Revisiting colimits in $\mathbf{Cat}$ and homotopy category

This paper establishes the (co)completeness of the category of small categories ($\mathbf{Cat}$) by explicitly constructing specific weighted colimits that are equivalent to the existence of the homotopy category functor, thereby providing a systematic elementary approach to justify these limits and reformulate constructions like coequalizers and localizations.

The Gist

Imagine you are an architect trying to build a massive, complex city called Cat (the world of all small categories). In this city, the buildings are "objects," the roads are "morphisms" (arrows), and the traffic rules are "composition" (how you combine roads).

For a long time, mathematicians knew this city was "cocomplete." In plain English, this means: No matter how many buildings or roads you want to add, or how you want to glue them together, you can always build the resulting structure without the city collapsing.

However, the blueprints for how to build these structures were either too complicated (like using a sledgehammer to crack a nut) or relied on circular logic (saying "we can build it because we can build it").

This paper, by Varinderjit Mann, offers a fresh, simpler, and more direct way to prove that the city of Cat is stable and can be built in any way you like. Here is the story of how they did it, using some creative analogies.

1. The Problem: The Circular Trap

The author points out a tricky loop in previous math papers. To prove the city of Cat is stable, people usually tried to prove that the "Nerve" functor (a way of translating a city into a different language called sSet, or "Simplicial Sets") has a "left adjoint" (a reverse translator).

But here's the catch: To prove the reverse translator exists, you usually need to know the city is already stable. To prove the city is stable, you need the reverse translator. It's a "chicken and egg" problem.

2. The Solution: The "Spine" and the "Skeleton"

The author's breakthrough is realizing we don't need to build the whole city from scratch every time. We can build it using a specific, simpler method involving Weighted Colimits.

Think of a Weighted Colimit as a "recipe" for gluing things together.

• The Recipe: "Take these shapes, glue them here, and squash these parts together."
• The Weight: The instructions on how to glue them.

The paper argues that if we can just prove we can follow a specific type of recipe (one based on the "spine" of a shape), we can build the whole city.

The Analogy of the Spine

Imagine a 3D model of a human body.

• The Full Model: The complete, detailed body (the full category).
• The Spine: Just the backbone.

In the world of simplicial sets (the "sSet" language), any complex shape can be built by gluing together simple triangles. The author discovered that to build a category, you only really need to pay attention to the "Spine" of these shapes.

• The spine is just a chain of points connected by lines (like a string of beads).
• If you can glue these beads together correctly, the rest of the "flesh" (the higher-dimensional details) automatically falls into place because of the rules of the city.

3. The Construction Process: Building the City

The paper constructs the city in three distinct phases, like building a house:

Phase 1: The Foundation (0-Skeleton)
First, we just lay out the land. We take a list of points (objects) and place them on the ground. This is easy; it's just a pile of disconnected dots.

Phase 2: The Framework (1-Skeleton)
Next, we start building roads between the dots. We take all the "non-degenerate" 1-simplices (the unique roads that aren't just loops or duplicates) and lay them down.

• The Magic: We build a Free Category. This is like building a city where you can drive from A to B, and if there's a road from B to C, you can drive A→B→C. We haven't added any traffic rules yet; we just have all possible paths.

Phase 3: The Traffic Rules (2-Skeleton)
Now we have a city with infinite paths, but it's chaotic. We need to enforce the rules.

• In the "sSet" language, a 2-simplex is a triangle. It represents a rule: "If you go A→B and then B→C, it is the same as going A→C directly."
• The author shows that we can take our chaotic "Free City" and glue the triangles together. We squash the paths that are supposed to be equal.
• This step is the "Homotopy Category" construction. It's like taking a tangled ball of yarn (the free paths) and smoothing it out into a neat, organized map where equivalent routes are identified.

4. The Big Reveal: The "Reflective" Mirror

Once the author proved that this 3-step construction (Points → Roads → Rules) always works for any shape, they solved the circular problem.

They showed that the Nerve Functor (translating City to Map) and the Realization Functor (translating Map back to City) are perfect partners.

