A criterion for existence of right-induced model structures

This paper establishes a succinct sufficient condition for the existence of right-induced model structures on a category $\mathcal{N}$ when the functor $F: \mathcal{N} \to \mathcal{M}$ admits both adjoints, demonstrating its applicability through examples like change-of-rings and anti-involutive structures on $(\infty, 1)$-categories where base Quillen equivalences are shown to lift.

The Gist

Imagine you are an architect trying to build a new, complex city (let's call it City N) based on the blueprints of an existing, well-established city (City M).

City M is a "Model City." It has strict rules about what counts as a "good road" (weak equivalences), what counts as a "strong foundation" (cofibrations), and what counts as a "safe building" (fibrations). These rules form what mathematicians call a Model Structure.

Your goal is to build City N so that it inherits these rules from City M. You want a road in City N to be considered "good" if and only if the corresponding road in City M is "good."

The Problem: The One-Way Street

Usually, you have a map (a functor) that takes you from City N to City M. If this map is just a one-way street (a right adjoint), you can sometimes copy the rules over. But often, the rules get messy, or the construction fails because the map doesn't have enough "handles" to grab onto the structure of City M.

The Solution: The Three-Handed Architect

This paper introduces a clever trick. The authors say: "What if your map isn't just a one-way street, but a three-way intersection?"

Imagine your map, F, is connected to two other maps:

1. L (The Left Hand): A map that goes from M back to N.
2. R (The Right Hand): Another map that goes from M back to N.

So, you have a chain: L ↔ F ↔ R.

The paper's main discovery (Theorem 2.3) is a simple test: If you combine the Left Hand and the Right Hand (L then F, or F then R) and they play nicely with the rules of City M, then you can successfully build City N with the exact same rules as City M.

Think of it like a sandwich. If the bread (L and R) holds the filling (F) together perfectly, the whole sandwich (the new model structure) holds its shape.

Why is this useful? (The "Real-World" Examples)

The authors show that this trick works for many weird and wonderful mathematical cities. Here are a few analogies:

1. The Mirror City (Anti-Involution)
Imagine a city where every street has a mirror image. If you walk down Main Street, there's a "Reverse Main Street" that looks exactly the same but flipped.

• The Math: This is called an "anti-involution."
• The Result: The authors show that if you have a model for "normal" cities (like Simplicial Sets, which model shapes and spaces), you can automatically build a model for "Mirror Cities." This is huge for understanding spaces that have a built-in symmetry or reversal, like time-reversal in physics or duality in algebra.

2. The Group Party (Group Actions)
Imagine you have a city, and a group of friends (a group $G$) comes in and starts rearranging the buildings.

• The Math: This is a "semidirect product."
• The Result: The paper gives a recipe to take the rules of the original city and apply them to the "Party City" where the buildings are being shuffled around by the group. This helps mathematicians study shapes that have symmetries (like a snowflake or a molecule).

3. The Ring Exchange (Change of Rings)
In algebra, you can change the "numbers" you use to build things (e.g., switching from integers to fractions).

• The Math: This is "change of rings."
• The Result: The paper proves that if you have a good set of rules for building with Integers, you can automatically get a good set of rules for building with Fractions, provided the switch is done in a specific, well-behaved way.

The Big Payoff: Connecting Two Worlds

The most exciting part of the paper is in the final section.

Mathematicians have two different ways of modeling "Infinity Categories" (which are like cities where you can travel between points in infinitely many ways).

• Model A: Uses "Simplicial Sets" (like building with triangles).
• Model B: Uses "Simplicial Categories" (like building with labeled roads).

For a long time, we knew these two models were equivalent (they describe the same underlying reality). But what if we want to study "Mirror Infinity Categories" (where the whole city is flipped)?

The authors use their "Three-Handed" trick to prove that the equivalence between Model A and Model B still holds even when you add the mirror symmetry.

The Takeaway

This paper is like a universal adapter. It says: "If you have a complex mathematical structure and you want to add a layer of symmetry (like a mirror, a group of friends, or a ring change), you don't need to reinvent the wheel. Just check if your 'Left Hand' and 'Right Hand' maps play nice with the original rules. If they do, the new, symmetrical structure exists automatically."

It turns a difficult, case-by-case construction problem into a simple checklist, allowing mathematicians to quickly build models for complex, symmetric worlds that were previously hard to define.

Technical Summary

Here is a detailed technical summary of the paper "A criterion for existence of right-induced model structures" by Gabriel C. Drummond-Cole and Philip Hackney.

1. Problem Statement

The central problem addressed in this paper is the construction of right-induced model category structures. Given a functor $F: \mathcal{N} \to \mathcal{M}$ where $\mathcal{M}$ is a known Quillen model category, the authors seek conditions under which $\mathcal{N}$ can be endowed with a model structure such that:

• A morphism $f$ in $\mathcal{N}$ is a weak equivalence (resp. fibration) if and only if $F(f)$ is a weak equivalence (resp. fibration) in $\mathcal{M}$.
• The functor $F$ "creates" these structures.

While classical theorems exist for cases where $F$ is a right adjoint and $\mathcal{M}$ is cofibrantly generated, they often require specific acyclicity conditions that are difficult to verify in practice. The authors aim to provide a succinct, sufficient condition that simplifies the verification process, particularly in scenarios where $F$ admits both a left and a right adjoint.

