Formal weakly enriched category theory
Abstract
A formal category theory is constructed (in the form of a proarrow equipment), encoding weak coherent enrichment over a monoidal model category mV. We describe how basic categorical concepts formulated via the equipment translate back to enriched categories. We characterize Dwyer-Kan equivalences of enriched categories as $2$-categorical equivalences. Specializing to either the Kan-Quillen model structure on simplicial sets, or the Quillen-Serre model structure on topological spaces, we prove that the resulting formal category theory is equivalent to the one associated with the ∞-cosmos of quasicategories, thereby extending the formal approach to (∞,1)-categories in the sense of Riehl-Verity to encompass both simplicial and topological categories. A notion of classifying object, formulated internally to the equipment of mV-categories, leads to enriched versions of Quillen's Theorem A.
Related
- Hanslick's Musical Formalism as a Mathematical Structure - Examining meaning in Music through the Lens of Category Theory
- Aligning Information Theory and Formal Verification: A Lean Formalization of Rate-Distortion Theory
- Limit theory for a class of non-stationary linear processes with weakly dependent heavy-tailed innovations
- Response theory at low temperature for weakly interacting quantum spin systems
- WEAKLY VISCOUS NONLOCAL WAVE EQUATIONS: FORWARD THEORY, LEBESGUE RUNGE APPROXIMATION, AND INVERSE PROBLEMS
BibTeX
@misc{leoncini2026formal,
title = {{Formal weakly enriched category theory}},
author = {Giuseppe Leoncini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.28435},
eprint = {2609.28435},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28435},
}
RIS
TY - UNPB
TI - Formal weakly enriched category theory
AU - Giuseppe Leoncini
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2609.28435
UR - https://arxiv.org/abs/2609.28435
AB - A formal category theory is constructed (in the form of a proarrow equipment), encoding weak coherent enrichment over a monoidal model category mV. We describe how basic categorical concepts formulated via the equipment translate back to enriched categories. We characterize Dwyer-Kan equivalences of enriched categories as $2$-categorical equivalences. Specializing to either the Kan-Quillen model structure on simplicial sets, or the Quillen-Serre model structure on topological spaces, we prove that the resulting formal category theory is equivalent to the one associated with the ∞-cosmos of quasicategories, thereby extending the formal approach to (∞,1)-categories in the sense of Riehl-Verity to encompass both simplicial and topological categories. A notion of classifying object, formulated internally to the equipment of mV-categories, leads to enriched versions of Quillen's Theorem A.
ER -