Cores and localizations of $(\infty,\infty)$-categories

This paper investigates $(\infty,\infty)$-categories by comparing the $(\infty,1)$-categories obtained via core and localization functors in the limit $d\to\infty$, demonstrating that the latter is a reflective localization of the former while also exploring intermediate localizations arising from coinductive notions of invertibility.

The Gist

Here is an explanation of the paper "Cores and Localizations of (∞, ∞)-Categories" using everyday language, analogies, and metaphors.

The Big Picture: The Infinite Ladder

Imagine you are building a ladder that goes up forever.

• Level 0 is just a collection of points (like a cloud of dust).
• Level 1 adds arrows connecting those points (like a road map).
• Level 2 adds arrows between the arrows (like traffic rules or "why" we are taking that road).
• Level 3 adds arrows between the traffic rules, and so on, forever.

In mathematics, this infinite structure is called an $(\infty, \infty)$-category. It's a way of organizing information where everything is connected, and those connections can be connected, and so on, infinitely.

The problem is: How do you handle infinity? When you have an infinite ladder, do you care about every single rung, or do you just care about the fact that you can climb up?

The authors of this paper explore two different ways to "flatten" this infinite ladder into something manageable. They call these two approaches The Core and The Localization.

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1. The Two Philosophies: The Librarian vs. The Optimist

The paper compares two ways of looking at this infinite structure. Think of them as two different ways of editing a massive, chaotic encyclopedia.

The "Core" Approach (The Strict Librarian)

Imagine a librarian who is very strict about what counts as "real."

• The Rule: "If you can't go back and forth perfectly, it doesn't count."
• The Action: The librarian throws away every arrow that isn't perfectly reversible. If you can go from A to B, but you can't get back to A with a perfect "undo" button, that arrow is deleted.
• The Result: You are left with a very clean, rigid structure where everything is an "equivalence." This is the Core. It's safe, but it might throw away interesting, messy details.

The "Localization" Approach (The Optimist)

Imagine an optimist who believes that if you try hard enough, you can make anything reversible.

• The Rule: "If you can't go back perfectly, we'll just pretend you can by adding a 'magic undo' button."
• The Action: Instead of deleting the one-way arrows, the optimist invents a new arrow that goes the other way and says, "Okay, now it's reversible!" They force the structure to behave as if everything is an equivalence.
• The Result: You get a structure where everything is connected, but it might be "squashed" or simplified. This is the Localization.

2. The Main Discovery: One is a Mirror of the Other

The authors ask: "Are these two approaches the same?"

The Answer: No, but they are related in a very specific way.

Think of the Core (the strict librarian) as a high-resolution, detailed photograph.
Think of the Localization (the optimist) as a blurry, simplified sketch of that photo.

The paper proves that the Localization is actually a "reflection" of the Core.

• If you take the detailed photo (Core) and look at it in a funhouse mirror (Localization), you get the sketch.
• However, if you take the sketch and try to turn it back into the photo, you can't recover all the lost details. The mirror is "faithful" (it doesn't lie), but the photo is "faithful" in a deeper, more complex way.

The "Weakly Surjective" Concept:
The authors introduce a fancy term called "weakly $\infty$-surjective."

• Imagine you are trying to cover a floor with tiles (arrows).
• Surjective: You cover every inch of the floor perfectly.
• Weakly Surjective: You cover the floor, but you might leave tiny gaps that are so small you can't see them unless you look with a microscope (or in this case, an infinite microscope).
• The paper proves that the "Optimist's" view (Localization) is exactly what you get when you take the "Librarian's" view (Core) and fill in those tiny, invisible gaps.

3. The "Coinductive" Twist: The Infinite Tower

The paper gets even more interesting when it talks about Coinductive Isomorphisms.

Imagine a game of "Telephone" that goes on forever.

• Person A says, "I can go from X to Y."
• Person B says, "I can go from Y to X."
• Person C says, "I can go from X to Y, but I need a proof that Person B can go back."
• Person D says, "I can prove Person B can go back, but I need a proof that Person C can go forward..."

This goes on forever. This is a Coinductive Isomorphism. It's a promise that "I can eventually get back to you," even if the chain of promises is infinite.

The authors found that:

1. The Core approach is very good at handling these infinite promises. It keeps them.
2. The Localization approach is stricter. It demands that these infinite promises actually resolve into a real, solid connection. If the promise chain is too long or weird, the Localization says, "No, that's not a real connection," and collapses it.

4. The "Spans" and "Cobordisms" Examples

To prove their point, the authors looked at two famous mathematical objects:

• Spans: Imagine a bridge connecting two islands. In the "Core" view, the bridge is just a bridge. In the "Localization" view, because you can cross it and come back, the bridge effectively disappears, and the two islands become the same place.
• Cobordisms: Imagine shapes morphing into other shapes (like a circle turning into a square). In the "Core," these morphisms are distinct and interesting. In the "Localization," because you can morph back and forth, the whole system collapses into a single, simple point.

