definitions of fine sheaves / partitions of unity

Wedge product and right derived functors

For basis $\mathcal{B}$ of the topology on $X$, the restriction functor $\mathbf{r}:{\rm Sh}(X)\to{\rm Sh}(\mathcal{B})$ is an equivalence.

Prove that $F$ is locally constant iff the associated étale space over $X$ is a covering.

Just what is Mac Lane & Moerdijk's $\Lambda$ from $\S II.5$?

How to show that these presheaves are sheaves

. . . and what about $\Gamma$ in $\S II.5$ of Mac Lane and Moerdijk?

Two equivalent notions of compactness?

Show that $\Gamma$, $\Lambda$, and the associated sheaf functor are all left exact.

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