2. Methodology

The authors develop a theoretical framework based on adjoint strings $(L, F, R)$, where $L \dashv F \dashv R$. Their methodology proceeds as follows:

• Refinement of Existence Criteria: They revisit the standard criteria for right-induced model structures (e.g., Hirschhorn's Theorem 11.3.2). They establish that if $F$ has both adjoints, the existence of the induced structure relies heavily on the behavior of the composite endofunctors $FL$ and $FR$.
• The Core Criterion (Theorem 2.3): The authors prove that if $(FL, FR)$ forms a Quillen adjunction on the base category $\mathcal{M}$, then a right-induced cofibrantly generated model structure exists on $\mathcal{N}$.
  • Specifically, if $FL$ is a left Quillen functor (preserves cofibrations and acyclic cofibrations) and $FR$ is a right Quillen functor (preserves fibrations and acyclic fibrations), the induced structure is guaranteed.
• Smallness and Factorization: They utilize the "small object argument" to ensure the existence of factorizations in $\mathcal{N}$. They prove that if objects in the base category are small, objects in the functor category (or related categories) remain small, satisfying the necessary conditions for the small object argument.
• Lifting Quillen Equivalences: A significant methodological contribution is Theorem 5.6, which provides conditions under which a Quillen equivalence between base categories $\mathcal{M}$ and $\mathcal{M}'$ lifts to a Quillen equivalence between the induced categories $\mathcal{N}$ and $\mathcal{N}'$.

3. Key Contributions and Results

A. Theoretical Framework

• Theorem 2.3: The primary result. It states that for a functor $F: \mathcal{N} \to \mathcal{M}$ with adjoints $L$ and $R$, if $(FL, FR)$ is a Quillen adjunction on $\mathcal{M}$, then $\mathcal{N}$ admits a right-induced model structure. This structure is cofibrantly generated, and both $(L, F)$ and $(F, R)$ become Quillen adjunctions.
• Proposition 2.4: If the base model category $\mathcal{M}$ is left proper, the induced model structure on $\mathcal{N}$ is also left proper.
• Lemma 3.1 & Corollary 3.2: They establish that in functor categories $K^C$ where $K$ is cocomplete and objects are small, all objects are small. This allows the application of their criteria to diagram categories and presheaves.

B. Applications to Specific Structures

The authors apply their criterion to several diverse examples, recovering known results and establishing new ones:

1. Groupoids and Categories: They show the canonical model structure on groupoids is right-induced from the canonical structure on categories ($Cat$).
2. Anti-involutive Structures:
   • Categories with Anti-involution ($iCat$): They construct a model structure on categories equipped with a strict anti-involution $\tau: X^{op} \to X$.
   • Simplicial Categories with Anti-involution ($isCat$): They lift the Bergner model structure (modeling $(\infty, 1)$-categories) to the category of simplicial categories with anti-involution.
   • Real Simplicial Sets: They analyze the category $\nabla = \Delta \rtimes C_2$ (where $C_2$ acts by reversing order). They show the Joyal model structure on simplicial sets lifts to a model structure on $\nabla$-presheaves (real simplicial sets).
3. Chain Complexes: They demonstrate how projective and injective model structures on chain complexes lift along ring homomorphisms under specific flatness/projectivity conditions.
4. Operads: They apply the theory to cyclic operads and modular operads, showing how model structures lift from operads to cyclic operads and from cyclic to modular operads under specific conditions.

C. Lifting Quillen Equivalences

• Theorem 5.5: The authors prove that the known Quillen equivalence between the Joyal model structure on simplicial sets ($sSet$) and the Bergner model structure on simplicial categories ($sCat$) lifts to a Quillen equivalence between their anti-involutive counterparts ($isSet$ and $isCat$).
• Theorem 5.6: A general theorem stating that if a Quillen equivalence exists between base categories and the relevant adjunctions lift, the induced model structures are also Quillen equivalent.
• Example 5.7: They apply this to show that the equivalence between the Joyal structure and the Complete Segal Space structure lifts to the context of $C_2$-actions (real structures).

4. Significance

• Unification of Models for $(\infty, 1)$-Categories with Duality: The paper provides a rigorous foundation for studying $(\infty, 1)$-categories equipped with anti-involutions (duality). By showing that the Joyal and Bergner models are Quillen equivalent even in the presence of anti-involution, the authors confirm that these different approaches to higher category theory with duality are consistent.
• Simplification of Verification: The criterion $(FL, FR)$ being a Quillen adjunction is often easier to verify than the raw acyclicity conditions required by classical theorems. This makes the construction of model structures on complex categories (like those with group actions or operadic structures) more accessible.
• New Models for Real Algebraic K-Theory: The construction of the model structure on real simplicial sets ($\nabla$-presheaves) provides a robust homotopy-theoretic framework for real algebraic K-theory and related fields.
• Generalizability: The methods are shown to apply to $(\infty, n)$-categories with twisted actions of $(C_2)^n$, suggesting a broad applicability to higher-dimensional categorical structures with symmetry.

In summary, the paper provides a powerful, general tool for constructing model structures on categories with additional algebraic or symmetry constraints, and it successfully demonstrates that fundamental equivalences in higher category theory are preserved under these constraints.