Working group on higher categories

One-day working group for young researchers, on Monday 19 October 2026, opening the Rencontre 2026 de topologie algébrique (Institut Fourier, Grenoble, 19–23 October 2026). The five talks go from presheaf categories and Cisinski's homotopy theory of presheaves to Joyal's category $\mathrm{\Theta }\backslash Theta$, $nn$-quasi-categories, and Campbell's homotopy coherent nerve for bicategories.

Programme

Talk 1 — Presheaf categories I

• Basic properties of presheaf categories: the Yoneda embedding, every presheaf is a canonical colimit of representables; the example of simplicial sets.
• Universal property (cocontinuous functors out of the presheaves on a small category); the examples of topological realization and the nerve.
• The functors $u_{!}u\_!$, $u^{\ast }u^*$, $u_{\ast }u\_*$ induced by $u\text{\hspace{0.17em}⁣}:A\to Bu\backslash colon\; A\; \backslash to\; B$.
• Recollections on model categories: homotopies, homotopy category, derived functors, Quillen adjunctions and equivalences; example: the Kan–Quillen model structure.

References: P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory; M. Hovey, Model categories.

Talk 2 — Presheaf categories II

• Cisinski's theory of $AA$-localizers.
• Examples: the Kan–Quillen model structure; the Joyal model structure (quasi-categories); the Rezk model structure on bisimplicial sets.

References: D.-C. Cisinski, Higher categories and homotopical algebra, section 2.4; A. Joyal and M. Tierney, Quasi-categories vs Segal spaces.

Talk 3 — The category $\mathrm{\Theta }\backslash Theta$ and $nn$-quasi-categories

• Strict $\omega \backslash omega$-categories; the category $\mathrm{\Theta }\backslash Theta$ as a subcategory of $\omega \text{-}\mathbf{C}\mathbf{a}\mathbf{t}\backslash omega\backslash text\{-\}\backslash mathbf\{Cat\}$; the cellular nerve.
• Definition as an iterated wreath product; horns in $\mathrm{\Theta }\backslash Theta$; the model structure for $nn$-quasi-categories.
• Possibly, the comparison with complete ${\mathrm{\Theta }}_n\backslash Theta\_n$-spaces.

References: D. Ara, Higher quasi-categories vs higher Rezk spaces; A. Joyal, Disks, duality and $\mathrm{\Theta }\backslash Theta$-categories; C. Berger, Iterated wreath product of the simplex category and iterated loop spaces.

Talk 4 — A homotopy coherent cellular nerve for bicategories I (my talk)

• Recollections on bicategories; the Duskin nerve, valued in simplicial sets.
• The homotopy coherent nerve and its full faithfulness.
• Lack's model structure on bicategories; the Quillen adjunction (Theorem 5.10).

Reference: A. Campbell, A homotopy coherent cellular nerve for bicategories.

Slides

Talk 5 — A homotopy coherent cellular nerve for bicategories II

• The homotopy bicategory of a 2-quasi-category; the adjunction (Theorem 6.29).
• Characterization of the equivalences of 2-quasi-categories (Theorem 7.25) and of the 2-quasi-categories that come from a bicategory (Theorem 7.28).
• The equivalence with bicategories (Theorem 8.7).

Reference: A. Campbell, A homotopy coherent cellular nerve for bicategories.