---
title: 'Models for dagger $(\infty,1)$-categories II: Dagger complete Segal spaces and the Joyal--Tierney equivalence'
authors:
  - "Keima Akasaka"
year: 2026
venue: "arXiv:math.CT"
arxiv: "2609.06935"
doi: null
source: "https://arxiv.org/abs/2609.06935"
topics:
  - "curiosity"
  - "maths-foundations"
seed_rank: 1777
seed_batch: "curiosity-2026-09-11"
reviewed: "2026-09-11"
pool: "maths-foundations"
relevance_score: 10
lineage: dagger-infinity-categories
cites:
  - title: 'Models for dagger $(\infty,1)$-categories II: Dagger complete Segal spaces and the Joyal--Tierney equivalence'
    url: "https://arxiv.org/abs/2609.06935"
    year: 2026
    arxiv: "2609.06935"
    doi: null
see:
  - "1239-quasi-categories-vs-segal-spaces"
  - "1373-the-infinity-category-of-infinity-categories-in-simplicial-type-theory"
---

# Models for dagger $(\infty,1)$-categories II: Dagger complete Segal spaces and the Joyal--Tierney equivalence

## One-sentence takeaway

Part II completes a Quillen-equivalence chain among dagger Bergner, dagger Joyal, and dagger Rezk models for dagger (∞,1)-categories.

## Why it's lovely

Why you might love this: dagger categories are the home of unitary / adjoint-aware homotopy — the place where "inverse" is a structure, not a property. Getting complete Segal spaces and a Joyal–Tierney equivalence to play nicely with the dagger is spectator-sport ∞-category theory with immediate payoffs for reversible and quantum-flavoured higher algebra.

## Key ideas

- Builds on Part I's dagger Bergner and dagger Joyal model structures.
- Homotopically constant resolution model plus unitarily complete dagger Segal spaces.
- Change-of-index adjunction comparing presentations of dagger simplicial spaces.
- Dagger analogue of the Joyal–Tierney Quillen equivalence.
- Together with Part I: Quillen equivalences linking Bergner / Joyal / Rezk dagger models.

## Caveats

Model-category foundations; does not yet compute a specific dagger (∞,1)-category of interest. Read beside the Segal-space vs quasicategory shelf (1239) and synthetic ∞-category cards.

## Links

- arXiv abs: https://arxiv.org/abs/2609.06935
- PDF: https://arxiv.org/pdf/2609.06935.pdf