[Paper Review] Simplicial radditive functors

Vladimir Voevodsky|ArXiv.org|May 28, 2008

Homotopy and Cohomology in Algebraic Topology2 references4 citations

TL;DR

This paper develops a homotopical framework for functors between categories of simplicial radditive functors, establishing conditions under which simplicial extensions of continuous functors preserve local equivalences. The key contribution is a criterion using $ar{\Delta}$-closed classes that ensures derived functors between localized homotopy categories preserve $E$-local equivalences, generalizing classical results on simplicial sets to broader contexts like algebraic geometry and motivic homotopy theory.

ABSTRACT

The simplicial extension of any functor from Sets to Sets which commutes with directed colimits takes weak equivalences to weak equivalences. The goal of the present paper is construct a framework which can be used to proof results of this kind for a wide class of closed model categories and functors between those categories.

Motivation & Objective

• To construct a general framework for proving that functors between closed model categories preserve weak equivalences, especially in contexts like motivic homotopy theory.
• To address the challenge that radditive functors do not commute with colimits, undermining standard homotopical techniques based on cofibrant replacement.
• To generalize the classical result that functors commuting with colimits preserve weak equivalences in simplicial sets to more complex categories such as those of presheaves and additive functors.
• To define and characterize conditions under which derived functors between localized homotopy categories preserve $E$-local equivalences.

Proposed method

• Introduces the notion of $ar{\Delta}$-closed classes of morphisms in $\Delta^{op}C$, which are closed under simplicial extensions of continuous functors.
• Uses the projective closed model structure on simplicial radditive functors to define homotopy categories $H(C)$ and their localizations $H(C,E)$ via $E$-local equivalences.
• Establishes that $E$-local equivalences in $\Delta^{op}C^{\#}$ are characterized by $\bar{\Delta}$-closure, enabling transfer of properties across functors.
• Applies Smith's localization theorem and Bousfield localization to relate $cl_l(E)$ to model structures, even without left properness.
• Develops a functoriality criterion: if $F^{rad}(f \amalg \mathrm{id}_X) \in cl_l(E')$ for all $f \in E$, then $F^{rad}$ preserves $E$-local equivalences.
• Uses adjoint pairs $(F^{rad}, F_{rad})$ and proves that $F_{rad}$ reflects $E$-local equivalences under surjectivity conditions on $F$.

Experimental results

Research questions

• RQ1 Under what conditions does the simplicial extension of a continuous functor $F: C^\# \to (C')^\#$ preserve $E$-local equivalences between objects in $\Delta^{op}C^\#$?
• RQ2 How can one ensure that the derived functor $\mathbf{L}F^{rad}: H(C) \to H(C')$ preserves $E$-local equivalences in the localized homotopy categories?
• RQ3 What role does the $\bar{\Delta}$-closure of morphism sets play in characterizing $E$-local equivalences and ensuring functoriality?
• RQ4 When does the right adjoint $F_{rad}$ reflect $E'$-local equivalences, and what conditions on $F$ guarantee this?
• RQ5 Can the adjunction $\mathbf{L}F^{rad} \dashv \mathbf{R}F_{rad}$ between localized homotopy categories be a Quillen adjunction, and what are the obstructions?

Key findings

• The simplicial extension of any continuous functor $F: C^\# \to (C')^\#$ preserves projective equivalences between objects in $\Delta^{op}C^\#$, leading to a well-defined derived functor $\mathbf{L}F^{rad}: H(C) \to H(C^\prime)$.
• Theorem 4.19 establishes that if $F^{rad}(E \amalg \mathrm{id}_C) \subset cl_l(E')$, then $\mathbf{L}F^{rad}(cl_l(E)) \subset cl_l(E')$, ensuring preservation of $E$-local equivalences.
• Theorem 4.20 shows that when $F$ commutes with finite coproducts, $F^{rad}$ preserves $E$-local equivalences between objects in $\Delta^{op}C^\#$, and $F_{rad}$ preserves $E'$-local equivalences in $\Delta^{op}Rad(C^\prime)$.
• Corollary 4.21 proves that $F_{rad}$ maps $E'$-local objects to $E$-local objects, and under surjectivity of $F$, the derived functors form a localization adjunction.
• If $F$ is surjective on isomorphism classes, then $\mathbf{R}F_{rad}$ reflects isomorphisms, so $cl_l(E') = F_{rad}^{-1}(cl_l(E))$, establishing a full characterization of local equivalences.
• The framework applies to motivic contexts: it generalizes the preservation of $\mathbf{A}^1$-equivalences by symmetric power functors and other constructions in algebraic geometry.