[Paper Review] Sheaves of ordered spaces and interval theories

Krzysztof Worytkiewicz|ArXiv.org|Aug 13, 2008

Homotopy and Cohomology in Algebraic Topology10 references3 citations

TL;DR

This paper establishes a topos-theoretic framework for directed homotopy theory by embedding the category of locally ordered spaces (local epo-spaces) into a topos of sheaves over a site of elementary partially ordered spaces. It shows that D-homotopy (with the standard interval equipped with natural order) is weaker than Di-homotopy (with discrete order), proving that the identity functor from the D-homotopy model to the Di-homotopy model is a left Quillen functor, thus relating two key notions of directed homotopy via sheaf-theoretic machinery.

ABSTRACT

We study the homotopy theory of locally ordered spaces, that is manifolds with boundary whose charts are partially ordered in a compatible way. Their category is not particularly well-behaved with respect to colimits. However, this category turns out to be a certain full subcategory of a topos of sheaves over a simpler site. A precise characterisation of this subcategory is provided. The ambient topos makes available some general homotopical machinery.

Motivation & Objective

• To provide a homotopical framework for locally ordered spaces, which are poorly behaved under colimits in their native category.
• To compare two notions of directed homotopy—Di-homotopy (using discrete order on the interval) and D-homotopy (using natural order)—in a unified setting.
• To show that the D-homotopy model is weaker than the Di-homotopy model by embedding both into a common topos of sheaves.
• To use interval-based model structures in Grothendieck topoi to compare weak equivalences induced by different ordered intervals.
• To establish that the identity functor from the D-homotopy to the Di-homotopy model is a left Quillen functor, implying a hierarchy of homotopical information.

Proposed method

• Embed the category of local epo-spaces into the topos of sheaves over a site of epo-spaces via the restriction of the Yoneda embedding.
• Characterize local epo-spaces as sheaves that are $π$-local with respect to étale dimaps, using a full subcategory of sheaves.
• Construct two interval objects in the sheaf topos: one with discrete order ($\mathcal{I}_d$) and one with natural order ($\mathcal{I}_D$), representing Di-homotopy and D-homotopy respectively.
• Define morphisms of intervals and analyze how weak equivalences in one model structure relate to those in another via component-wise weak equivalences.
• Use a contracting homotopy $h(f,g) = f \cdot g$ from $h_{\mathbb{P}}(\Delta_d) \times h_{\mathbb{P}}(\Delta_d) \to h_{\mathbb{P}}(\Delta_D)$ to show that $!_D$ is an $\mathcal{I}_d$-weak equivalence.
• Apply results from Cisinski’s interval-based model structures to compare the weak equivalences of the two model structures induced by $\mathcal{I}_d$ and $\mathcal{I}_D$.

Experimental results

Research questions

• RQ1 How can the category of locally ordered spaces be embedded into a more well-behaved homotopical setting?
• RQ2 What is the relationship between Di-homotopy and D-homotopy in the context of locally ordered spaces?
• RQ3 Are the weak equivalences in the D-homotopy model contained within those of the Di-homotopy model?
• RQ4 Can a sheaf-theoretic framework unify and compare different notions of directed homotopy?
• RQ5 Is there a Quillen functor relationship between the model structures induced by the discrete and natural order intervals?

Key findings

• The category of local epo-spaces fully embeds into the topos of sheaves over the site of epo-spaces via the restricted Yoneda embedding.
• A sheaf is a local epo-space if and only if it admits a family of $\mathbb{P}$-local monomorphisms from representable sheaves whose coproduct maps are epimorphisms.
• The morphism of intervals $\iota: \mathcal{I}_d \to \mathcal{I}_D$ is a component-wise $\mathcal{I}_d$-weak equivalence.
• The weak equivalences of the D-homotopy model ($\mathcal{W}_{\mathcal{I}_D}$) are contained within those of the Di-homotopy model ($\mathcal{W}_{\mathcal{I}_d}$).
• The identity functor from the D-homotopy model to the Di-homotopy model is a left Quillen functor, indicating a hierarchy of homotopical information.
• The contracting homotopy $h(f,g) = f \cdot g$ is essential in showing $!_D$ is an $\mathcal{I}_d$-weak equivalence, but no such homotopy exists in the reverse direction due to order-connectedness constraints.