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<!DOCTYPE html>
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<title>Two Conjectures on EqCat∞</title>
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<body>
<!-- Slide 1: Title -->
<div class="slide active" id="slide-1">
<h1>Two Conjectures on EqCat∞</h1>
<p class="subtitle">Open questions about equational reasoning in higher categories</p>
<div style="margin-top: 40px;">
<div class="step-box" style="max-width: 800px; margin: 15px auto;">
<strong>Conjecture 1:</strong> Weak Adequacy — Do the axioms characterize (∞,1)-categories?
</div>
<div class="step-box" style="max-width: 800px; margin: 15px auto;">
<strong>Conjecture 2:</strong> Eff Model — Does the effective topos support an internal ∞-category?
</div>
</div>
<div style="margin-top: 40px;">
<p style="font-size: 1.1em; color: #888;">
Based on "Equational Reasoning in ∞-Categories"<br>
(Anonymous, LICS '26)
</p>
</div>
</div>
<!-- Slide 2: Background -->
<div class="slide" id="slide-2">
<h2>Background: The Theory EqCat∞</h2>
<div class="content">
<p>
EqCat∞ is a compact, typed equational theory for ∞-categories.
Its language has:
</p>
<ul>
<li><span class="highlight">n-cells</span> at every dimension n ≥ 0</li>
<li><span class="highlight">src, tgt</span> — source and target operators (dimension n+1 → n)</li>
<li><span class="highlight">id</span> — identity operator (dimension n → n+1)</li>
<li><span class="highlight">ˆ<sub>k</sub></span> — composition through a shared k-dimensional boundary</li>
</ul>
<p style="margin-top: 25px;">
The axioms IC1–IC13 encode globularity, typing of composition,
associativity, exchange, unit laws, and the interaction of identity
with composition.
</p>
</div>
</div>
<!-- Slide 3: The Axioms -->
<div class="slide" id="slide-3">
<h2>The 13 Axioms (IC1–IC13)</h2>
<div class="content">
<div class="axiom-box">
<strong>Globularity:</strong> IC1–IC2 — src and tgt compose correctly across dimensions<br><br>
<strong>Identity typing:</strong> IC3–IC4 — src(id a) = a, tgt(id a) = a<br><br>
<strong>Composition typing:</strong> IC5–IC8 — src and tgt of compositions<br><br>
<strong>Associativity:</strong> IC9 — (z ˆ<sub>k</sub> y) ˆ<sub>k</sub> x = z ˆ<sub>k</sub> (y ˆ<sub>k</sub> x)<br><br>
<strong>Exchange:</strong> IC10 — two composition operators at different levels interchange<br><br>
<strong>Unit laws:</strong> IC11–IC12 — composing with an identity gives back the original cell<br><br>
<strong>Id & composition:</strong> IC13 — id(y ˆ<sub>k</sub> x) = id y ˆ<sub>k+1</sub> id x
</div>
<p class="small-note">These 13 axioms are the entire theory. Everything else must follow from them.</p>
</div>
</div>
<!-- Slide 4: Strong Adequacy -->
<div class="slide" id="slide-4">
<h2>What Is Already Proved</h2>
<div class="content">
<div class="conjecture-box">
<h3>Theorem 6.1 — Strong Adequacy</h3>
<p>Under the <span class="highlight">strong reading</span>, where every equation is a literal identity,
the models of EqCat∞ are <span class="green">exactly</span> the strict ω-categories.</p>
<p style="margin-top: 10px;">Both directions are proved. The axioms are complete.</p>
</div>
<p>
Additionally, the paper constructs a <span class="green">homotopy model</span> (Section 7)
and proves <span class="green">soundness</span> (Section 8): (∞,1)-categories
satisfy the axioms under the weak reading.
</p>
<p style="margin-top: 15px;">
<span class="highlight">What is missing:</span> the converse for the weak reading.
Does satisfying the axioms weakly <em>suffice</em> to be an (∞,1)-category?
