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Abstract We study the simplicial coalgebra of chains on a simplicial set with respect to three notions of weak equivalence. To this end, we construct three model structures on the category of reduced simplicial sets for any commutative ring $$R$$. The weak equivalences are given by: (1) an $$R$$-linearized version of categorical equivalences, (2) maps inducing an isomorphism on fundamental groups and an $$R$$-homology equivalence between universal covers, and (3) $$R$$-homology equivalences. Analogously, for any field $${\mathbb{F}}$$, we construct three model structures on the category of connected simplicial cocommutative $${\mathbb{F}}$$-coalgebras. The weak equivalences in this context are (1′) maps inducing a quasi-isomorphism of dg algebras after applying the cobar functor, (2′) maps inducing a quasi-isomorphism of dg algebras after applying a localized version of the cobar functor, and (3′) quasi-isomorphisms. Building on a previous work of Goerss in the context of (3)–(3′), we prove that, when $${\mathbb{F}}$$ is algebraically closed, the simplicial $${\mathbb{F}}$$-coalgebra of chains defines a homotopically full and faithful left Quillen functor for each one of these pairs of model categories. More generally, when $${\mathbb{F}}$$ is a perfect field, we compare the three pairs of model categories in terms of suitable notions of homotopy fixed points with respect to the absolute Galois group of $${\mathbb{F}}$$.
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On the construction of limits and colimits in ∞-categories
In previous work, we introduce an axiomatic framework within which to prove theorems about many varieties of infinite-dimensional categories simultaneously. In this paper, we establish criteria implying that an ∞-category --- for instance, a quasi-category, a complete Segal space, or a Segal category --- is complete and cocomplete, admitting limits and colimits indexed by any small simplicial set. Our strategy is to build (co)limits of diagrams indexed by a simplicial set inductively from (co)limits of restricted diagrams indexed by the pieces of its skeletal filtration. We show directly that the modules that express the universal properties of (co)limits of diagrams of these shapes are reconstructible as limits of the modules that express the universal properties of (co)limits of the restricted diagrams. We also prove that the Yoneda embedding preserves and reflects limits in a suitable sense, and deduce our main theorems as a consequence.
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- Award ID(s):
- 1652600
- PAR ID:
- 10288977
- Date Published:
- Journal Name:
- Theory and applications of categories
- Volume:
- 35
- Issue:
- 30
- ISSN:
- 1201-561X
- Page Range / eLocation ID:
- 1101-1158
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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