Injective and projective model structures on enriched diagram categories

Abstract

In the enriched setting, the notions of injective and projective model structures on a category of enriched diagrams also make sense. In this paper, we prove the existence of these model structures on enriched diagram categories under local presentability, accessibility, and "acyclicity" conditions, using the methods of lifting model structures from an adjunction introduced

Enriched diagrams of topological spaces over locally contractible enriched categories

“…This section discusses the interactions between the two model structures. The existence of the projective model structure is a straightforward consequence of Moser's work [Mos19]. We are also able to prove that this model structure is left proper in Theorem 6.6 (it is right proper because all objects are fibrant).…”

A Künneth theorem for configuration spaces

“…For any simplicial operad $$\mathcal {O}$$, this equivalence extends to an equivalence of module categories $$\begin{equation*} N:(\mathsf {grMod}_{R}^{ {\Delta }^{op}})^{\mathsf {F}(\mathcal {O})_R^{op}}\xrightarrow {\simeq } (\mathsf {grCh}_R^{\geqslant 0})^{\mathsf {F}(\mathcal {O})_R^{op}}. \end{equation*}$$A slight variant of [29, Example 7.9] shows that, if we equip $$\mathsf {grCh}_R^{\geqslant 0}$$ with the model structure in which fibrations are surjections in positive degrees and weak equivalences quasi‐isomorphisms, then $$(\mathsf {grCh}_R^{\geqslant 0})^{\mathsf {F}(\mathcal {O})_R^{op}}$$ admits the projective model structure, which transfers via the equivalence $$N$$ to $$(\mathsf {grMod}_{R}^{ {\Delta }^{op}})^{\mathsf {F}(\mathcal {O})_R^{op}}\cong \mathsf {Mod}_{\mathcal {O}_R}^{ {\Delta }^{op}}$$.…”

A shadow framework for equivariant Hochschild homologies

“…This hypothesis holds if, for example, V is a locally presentable base [28, Definition 2.1], and tensoring with A op ⊗ B a ⊗ b, a ′ ⊗ b ′ preserves acyclic cofibrations in V for every pair of V-categories A and B. These conditions guarantee the existence of a projective model structure on the category of (A, B)-bimodules [28,Remark 4.5]. Moreover, the proof of [18,Theorem 11.7.3] applies essentially verbatim in this case, to show that objectwise tensoring and cotensoring by simplicial sets endows A Mod B with the structure of a simplicial model category.…”

“…Examples of locally presentable bases include the categories of simplicial sets, of symmetric spectra, and of chain complexes over a commutative ring [28,Examples 5.6,6.6,6.7]. The condition on preservation of acyclic cofibrations holds if, for example, all objects in V are cofibrant or we consider only those V-categories such that the morphism objects are cofibrant in V. Definition 3.2.7.…”