</p>
</div>
</div>
<!-- Slide 5: The Weak Reading -->
<div class="slide" id="slide-5">
<h2>The Weak Reading</h2>
<div class="content">
<p>In the <span class="highlight">strong reading</span>, an equation a = b between n-cells means:</p>
<div class="step-box">a and b are the same n-cell. Literally identical.</div>
<p>In the <span class="highlight">weak reading</span>, an equation a = b between n-cells means:</p>
<div class="step-box">
There exists an <span class="green">(n+1)-dimensional cell</span> φ connecting a to b.<br>
The equation holds <em>up to a witness one dimension higher</em>.
</div>
<p style="margin-top: 20px;">
The witness φ is itself a cell, and <em>its</em> equations with other cells
are witnessed by (n+2)-cells, and so on — witnesses all the way up.
</p>
</div>
</div>
<!-- Slide 6: (∞,1)-categories -->
<div class="slide" id="slide-6">
<h2>What Is an (∞,1)-Category?</h2>
<div class="content">
<p>An <span class="highlight">(∞,1)-category</span> is an ∞-category where:</p>
<ul>
<li><strong>0-cells</strong> (objects) — no invertibility requirement</li>
<li><strong>1-cells</strong> (morphisms) — no invertibility requirement</li>
<li><strong>2-cells and above</strong> — all <span class="green">invertible</span></li>
</ul>
<p style="margin-top: 25px;">
Intuition from topology: given a space X, paths (1-cells) have a direction,
but homotopies between paths (2-cells) and all higher structure are
always reversible.
</p>
<p style="margin-top: 15px;">
Standard models include quasicategories (Joyal/Lurie), complete Segal
spaces (Rezk), and Segal categories — all known to be equivalent.
</p>
</div>
</div>
<!-- Slide 7: The Conjecture -->
<div class="slide" id="slide-7">
<h2>The Weak Adequacy Conjecture</h2>
<div class="content">
<div class="conjecture-box">
<h3>Conjecture</h3>
<p>
Consider the EqCat∞ axioms augmented with the requirement that all cells
of dimension ≥ 2 are invertible. Under the weak reading — where an
equation between n-dimensional terms is witnessed by an (n+1)-dimensional
cell rather than being a strict identity — the models of this theory are
<span class="highlight">exactly</span> the (∞,1)-categories.
</p>
</div>
<p>This requires proving <strong>two directions</strong>:</p>
<div class="step-box">
<span class="step-label">Soundness:</span> Every (∞,1)-category satisfies the axioms.
<span class="green">(Largely done — Sections 7 & 8 of the paper.)</span>
</div>
<div class="step-box">
<span class="step-label">Completeness:</span> Every model satisfying the axioms is an (∞,1)-category.
<span class="highlight">(Open.)</span>
</div>
</div>
</div>
<!-- Slide 8: The Coherence Tower -->
<div class="slide" id="slide-8">
<h2>The Central Difficulty: The Coherence Tower</h2>
<div class="content">
<p>An (∞,1)-category requires an infinite tower of coherence data:</p>
<table class="tower-table">
<tr><th>Level</th><th>What lives here</th><th>Example</th></tr>
<tr>
<td>0</td>
<td>Operations</td>
<td>Composition, identity, src, tgt</td>
</tr>
<tr>
<td>1</td>
<td>Witnesses that axioms hold</td>
<td>Associator: (hˆg)ˆf ≅ hˆ(gˆf)</td>
</tr>
<tr>
<td>2</td>
<td>Coherences between witnesses</td>
<td>Pentagonator for 4-fold composition</td>
</tr>
<tr>
<td>3</td>
<td>Coherences between coherences</td>
<td>Higher associahedron data</td>
</tr>
<tr>
<td>n</td>
<td>Coherences between level-(n−1)</td>
<td>…and so on, forever</td>
</tr>
</table>
<p style="margin-top: 15px;">
IC1–IC13 give you levels 0 and 1 directly.
The conjecture claims levels 2, 3, 4, … are
<span class="highlight">generated automatically</span> by the weak reading.