• Nerve: Takes a city and turns it into a blueprint (a simplicial set).
• Realization: Takes a blueprint and builds the city using the 3-step method above.

Because this construction works for any blueprint, the city of Cat is guaranteed to be cocomplete. You can glue any two cities together, or squash any two roads, and the result will always be a valid city.

5. Why This Matters (The "So What?")

Before this paper, if you wanted to build a specific type of city (like a "Coequalizer," which is a way of merging two cities that look similar), you had to use incredibly complex, messy math that was hard to follow.

This paper says: "Don't worry about the messy details. Just follow the spine."

• It provides a clear, step-by-step recipe for building these complex structures.
• It proves that the "Homotopy Category" (the process of smoothing out the city) is a real, well-defined machine.
• It opens the door to easier proofs for other complex mathematical structures, like "Localizations" (which is like taking a city and declaring certain one-way streets to be two-way).

Summary

Think of this paper as a master architect who, instead of trying to prove a skyscraper is stable by analyzing every single brick, realized: "If we just make sure the steel beams (the spines) are connected correctly, the whole building will stand up."

They showed that by focusing on the simple "spine" of shapes, we can construct the entire universe of categories in a way that is logical, non-circular, and surprisingly elegant.

Technical Summary

Here is a detailed technical summary of the paper "Revisiting colimits in Cat and homotopy category" by Varinderjit Mann.

1. Problem Statement

The paper addresses a foundational gap in the literature regarding the cocompleteness of the category of small categories ($\mathbf{Cat}$).

• Existing Approaches: Standard proofs of cocompleteness often rely on the fact that the forgetful functor $U: \mathbf{Cat} \to \mathbf{Graph}$ is finitary monadic. However, these proofs are often viewed as inefficient for the specific case of $\mathbf{Cat}$ due to their reliance on general enriched category theory ($\mathbf{V}$-categories).
• Elementary but Complex: Other elementary approaches (e.g., using generalized congruences) provide explicit constructions for coequalizers but are described as "fairly intricate" and difficult to follow.
• The Circular Argument: A common heuristic is that the nerve functor $N: \mathbf{Cat} \to \mathbf{sSet}$ is a reflective embedding (having a left adjoint $h$, the homotopy category functor). However, proving that $N$ is reflective typically assumes the cocompleteness of $\mathbf{Cat}$ to construct the left adjoint. The author notes a lack of direct, self-contained proofs that establish the existence of this adjunction without circular reasoning.

Goal: To provide a direct, elementary, and self-contained proof of the cocompleteness of $\mathbf{Cat}$ by constructing the left adjoint to the nerve functor without assuming the existence of all coequalizers a priori.

2. Methodology

The author employs a strategy centered on weighted colimits and the nerve-realization adjunction between $\mathbf{Cat}$ and simplicial sets ($\mathbf{sSet}$).

• Reduction to Weighted Colimits: The paper establishes that the cocompleteness of $\mathbf{Cat}$ is equivalent to the existence of a specific class of weighted colimits of the form $X \star [-]$, where $X \in \mathbf{sSet}$ is a simplicial set and $[-]: \Delta \hookrightarrow \mathbf{Cat}$ is the inclusion of the simplex category.
• The Nerve-Realization Pair:
  • The Nerve $N: \mathbf{Cat} \to \mathbf{sSet}$ is defined as the right Kan extension of the inclusion $[-]: \Delta \to \mathbf{Cat}$ along the Yoneda embedding.
  • The Realization (Homotopy Category) $h: \mathbf{sSet} \to \mathbf{Cat}$ is the left Kan extension.
  • The existence of $h$ as a left adjoint to $N$ is contingent on the existence of the weighted colimits $X \star [-]$.
• 2-Dimensional Reduction: A crucial technical insight is that the nerve of a category is 2-coskeletal. This means the nerve functor is determined entirely by its behavior on simplices up to dimension 2. Consequently, the existence of $X \star [-]$ for any simplicial set $X$ reduces to the existence of $sk_2(X) \star [-]$, where $sk_2$ is the 2-skeleton functor.
• Skeletal Filtration: The author constructs these colimits by filtering simplicial sets into their skeletal components ($sk_0, sk_1, sk_2$). Since weighted colimits are cocontinuous, the existence of $X \star [-]$ is reduced to constructing colimits for:
  1. Discrete sets ($sk_0$).
  2. Free categories generated by graphs ($sk_1$).
  3. Quotients imposing composition relations ($sk_2$).

