Paper Citation Record · LEDGER

Generation of Grothendieck topologies, provability and operations on subtoposes

As of 8 August 2026, this Paper Citation Record lists 32 of 32 outbound references and 2 inbound Pith citation observations for arXiv:2508.21134.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2508.21134 v1

Coverage vector

measured 32 of 32 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-05T14:46:05.967713Z

measured 34 of 34 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-08T06:32:00.761636+00:00

measured 2 of 2 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-07-05T14:25:18.768204Z

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: pith, observed 2026-07-05T14:31:09.375499Z

Reference resolution

32 of 32 outbound references displayed

verified exact0
verified fuzzy18
unresolved14
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 3929cc01-f180-4855-bddc-8c819f47767c · outbound

This paper cites horizontal.

Generation of Grothendieck topologies, provability and operations on subtoposes horizontal

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T14:46:06.968316Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 2dc63996-6d37-4131-9785-2a87be2e2b3c · outbound

This paper cites an unresolved cited work.

Generation of Grothendieck topologies, provability and operations on subtoposes Unresolved cited work

Reference 2

Resolution: unresolved; raw_fallback, observed 2026-08-05T14:46:07.132012Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 6c769da2-d978-47c2-8d1c-94cd2b146690 · outbound

This paper cites pr´ esent´ e.

Generation of Grothendieck topologies, provability and operations on subtoposes pr´ esent´ e

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T14:46:07.163804Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 7f4f4ceb-9ce7-444e-8fc4-d4bc8ac420e7 · outbound

This paper cites an unresolved cited work.

Generation of Grothendieck topologies, provability and operations on subtoposes Unresolved cited work

Reference 4

Resolution: unresolved; raw_fallback, observed 2026-08-05T14:46:07.095607Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 67c9caf0-107a-4d6e-a970-843327f18b64 · outbound

This paper cites an unresolved cited work.

Generation of Grothendieck topologies, provability and operations on subtoposes Unresolved cited work

Reference 5

Resolution: unresolved; raw_fallback, observed 2026-08-05T14:46:07.074748Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 233fd3ef-ecad-4de6-83bb-1cd8bb4cba6d · outbound

This paper cites Apr` es un nombre fini de ces r´ eductions, tout tel objetφ(⃗ x) de Ccar Σ se r´ e´ ecrit sous la forme de (i).

Generation of Grothendieck topologies, provability and operations on subtoposes Apr` es un nombre fini de ces r´ eductions, tout tel objetφ(⃗ x) de Ccar Σ se r´ e´ ecrit sous la forme de (i)

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T14:46:07.051376Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation ec9c3a54-bc55-4c05-b6c7-da8c5601510d · outbound

This paper cites an unresolved cited work.

Generation of Grothendieck topologies, provability and operations on subtoposes Unresolved cited work

Reference 7

Resolution: unresolved; raw_fallback, observed 2026-08-05T14:46:07.030701Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 45d089ab-d3cf-41ac-89c9-f21569c1f710 · outbound

This paper cites horizontal.

Generation of Grothendieck topologies, provability and operations on subtoposes horizontal

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T14:46:06.992181Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 49394ae1-6c24-4c22-a3fb-2966cab913fd · outbound

This paper cites fibration.

Generation of Grothendieck topologies, provability and operations on subtoposes fibration

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T14:46:06.932118Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 35b16e50-8502-4353-bc8e-b7851d1b2cbb · outbound

This paper cites 208 CHAPITRE IV.

Generation of Grothendieck topologies, provability and operations on subtoposes 208 CHAPITRE IV

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T14:46:06.904464Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 174f0e4a-b1f9-4e87-a550-ffc6d00ce5f9 · outbound

This paper cites D´ emonstration : Grothendieck Supposons d’abord que F soit une fibration et consid´ erons deux morphismes deB de la forme F (X) Y ′ // Y pour un objet X de C.

Generation of Grothendieck topologies, provability and operations on subtoposes D´ emonstration : Grothendieck Supposons d’abord que F soit une fibration et consid´ erons deux morphismes deB de la forme F (X) Y ′ // Y pour un objet X de C

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T14:46:06.875174Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 27585e44-b728-49db-a6df-565e9a98ca51 · outbound

This paper cites essentiel.

Generation of Grothendieck topologies, provability and operations on subtoposes essentiel

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T14:46:06.837339Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 7f56bbc5-d82f-4802-9a49-8d472e4d16a5 · outbound

This paper cites p(X) − →Y ], • les morphismes sont les morphismes de C X2 x − − →X1 qui rendent commutatif le triangle p(X2) p(x) Y << "" p(X1) p(X2) p(x)[resp.