</p>
</div>
</div>
<!-- Slide 9: Why the tower might be generated -->
<div class="slide" id="slide-9">
<h2>Why the Tower Might Be Automatic</h2>
<div class="content">
<p>
The key idea: the axioms apply <em>at every dimension</em>.
</p>
<p>
IC9 says composition is associative. Under the weak reading, this gives
an associator (a 2-cell). But the associator is itself a cell, and IC9
applies to <em>it</em> too, potentially generating the pentagonator (a 3-cell).
</p>
<div class="step-box">
<span class="step-label">Level 1:</span> IC9 applied to f, g, h gives associator α<br>
<span class="step-label">Level 2:</span> IC9 applied to associators gives pentagonator π<br>
<span class="step-label">Level 3:</span> IC9 applied to pentagonators gives the next coherence<br>
<span class="step-label">Level n:</span> Each level arises from applying the axioms to the previous level's witnesses
</div>
<p style="margin-top: 20px;">
If this bootstrapping works, then 13 finitely-stated axioms generate
the full infinite tower — a striking case of
<span class="green">finite presentation of infinite coherence</span>.
</p>
</div>
</div>
<!-- Slide 10: Evidence For -->
<div class="slide" id="slide-10">
<h2>Evidence For the Conjecture</h2>
<div class="content">
<ul class="evidence-for">
<li>
<strong>Strong adequacy works.</strong>
The same 13 axioms, read strictly, characterize strict ω-categories
exactly (Theorem 6.1). The axioms are the "right" axioms — they
capture all the structural laws.
</li>
<li>
<strong>The homotopy model validates soundness.</strong>
A natural class of (∞,1)-categories (boundary-preserving homotopies
on topological spaces) satisfies the axioms, with witnesses provided
by reparameterizations (Sections 7–8).
</li>
<li>
<strong>Normalization succeeds for the free model.</strong>
The decidability result (Theorem 4.6) shows the free model has clean
normal forms. Missing coherences would make the free model "too free,"
distinguishing terms that should be identified.
</li>
<li>
<strong>Precedent: Mac Lane coherence.</strong>
For monoidal categories, finitely many conditions (pentagon + triangle)
generate all coherences. IC1–IC13 could play an analogous role.
</li>
</ul>
</div>
</div>
<!-- Slide 11: Reasons for Caution -->
<div class="slide" id="slide-11">
<h2>Reasons for Caution</h2>
<div class="content">
<ul class="evidence-against">
<li>
<strong>Finite axioms, infinite tower.</strong>
It is not obvious that 13 axioms generate all higher coherences.
The pentagonator, and every level above it, must arise without
being explicitly axiomatized.
</li>
<li>
<strong>No bridge to standard models yet.</strong>
(∞,1)-categories are usually defined via horn-filling (quasicategories)
or limit conditions (Segal spaces) — very different from equational
axioms. No translation between EqCat∞ and these frameworks exists.
</li>
<li>
<strong>Simpson's conjecture.</strong>
If true (every weak ∞-category is semi-strict), it suggests the
coherence landscape is uneven — some axioms need weakening more
than others, complicating a uniform weak reading.
</li>
<li>
<strong>No algebraic comparison.</strong>
Batanin–Leinster algebraic weak ∞-categories use globular operads.
Relating EqCat∞ to these is an open problem in itself.
</li>
</ul>
</div>
</div>
<!-- Slide 12: Proof Strategy -->
<div class="slide" id="slide-12">
<h2>Possible Proof Strategy</h2>
<div class="content">
<p>A proof of the completeness direction might proceed as follows:</p>
<div class="step-box">
<span class="step-label">Step 1.</span>
Given a weak model M of EqCat∞ + invertibility, construct
from it a simplicial set (or other combinatorial object).
</div>
<div class="step-box">
<span class="step-label">Step 2.</span>
Show this simplicial set satisfies the inner horn-filling condition,
making it a quasicategory.
</div>
<div class="step-box">
<span class="step-label">Step 3.</span>
Show this construction is an equivalence: every quasicategory arises
(up to equivalence) from a weak model.