3. Key Contributions and Results

A. Equivalence of Cocompleteness and Weighted Colimits

The paper proves the following proposition:

$\mathbf{Cat}$ is cocomplete if and only if for every simplicial set $X$, the weighted colimit $X \star [-]$ exists.

This reduces the problem of constructing arbitrary colimits in $\mathbf{Cat}$ to constructing specific weighted colimits in $\mathbf{sSet}$.

B. Explicit Construction of the Left Adjoint ($h$)

The author explicitly constructs the left adjoint $h: \mathbf{sSet} \to \mathbf{Cat}$ (the homotopy category functor) without circularity. The construction proceeds in two main steps (Lemmas 4.1.3 and 4.1.4):

1. Free Category Construction ($F_X$): For a simplicial set $X$, the colimit $sk_1(X) \star [-]$ is constructed as the free category generated by the 0-simplices (objects) and non-degenerate 1-simplices (arrows) of $X$.
2. Quotient Construction ($h_X$): The colimit $sk_2(X) \star [-]$ is constructed by taking $F_X$ and quotienting by an equivalence relation generated by the non-degenerate 2-simplices. Specifically, if a 2-simplex has boundary $(f, g, h)$, the relation $g \circ f \sim h$ is imposed.
   • This yields the category $h_X$ where morphisms are zig-zags of arrows in $X$ modulo the relations imposed by 2-simplices.

C. Proof of Reflective Embedding

Using the explicit construction of $h$, the paper demonstrates:

• $h$ is a left adjoint to the nerve functor $N$.
• $N$ is fully faithful (Corollary 3.1.9).
• Therefore, $N$ is a reflective embedding.
• By standard category theory results, the existence of a reflective embedding from a complete and cocomplete category ($\mathbf{sSet}$) into $\mathbf{Cat}$ implies that $\mathbf{Cat}$ is both complete and cocomplete.

D. Explicit Formulas for Coequalizers and Localization

The paper derives concrete descriptions for complex constructions in $\mathbf{Cat}$:

• Coequalizers: The coequalizer of two functors $F, G: C \to D$ is described as the homotopy category of the levelwise coequalizer of their nerves. The morphisms are zig-zags in the coequalizer of the underlying graphs, subject to relations derived from 2-simplices in the nerve of $D$.
• Localization: The total localization (inverting all morphisms) and relative localization (inverting a specific subset of morphisms) are constructed as pushouts in $\mathbf{Cat}$. The paper provides a concrete description of the resulting morphisms as zig-zags where inverted arrows can be traversed backward.

4. Significance

• Resolution of Circular Reasoning: The paper provides the first direct, self-contained proof that the nerve functor is reflective, removing the need to assume the cocompleteness of $\mathbf{Cat}$ to prove it.
• Elementary Approach: It offers a more accessible alternative to the monadicity approach or the intricate congruence-based constructions found in standard literature. By leveraging the 2-dimensional nature of categories, it simplifies the construction of colimits.
• Computational Utility: The explicit construction of $h$ (via pushouts of free categories and quotients) provides a practical algorithm for computing colimits in $\mathbf{Cat}$, coequalizers, and localizations, which are often difficult to handle abstractly.
• Structural Insight: The work highlights the deep connection between the 2-coskeletal nature of categories and the existence of colimits, emphasizing that the "shape" of colimits in $\mathbf{Cat}$ is fundamentally determined by 2-simplices.

In summary, Mann's paper successfully bridges the gap between the abstract theory of weighted colimits and the concrete construction of limits/colimits in $\mathbf{Cat}$, providing a rigorous, non-circular foundation for the cocompleteness of the category of categories.