Generation of Grothendieck topologies, provability and operations on subtoposes p(X) − →Y ], • les morphismes sont les morphismes de C X2 x − − →X1 qui rendent commutatif le triangle p(X2) p(x) Y << "" p(X1) p(X2) p(x)[resp

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T14:46:06.816753Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 987877c4-d3ab-41c4-8c11-146955b8bbc4 · outbound

This paper cites Cela montre que l’application p!(P ×p∗Q1 p∗Q2) − →p!P (Y ) ×Q1(Y ) Q2(Y ) est une bijection, comme annonc´ e.

Generation of Grothendieck topologies, provability and operations on subtoposes Cela montre que l’application p!(P ×p∗Q1 p∗Q2) − →p!P (Y ) ×Q1(Y ) Q2(Y ) est une bijection, comme annonc´ e

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T14:46:06.794937Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 6c3b66d5-e7bb-4079-9790-51537364a571 · outbound

This paper cites an unresolved cited work.

Generation of Grothendieck topologies, provability and operations on subtoposes Unresolved cited work

Reference 15

Resolution: unresolved; raw_fallback, observed 2026-08-05T14:46:06.731136Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation b77b39e2-c01b-4e02-8feb-80b8585a5a3f · outbound

This paper cites Comme tout morphisme Y → p(X) se rel` eve en un morphisme horizontal dans C, on voit que pour un tel crible C ′ = p−1C d’un objet X, on a C ′ p = C.

Generation of Grothendieck topologies, provability and operations on subtoposes Comme tout morphisme Y → p(X) se rel` eve en un morphisme horizontal dans C, on voit que pour un tel crible C ′ = p−1C d’un objet X, on a C ′ p = C

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T14:46:06.710637Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 4c2c8a11-d68a-447b-ba59-e2a5fcd3a688 · outbound

This paper cites topologies.

Generation of Grothendieck topologies, provability and operations on subtoposes topologies

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T14:46:06.662604Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 6a3e7536-c521-4f42-8581-02c880597e35 · outbound

This paper cites an unresolved cited work.

Generation of Grothendieck topologies, provability and operations on subtoposes Unresolved cited work

Reference 18

Resolution: unresolved; raw_fallback, observed 2026-08-05T14:46:06.624114Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation c4a15961-e535-4b5a-acd9-769eca2f0fe0 · outbound

This paper cites (ii) Il y a mˆ eme ´ egalit´ e de sous-objets C ′ f = Cf ×f!X f!X ′ si le morphisme X ′ x − − →X est un ´ epimorphisme.

Generation of Grothendieck topologies, provability and operations on subtoposes (ii) Il y a mˆ eme ´ egalit´ e de sous-objets C ′ f = Cf ×f!X f!X ′ si le morphisme X ′ x − − →X est un ´ epimorphisme

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T14:46:06.595596Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 96463922-eeed-4a30-950e-675466b4a4d2 · outbound

This paper cites Grothendieck Fin de la d´ emonstration du th´ eor` eme IV.1.8 (ii).

Generation of Grothendieck topologies, provability and operations on subtoposes Grothendieck Fin de la d´ emonstration du th´ eor` eme IV.1.8 (ii)

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T14:46:06.568083Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation c3ff0fc5-8134-4c7c-8238-478c827df23d · outbound

This paper cites Comme le morphisme X ′ − →X est un ´ epimorphisme, on obtient d’apr` es le lemme IV.1.9 (ii) la formule C ′ f = Cf ×f!X f!X ′.

Generation of Grothendieck topologies, provability and operations on subtoposes Comme le morphisme X ′ − →X est un ´ epimorphisme, on obtient d’apr` es le lemme IV.1.9 (ii) la formule C ′ f = Cf ×f!X f!X ′

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T14:46:06.541093Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation b91fe996-4502-49de-bf9f-8da78c745480 · outbound

This paper cites D’o` u il r´ esulte comme voulu que l’application Hom(X ′, f∗Y ) − →Hom(C ′, f∗Y ) est une bijection si Y est un objet de EJ j∗ // E.

Generation of Grothendieck topologies, provability and operations on subtoposes D’o` u il r´ esulte comme voulu que l’application Hom(X ′, f∗Y ) − →Hom(C ′, f∗Y ) est une bijection si Y est un objet de EJ j∗ // E

Reference 22

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T14:46:06.493565Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation c0893d19-2f52-498d-8a8e-77f526db195a · outbound

This paper cites an unresolved cited work.

