[Paper Review] Limits in $n$-categories

Carlos Simpson|ArXiv.org|Aug 7, 1997

Homotopy and Cohomology in Algebraic Topology19 references16 citations

TL;DR

This paper introduces a unified framework for limits in $n$-categories, generalizing categorical and homotopical limits by defining direct and inverse limits via universal properties in fibrant $n$-categories. The key result establishes the existence of arbitrary inverse and direct limits in the $n+1$-category $nCAT'$, extending classical category-theoretic limits to higher homotopical settings.

ABSTRACT

We define notions of direct and inverse limits in an $n$-category. We prove that the $n+1$-category $nCAT'$ of fibrant $n$-categories admits direct and inverse limits. At the end we speculate (without proofs) on some applications of the notion of limit, including homotopy fiber product and homotopy coproduct for $n$-categories, the notion of $n$-stack, representable functors, and finally on a somewhat different note, a notion of relative Malcev completion of the higher homotopy at a representation of the fundamental group.

Motivation & Objective

• To develop a coherent notion of limits in $n$-categories that unifies categorical limits and homotopical holim.
• To address the lack of systematic treatment of higher limits in $n$-categorical homotopy theory.
• To establish foundational existence theorems for inverse and direct limits in the $n+1$-category $nCAT'$ of fibrant $n$-categories.
• To propose applications in higher stack theory, representable functors, and relative Malcev completion of homotopy types.
• To provide a conceptual bridge between classical category theory and modern homotopical algebra via $n$-categorical limits.

Proposed method

• Define direct and inverse limits in an $n$-category $C$ using universal properties involving equivalences of $(n-1)$-categories of morphisms.
• Use the $n+1$-precat $\Upsilon^k(E_1,\ldots,E_k)$ as a higher-dimensional simplex with $n$-precats on edges, serving as a technical tool for limit constructions.
• Construct inverse limits in $nCAT'$ as $\operatorname{Hom}_{\underline{Hom}(A,C)}(\ast,\varphi)$, generalizing the set-theoretic inverse limit.
• For direct limits, employ a duality trick: construct an auxiliary $n+1$-category $D$ parametrizing morphisms $\psi \to B$, then take the inverse limit of the evaluation functor $D \to nCAT'$.
• Handle set-theoretic issues by restricting to a bounded cardinality subcategory $D_\alpha$ to ensure smallness and avoid foundational obstructions.
• Apply the framework to define relative Malcev completion of higher homotopy types via inverse limits in suitable $n+1$-categories of stacks and groupoids.

Experimental results

Research questions

• RQ1 How can the classical notion of limit in category theory be generalized to $n$-categories to incorporate higher homotopical information?
• RQ2 What conditions ensure the existence of inverse and direct limits in the $n+1$-category $nCAT'$ of fibrant $n$-categories?
• RQ3 Can the notion of limit be used to define and study higher analogues of Malcev completion in homotopy theory?
• RQ4 How do homotopy coproducts and fiber products in $nCAT'$ relate to the closed model structure on $n$-categories?
• RQ5 What is the role of representable functors and internal homs in $n$-categories, and when can they be constructed via limits?

Key findings

• The $n+1$-category $nCAT'$ admits arbitrary inverse and direct limits, generalizing the classical existence of limits in $\mathbf{Sets}$ (the $n=0$ case).
• Inverse limits in $nCAT'$ are constructed as $\operatorname{Hom}_{\underline{Hom}(A,C)}(\ast,\varphi)$, satisfying the universal property via equivalence of morphism $(n-1)$-categories.
• Direct limits are constructed via a duality trick: by forming a category $D$ of morphisms $\psi \to B$ and taking the inverse limit of the evaluation functor $D \to nCAT'$, with bounded cardinality to resolve set-theoretic issues.
• The relative Malcev completion $Malc(X,\rho)$ of the higher homotopy type of a space $X$ at a representation $\rho$ is defined as the inverse limit in $nCAT'$ of a suitable subcategory of $n$-groupoids with unipotent fundamental groups.
• For a variety $X$ over $\mathbb{C}$, the Betti, de Rham, and Dolbeault versions of the Malcev completion are naturally equivalent: $Malc(X_B,\rho)/\mathbb{C} \cong Malc(X_{DR},P)/\mathbb{C} \cong Malc(X_{Dol},Q)/\mathbb{C}$.
• Conjecturally, $Malc(X_B,\rho)/\mathbb{R}$ carries a mixed Hodge structure, and $Malc(X_B,\rho)/\mathbb{C}$ carries a mixed twistor structure, extending classical Hodge-theoretic ideas to higher homotopy types.