This shows that the Localization is a "trivializer." It takes complex, interesting structures and turns them into boring, simple ones. The Core preserves the complexity.

Summary: The Takeaway

• The Problem: How do we mathematically handle structures with infinite layers of arrows?
• The Solution: We have two main tools: Cores (keep only the reversible parts) and Localizations (force everything to be reversible).
• The Relationship: The Core is the "true" version. The Localization is a simplified, "reflected" version of the Core.
• The Insight: The Localization is a "reflective localization" of the Core. This means the Core contains all the information, and the Localization is a specific way of filtering that information that loses some details but keeps the essential structure.
• The "Coinductive" Surprise: There is a middle ground called "Coinductive Completeness." It's a state where infinite promises are accepted if they make logical sense. The authors show that the "Localization" is actually a subset of this middle ground—it's the most strict version of it.

In one sentence: The paper maps out the relationship between a "strict" view of infinite mathematical structures and a "forced" view, proving that the strict view contains the forced view like a detailed map contains a simplified sketch, and identifying exactly which details get lost in the translation.

Technical Summary

This is a detailed technical summary of the paper "Cores and Localizations of $(\infty, \infty)$-Categories" by Viktoriya Ozornova, Martina Rovelli, and Tashi Walde.

1. Problem Statement

The central problem addressed is the rigorous definition and comparison of $(\infty, \infty)$-categories. While $(\infty, d)$-categories are well-understood for finite $d$ (where arrows of dimension $>d$ are invertible), the limit $d \to \infty$ presents a conundrum:

• The Inductive Loop: Defining an $(\infty, \infty)$-category requires defining invertibility for arrows of all dimensions. However, invertibility itself is defined inductively (an arrow is invertible if it has an inverse up to higher invertible arrows). In the infinite limit, this leads to a self-referential definition where the notion of "sameness" (isomorphism) becomes ambiguous.
• Two Approaches: There are two natural, universal ways to construct the $(\infty, 1)$-category of $(\infty, \infty)$-categories, denoted as $\text{Cat}_{\infty}$:
  1. The Core Limit ($\text{Cat}^R_{\infty}$): Taking the limit of the sequence of $(\infty, d)$-categories via core functors ($R_d$), which remove non-invertible arrows of dimension $d+1$. This yields "right" $\infty$-categories.
  2. The Localization Limit ($\text{Cat}^L_{\infty}$): Taking the limit via localization functors ($L_d$), which formally invert all arrows of dimension $d+1$. This yields "left" $\infty$-categories.

The authors investigate the relationship between these two resulting categories. It is known that $L$ is not fully faithful (e.g., it collapses the category of spans to a point), while $R$ is fully faithful. The paper asks: What is the precise relationship between $\text{Cat}^R_{\infty}$ and $\text{Cat}^L_{\infty}$, and what specific maps are inverted in the transition?

2. Methodology

The authors employ a combination of abstract axiomatic category theory and concrete model constructions:

• Biinductive Systems: They abstract the structure of the tower of categories $\text{Cat}_0 \subset \text{Cat}_1 \subset \dots$ into a biinductive system. This system consists of a sequence of full subcategories with left adjoints ($L_d$) and right adjoints ($R_d$) to the inclusions.
• Bias and Surjectivity: They introduce the concept of a bias on a biinductive system. This is a sequence of wide subcategories $S_n$ (representing $n$-surjective maps) that satisfy specific axioms regarding composition, cancellation, and interaction with $L_d$ and $R_d$.
  • $n$-Surjectivity: A map is $n$-surjective if it is surjective on objects and induces $(n-1)$-surjective maps on hom-categories.
  • Weak $\infty$-Surjectivity: A new notion defined specifically for the limit, where a map is weakly $\infty$-surjective if its components satisfy specific surjectivity conditions relative to the localization functors.
• Reflective Localization: They prove that the transition from the core-limit to the localization-limit is a reflective localization. This involves showing that the right adjoint $R: \text{Cat}^L_{\infty} \to \text{Cat}^R_{\infty}$ is fully faithful and identifying the class of maps inverted by the left adjoint $L$.
• Internal vs. External Invertibility: In Section 5, they shift from the external properties of the categories (cores/localizations) to internal notions of invertibility within an $\infty$-category. They define:
  • Coinductive Isomorphisms: Arrows that admit an infinite tower of "inverse" data (coinductive definition).
  • $\omega$-Isomorphisms: Arrows that admit finite towers of inverses of arbitrary height but not necessarily a single consistent infinite tower.
  • Completeness: A category is complete if its only coinductive (or $\omega$) isomorphisms are actual isomorphisms.