Generation of Grothendieck topologies, provability and operations on subtoposes Unresolved cited work

Reference 23

Resolution: unresolved; raw_fallback, observed 2026-08-05T14:46:06.454751Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 3e698f91-8484-4958-b00d-a2ec8d7faa72 · outbound

This paper cites an unresolved cited work.

Generation of Grothendieck topologies, provability and operations on subtoposes Unresolved cited work

Reference 24

Resolution: unresolved; raw_fallback, observed 2026-08-05T14:46:06.389920Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 18dcc6eb-6ab6-4d5b-a309-469da37be681 · outbound

This paper cites an unresolved cited work.

Generation of Grothendieck topologies, provability and operations on subtoposes Unresolved cited work

Reference 25

Resolution: unresolved; raw_fallback, observed 2026-08-05T14:46:06.349791Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation fc909808-2551-4a05-8f52-8fb2b63b6b19 · outbound

This paper cites Grothendieck La conclusion r´ esulte du th´ eor` eme de factorisation suivant : Th´ eor` eme IV.2.4.

Generation of Grothendieck topologies, provability and operations on subtoposes Grothendieck La conclusion r´ esulte du th´ eor` eme de factorisation suivant : Th´ eor` eme IV.2.4

Reference 26

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T14:46:06.322270Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 1ce91bbb-fcf3-476e-8c57-5887199f158e · outbound

This paper cites an unresolved cited work.

Generation of Grothendieck topologies, provability and operations on subtoposes Unresolved cited work

Reference 27

Resolution: unresolved; raw_fallback, observed 2026-08-05T14:46:06.284207Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 3a887544-f964-4d3d-808e-a58f762bba22 · outbound

This paper cites an unresolved cited work.

Generation of Grothendieck topologies, provability and operations on subtoposes Unresolved cited work

Reference 28

Resolution: unresolved; raw_fallback, observed 2026-08-05T14:46:06.255135Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 8c03437d-8a16-481e-94c9-f53d7096b682 · outbound

This paper cites an unresolved cited work.

Generation of Grothendieck topologies, provability and operations on subtoposes Unresolved cited work

Reference 29

Resolution: unresolved; raw_fallback, observed 2026-08-05T14:46:06.222250Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 0769d446-6873-4900-aa8b-714c1bdeb204 · outbound

This paper cites Il est aussi K-continu puisque le foncteur r : C → C′ transforme toute famille K-couvrante en une famille K ′-couvrante.

Generation of Grothendieck topologies, provability and operations on subtoposes Il est aussi K-continu puisque le foncteur r : C → C′ transforme toute famille K-couvrante en une famille K ′-couvrante

Reference 30

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T14:46:06.184020Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 66f8b749-9a02-485f-a475-dbaa3f016df9 · outbound

This paper cites an unresolved cited work.

Generation of Grothendieck topologies, provability and operations on subtoposes Unresolved cited work

Reference 31

Resolution: unresolved; raw_fallback, observed 2026-08-05T14:46:06.120654Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 77075e70-b2b1-4931-84d2-fdbf4b904177 · outbound

This paper cites Relative topos theory via stacks.

Generation of Grothendieck topologies, provability and operations on subtoposes Relative topos theory via stacks

Reference 32

Resolution: unresolved; no resolver link, observed 2026-08-05T14:46:05.967713Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

Observation 31edc888-4527-481e-856f-039e0993c908 · inbound

The Topological Dual of a Dataset: A Logic-to-Topology Encoding for AlphaGeometry-Style Data cites this paper.

The Topological Dual of a Dataset: A Logic-to-Topology Encoding for AlphaGeometry-Style Data Generation of Grothendieck topologies, provability and operations on subtoposes

Reference 5

Resolution: verified exact; arxiv_id, observed 2026-05-10T10:29:25.473283Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation d581ec2a-b227-4f42-a4e3-06294bb0baa2 · inbound

The Topological Dual of a Dataset: A Logic-to-Topology Encoding for AlphaGeometry-Style Data cites this paper.

The Topological Dual of a Dataset: A Logic-to-Topology Encoding for AlphaGeometry-Style Data Generation of Grothendieck topologies, provability and operations on subtoposes

Reference 5

Resolution: verified exact; local_arxiv, observed 2026-07-05T14:31:09.378047Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.