[Paper Review] Exact completion of path categories and algebraic set theory -- Part I: Exact completion of path categories

Benno van den Berg, Ieke Moerdijk|arXiv (Cornell University)|Mar 8, 2016

Homotopy and Cohomology in Algebraic Topology39 references3 citations

TL;DR

This paper introduces 'path categories'—a refinement of Brown's categories of fibrant objects with an added cofibrancy condition—and constructs their homotopy exact completion, a new category that freely adjoins homotopy quotients. The key result is that the exact completion of the homotopy category of a path category is itself the homotopy category of a larger path category, preserving and enhancing homotopical structures such as weak $Π$-types and local cartesian closure.

ABSTRACT

We introduce the notion of a "category with path objects", as a slight strengthening of Kenneth Brown's classic notion of a "category of fibrant objects". We develop the basic properties of such a category and its associated homotopy category. Subsequently, we show how the exact completion of this homotopy category can be obtained as the homotopy category associated to a larger category with path objects, obtained by freely adjoining certain homotopy quotients. In a second part of this paper, we will present an application to models of constructive set theory. Although our work is partly motivated by recent developments in homotopy type theory, this paper is written purely in the language of homotopy theory and category theory, and we do not presuppose any familiarity with type theory on the side of the reader.

Motivation & Objective

• To introduce and formalize the notion of a 'path category' as a strengthening of Brown's categories of fibrant objects, ensuring all objects are cofibrant.
• To develop the homotopy theory of such categories, including path objects with connection structures and diagonal fillers up to homotopy.
• To construct a homotopy exact completion that freely adjoins homotopy quotients, generalizing ordinary exact completion in category theory.
• To show that this completion preserves and enhances homotopical properties, such as the existence of exponentials and function extensionality.
• To lay categorical foundations for interpreting constructive set theory and type theory in homotopical settings, particularly via syntactic models.

Proposed method

• Define a path category as a category with fibrations, weak equivalences, and a path object construction satisfying additional axioms, including cofibrancy of all objects.
• Introduce a new construction, the homotopy exact completion $\mathrm{Hex}(\mathcal{C})$, which freely adjoins homotopy quotients to the homotopy category of a path category $\mathcal{C}$.
• Show that $\mathrm{Hex}(\mathcal{C})$ is equivalent to the homotopy category of a new path category $\mathrm{Ex}(\mathcal{C})$, via a construction that preserves fibrations and weak equivalences.
• Prove that if $\mathcal{C}$ has weak homotopy $\Pi$-types (i.e., fibrewise up-to-homotopy internal homs), then $\mathrm{Hex}(\mathcal{C})$ is locally cartesian closed.
• Use descent techniques in exact completions to compute exponentials in slices of $\mathrm{Hex}(\mathcal{C})$, relying on coequalizers over pseudo-equivalence relations.
• Demonstrate that stability of objects under transport along loops is preserved under exponential constructions, ensuring the completion satisfies function extensionality even when the original category does not.

Experimental results

Research questions

• RQ1 How can Brown’s notion of a category of fibrant objects be strengthened to ensure all objects are cofibrant, and what are the homotopical consequences of this modification?
• RQ2 Can the exact completion of a homotopy category be constructed in a way that preserves its homotopical structure, and if so, how?
• RQ3 Under what conditions does the homotopy exact completion of a path category inherit local cartesian closure and exponentials in slices?
• RQ4 To what extent does the homotopy exact completion improve extensionality properties, such as function extensionality, in the absence of such properties in the original category?
• RQ5 How does this construction relate to known categorical models in type theory and constructive set theory, particularly the setoid construction and syntactic models?

Key findings

• The homotopy exact completion $\mathrm{Hex}(\mathcal{C})$ of a path category $\mathcal{C}$ is equivalent to the homotopy category of a larger path category $\mathrm{Ex}(\mathcal{C})$, thus preserving the homotopical structure.
• If $\mathcal{C}$ has weak homotopy $\Pi$-types, then $\mathrm{Hex}(\mathcal{C})$ is locally cartesian closed, meaning every slice category has exponentials.
• The construction of exponentials in $\mathrm{Hex}(\mathcal{C})$ relies on descent via coequalizers over pseudo-equivalence relations, with stability of transport structures preserved under exponentiation.
• The completion ensures that function extensionality holds in $\mathrm{Hex}(\mathcal{C})$ even when $\mathcal{C}$ does not, due to the preservation of transport and equivalence structures.
• The path category $\mathrm{Ex}(\mathcal{C})$ admits a diagonal filler construction that is half strict and half up-to-homotopy, a key technical tool for proving stability and extensionality.
• When $\mathcal{C}$ has stable and disjoint homotopy sums, the homotopy exact completion $\mathrm{Hex}(\mathcal{C})$ is a pretopos, indicating strong logical closure properties.