3. Key Contributions and Results

A. The Main Theorem (The Relationship between Limits)

The paper establishes that $\text{Cat}^L_{\infty}$ is a reflective localization of $\text{Cat}^R_{\infty}$.

• Theorem 4.25: There is an adjunction $L: \text{Cat}^R_{\infty} \rightleftarrows \text{Cat}^L_{\infty} : R$ where $R$ is fully faithful.
• Inverted Maps: The functor $L$ inverts precisely the weakly $\infty$-surjective maps.
• Significance: This clarifies that the "left" $(\infty, \infty)$-categories are a specific subcategory of the "right" ones, obtained by forcing a specific class of maps (weakly $\infty$-surjective) to become equivalences.

B. Coinductive Completeness and Localizations

The authors analyze the subcategories of $\text{Cat}^R_{\infty}$ defined by internal invertibility:

• Coinductive Completion ($L_{\text{coind}}$): They construct a reflective localization of $\text{Cat}^R_{\infty}$ onto the subcategory of coinductively complete $\infty$-categories (where all coinductive isomorphisms are actual isomorphisms).
• Relation to Main Theorem: The localization $L: \text{Cat}^R_{\infty} \to \text{Cat}^L_{\infty}$ factors through the coinductive completion. That is, $\text{Cat}^L_{\infty}$ is a further localization of the coinductively complete categories.
• Characterization: $L$-equivalences are characterized as maps that are $\infty$-surjective up to $n$-isomorphisms for every finite $n$ (Proposition 5.53).

C. $\omega$-Completeness and the Henry-Loubaton Counterexample

• $\omega$-Completeness: The authors define $\omega$-complete categories (where $\omega$-isomorphisms are invertible).
• Strict Hierarchy: They prove that $\text{Cat}^L_{\infty} \subseteq \text{Cat}^{\omega}_{\infty} \subsetneq \text{Cat}^{\text{coind}}_{\infty}$.
• Counterexample: Using a construction by Henry and Loubaton (the "walking $\omega$-isomorphism" $E(\omega)$), they show that there exist coinductively complete categories that are not $\omega$-complete, and thus do not lie in the image of $R$. This proves that the inclusion $\text{Cat}^L_{\infty} \hookrightarrow \text{Cat}^R_{\infty}$ is not an equivalence even after restricting to coinductively complete objects.

D. Explicit Constructions

• Walking Coinductive Isomorphism: They provide an explicit cell-complex construction ($E$) of a "walking coinductive isomorphism," which serves as the generator for the coinductive completion.
• Non-Uniqueness: They demonstrate that coinductive invertibility data is not unique (Example 5.41), highlighting the complexity of the infinite tower of inverses.

4. Significance

• Resolution of Ambiguity: The paper resolves the ambiguity of what an $(\infty, \infty)$-category is by rigorously distinguishing between the "right" (core-based) and "left" (localization-based) definitions and mapping the precise gap between them.
• New Invariants: The introduction of weak $\infty$-surjectivity and the hierarchy of $n$-isomorphisms vs. coinductive isomorphisms provides new tools for analyzing infinite-dimensional categorical structures.
• Connection to Homotopy Theory: The work connects abstract categorical limits to concrete homotopical properties (surjectivity, connectivity) and relates to the theory of spectra and cobordism (e.g., showing the cobordism category collapses under $L$).
• Open Questions: The paper leaves open the question of whether the inclusion $\text{Cat}^L_{\infty} \subseteq \text{Cat}^{\omega}_{\infty}$ is proper, and whether the localization $L$ can be generated solely by maps of the form $\Sigma^n(C \to *)$.

Summary of the Hierarchy

The paper establishes the following hierarchy of full subcategories within the universe of right $\infty$-categories ($\text{Cat}^R_{\infty}$):
$$\text{Cat}^L_{\infty} \subseteq \text{Cat}^{\omega}_{\infty} \subsetneq \text{Cat}^{\text{coind}}_{\infty} \subsetneq \text{Cat}^R_{\infty}$$

• $\text{Cat}^L_{\infty}$: The "left" limit (localization at weakly $\infty$-surjective maps).
• $\text{Cat}^{\omega}_{\infty}$: Categories where $\omega$-isomorphisms are invertible.
• $\text{Cat}^{\text{coind}}_{\infty}$: Categories where coinductive isomorphisms are invertible (localization at $\infty$-surjective maps).
• $\text{Cat}^R_{\infty}$: The "right" limit (core limit).

The main result is that $\text{Cat}^L_{\infty}$ is obtained from $\text{Cat}^R_{\infty}$ by inverting weakly $\infty$-surjective maps, and this process passes through the coinductive completion.