</div>
<p style="margin-top: 20px; color: #a0a0c0; font-style: italic;">
The hard part is Step 2: extracting horn-fillers from the equational
witnesses. The reparameterization mechanism (Section 7.4) suggests
how this might work in the homotopy model, but a general argument
is needed.
</p>
</div>
</div>
<!-- Slide 13: Summary of Conjecture 1 -->
<div class="slide" id="slide-13">
<h2>Summary: Weak Adequacy Conjecture</h2>
<div class="content">
<div class="two-col">
<div class="col-for">
<div class="col-header">What we know</div>
<ul>
<li>IC1–IC13 characterize strict ω-categories (strong reading)</li>
<li>(∞,1)-categories satisfy IC1–IC13 weakly (soundness)</li>
<li>Equality in the free model is decidable</li>
</ul>
</div>
<div class="col-against">
<div class="col-header">What we don't know</div>
<ul>
<li>Whether weak models are necessarily (∞,1)-categories (completeness)</li>
<li>Whether the finite axioms generate the full coherence tower</li>
<li>How to bridge equational and horn-filling formulations</li>
</ul>
</div>
</div>
<p style="text-align: center; margin-top: 40px; font-size: 1.3em; color: #e94560;">
The Weak Adequacy Conjecture proposes that the answer to all three is yes.
</p>
</div>
</div>
<!-- Slide 14: Transition to Conjecture 2 -->
<div class="slide" id="slide-14">
<h2>A Second Question: The Effective Topos</h2>
<div class="content">
<p>
The weak adequacy conjecture asks whether EqCat∞ characterizes
<span class="highlight">(∞,1)-categories in general</span>.
</p>
<p style="margin-top: 20px;">
But there's a related question: can we find models of EqCat∞ in
<span class="highlight">specific mathematical structures</span> beyond
topological spaces?
</p>
<p style="margin-top: 25px;">
One particularly interesting candidate is the <span class="green">effective topos</span> (Eff),
a fundamental construction in realizability theory and constructive mathematics.
</p>
</div>
</div>
<!-- Slide 15: What is Eff? -->
<div class="slide" id="slide-15">
<h2>What Is the Effective Topos?</h2>
<div class="content">
<p>The <span class="highlight">effective topos</span> (Eff) is a topos constructed from:</p>
<ul>
<li><strong>Realizability:</strong> Objects are sets equipped with "realizability" structure from partial recursive functions</li>
<li><strong>Constructive logic:</strong> Internal logic is intuitionistic, not classical</li>
<li><strong>Computational content:</strong> Proofs carry algorithmic information</li>
</ul>
<p style="margin-top: 25px;">
Eff is a <span class="highlight">1-topos</span> — it has objects and morphisms with
good categorical structure (limits, colimits, exponentials). But it doesn't
inherently have higher-dimensional structure.
</p>
<p style="margin-top: 15px;">
<span class="green">Key property:</span> Eff is neither Set (classical sets) nor Top
(topological spaces), yet it's a rich mathematical universe with its own
internal logic.
</p>
</div>
</div>
<!-- Slide 16: The Eff Conjecture -->
<div class="slide" id="slide-16">
<h2>Conjecture 2: Internal ∞-Category in Eff</h2>
<div class="content">
<div class="conjecture-box">
<h3>Conjecture</h3>
<p>
There exists an <span class="highlight">internal ∞-category</span> in the
effective topos Eff that validates the axioms of EqCat∞ under the weak reading.
</p>
<p style="margin-top: 15px;">
More precisely: there is a model of EqCat∞ whose underlying structure
lives internally in Eff, with cells at each dimension being objects of Eff,
and composition/identity operations being morphisms in Eff.
</p>
</div>
<p style="margin-top: 20px;">
This would show that equational ∞-categorical reasoning is compatible
with <span class="green">realizability</span> and <span class="green">constructive mathematics</span>,
not just classical topology and homotopy theory.
</p>
</div>
</div>
<!-- Slide 17: Why This Matters -->
<div class="slide" id="slide-17">
<h2>Why the Eff Conjecture Matters</h2>
<div class="content">
<ul class="evidence-for">
<li>
<strong>Bridges realizability and higher categories.</strong>
Eff embodies computational/constructive thinking. An ∞-category
in Eff would mean higher-categorical reasoning has computational content.
