Pith Number

pith:WHZTUBB5

pith:2025:WHZTUBB57J4XQPM7ULA26DTBV6

not attested not anchored not stored refs pending

Elementary $\infty$-toposes from type theory

Dani\"el Apol, Maximilian Petrowitsch

Every categorical model of dependent type theory with univalent universes becomes an elementary ∞-topos after ∞-localization.

arxiv:2512.18891 v3 · 2025-12-21 · math.CT · math.AT · math.LO

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{WHZTUBB57J4XQPM7ULA26DTBV6}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

every categorical model of dependent type theory with dependent sums and products, intensional identity types and univalent universes presents via its ∞-localisation an elementary ∞-topos, that is, a finitely complete, locally cartesian closed ∞-category with enough univalent universal morphisms

C2weakest assumption

The ∞-localization functor applied to a categorical model of the given type theory preserves finite completeness, local cartesian closure, and the existence of enough univalent universal morphisms; this relies on the newly introduced notions of univalent tribe and univalent fibration being correctly defined and interacting properly with the localization.

C3one line summary

Categorical models of dependent type theory with sums, products, identity types and univalent universes present elementary ∞-toposes via ∞-localization, which also possess small subobject classifiers.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-06-02T03:05:04.554130Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

b1f33a043dfa79783d9fa2c1af0e61afa18d6d62d40745ad98894ec65b571ef6

Aliases

arxiv: 2512.18891 · arxiv_version: 2512.18891v3 · doi: 10.48550/arxiv.2512.18891 · pith_short_12: WHZTUBB57J4X · pith_short_16: WHZTUBB57J4XQPM7 · pith_short_8: WHZTUBB5

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/WHZTUBB57J4XQPM7ULA26DTBV6 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: b1f33a043dfa79783d9fa2c1af0e61afa18d6d62d40745ad98894ec65b571ef6

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "91459ca05b31445d2263e576b55e3b4e2cd29fa6f713be464350a265fcf30ee3",
    "cross_cats_sorted": [
      "math.AT",
      "math.LO"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CT",
    "submitted_at": "2025-12-21T21:20:13Z",
    "title_canon_sha256": "f1c16a0ff352afb0423f95a050073d82c9f81a6013c9f5d523db96fc1be71c03"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2512.18891",
    "kind": "arxiv",
    "version": 3
  }
}