</li>
<li>
<strong>Tests the generality of EqCat∞.</strong>
The axioms were designed for topological models. If they work in Eff,
they're more universal than initially apparent.
</li>
<li>
<strong>Potential for mechanization.</strong>
Eff's constructive nature aligns with proof assistants. A model in Eff
could lead to computational interpretations of ∞-categorical proofs.
</li>
<li>
<strong>Connection to HoTT.</strong>
Homotopy Type Theory seeks to combine higher categories with type theory.
Realizability toposes like Eff provide constructive models of type theories,
suggesting potential bridges.
</li>
</ul>
</div>
</div>
<!-- Slide 18: Challenges for the Eff Conjecture -->
<div class="slide" id="slide-18">
<h2>Challenges</h2>
<div class="content">
<ul class="evidence-against">
<li>
<strong>Eff is a 1-topos, not ∞-topos.</strong>
There's no built-in higher structure. Would need to construct simplicial
objects or some other internal model of ∞-categories.
</li>
<li>
<strong>The weak reading assumes topological homotopy.</strong>
The paper's weak semantics uses continuous maps I<sup>n</sup> → X.
Eff doesn't have this structure — need a different notion of "witness."
</li>
<li>
<strong>No obvious construction.</strong>
Unlike topological spaces (which naturally have homotopies), there's no
clear recipe for building cells and witnesses in Eff.
</li>
<li>
<strong>Eff doesn't model HoTT.</strong>
Eff doesn't satisfy univalence. If HoTT is the "right" internal language
for ∞-categories, Eff might not have the necessary structure.
</li>
</ul>
</div>
</div>
<!-- Slide 19: Approaches to the Eff Conjecture -->
<div class="slide" id="slide-19">
<h2>Possible Approaches</h2>
<div class="content">
<p>Several strategies could be attempted:</p>
<div class="step-box">
<span class="step-label">Approach 1: Simplicial objects.</span>
Construct the ∞-category as a simplicial object internal to Eff.
Prove it satisfies Segal conditions and witnesses the EqCat∞ axioms.
</div>
<div class="step-box">
<span class="step-label">Approach 2: Algebraic weak ∞-categories.</span>
Use Batanin/Leinster's globular operads framework. Show Eff supports
the necessary operad structure.
</div>
<div class="step-box">
<span class="step-label">Approach 3: Realizability interpretation.</span>
Define cells as realizability structures. Witnesses are realized by
partial recursive functions. Composition is a computable operation.
</div>
<div class="step-box">
<span class="step-label">Approach 4: Bridge via strong reading.</span>
The free model (strong reading) is already computational. Perhaps
a realizability interpretation exists there, and the weak reading
emerges by internalizing the quotient in Eff.
</div>
</div>
</div>
<!-- Slide 20: Overall Summary -->
<div class="slide" id="slide-20">
<h2>Two Conjectures, One Framework</h2>
<div class="content">
<div class="conjecture-box" style="margin-bottom: 25px;">
<h3>Conjecture 1: Weak Adequacy</h3>
<p>
Under the weak reading, EqCat∞ + invertibility ≥ 2 characterizes
exactly the (∞,1)-categories.
</p>
<p style="margin-top: 8px; font-size: 0.95em; color: #a0a0c0;">
<em>Status:</em> Soundness largely proved. Completeness open.
</p>
</div>
<div class="conjecture-box">
<h3>Conjecture 2: Eff Model</h3>
<p>
There exists an internal ∞-category in Eff validating EqCat∞.
</p>
<p style="margin-top: 8px; font-size: 0.95em; color: #a0a0c0;">
<em>Status:</em> Completely open. No construction known.
</p>
</div>
<p style="margin-top: 30px; text-align: center; font-size: 1.2em;">
Both conjectures test the <span class="green">scope and power</span> of
EqCat∞ as a foundation for higher-categorical reasoning.
</p>
</div>
</div>
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