Model Structures on Preorders

• Model Structures on Preorders are frameworks that extend Quillen’s model categories to order-theoretic contexts by defining cofibrant, fibrant, and weak equivalence classes.
• They employ idempotent monads and comonads to construct fibrant and cofibrant replacements, establishing unique and systematic factorization systems.
• Explicit computations on Boolean algebras and applications to type spaces in model theory demonstrate the practical utility and broad impact of these structures.

A model structure on a preorder extends the classical model category framework of Quillen to the setting where the category admits at most one morphism between each pair of objects. This paradigm identifies abstract homotopical behavior within purely order-theoretic environments. Model structures on preorders enable the systematic study and classification of (co)fibrant and (co)acyclic objects, factorization systems, and weak equivalences using categorical, topological, and matroidal input data, with applications including the construction of model categories on Boolean algebras and type spaces in model theory (Salch et al., 27 Dec 2025, Baglini et al., 11 Sep 2025).

1. Foundations of Model Structures on Preorders

Let $A$ be a preorder (i.e., a small category in which every hom-set has at most one element). A model structure on $A$ comprises three distinguished classes of morphisms:

• $cof$ (cofibrations),
• $fib$ (fibrations),
• $W$ (weak equivalences),

subject to the following axioms:

• $W$ satisfies the two‐out‐of‐three property,
• $(cof \cap W, fib)$ and $(cof, fib \cap W)$ induce orthogonal factorization systems,
• both $cof$ and $fib$ are closed under retracts.

The identification of cofibrant and fibrant objects controls the morphism classifications. Denote $C\subseteq Ob(A)$ as the set of cofibrant objects and $F\subseteq Ob(A)$ as the set of fibrant objects. For $f:X\to Y$,

• $f$ is a cofibration iff $X\in C$,
• $f$ is a fibration iff $Y\in F$,
• $f$ is a weak equivalence iff there exists $Z\in C\cap F$ such that $X\rightarrow Z \leq Y$ in $A$ (factoring through a bifibrant object) (Salch et al., 27 Dec 2025).

2. Order-Theoretic Characterization

Necessary and sufficient conditions for $(C,F)$ to define a model structure (Corollary B, (Salch et al., 27 Dec 2025)) are:

1. $C$ and $F$ are replete (closed under isomorphism).
2. $F$ is reflective: for each $x\in A$, the comma-poset $x/F$ has a terminal object, denoted $F(x)$.
3. $C$ is coreflective: for each $x\in A$, $C/x$ has an initial object, denoted $C(x)$.
4. For $x\leq C(y)$ and $F(x)\leq y$, it must hold that $F(x)\leq C(y)$.
5. Choice-free reformulation: if $x\leq$ every lower bound in $C$ of $y$ and $y\geq$ every upper bound in $F$ of $x$, then some upper bound in $F$ of $x$ lies below some lower bound in $C$ of $y$.

This structure ensures that the weak equivalences are precisely those maps $f:X\to Y$ with $C(Y)\leq X$ and $F(X)\leq Y$.

3. Classification via Fibrant and Cofibrant Replacement Functors

Model structures on preorders are classified through idempotent (co)monads representing (co)fibrant replacements. In any such structure on $A$, the two factorization procedures:

• $X \to (cof\cap W)\to F(X) \to fib \to Y$,
• $X \to cof \to C(Y) \to (fib\cap W) \to Y$,

are unique up to isomorphism. Define endofunctors:

• $F(A):=$ “acyclic-cof then fib” replacement,
• $C(A):=$ “cof then acyclic-fib” replacement.

Properties:

• $F$ is an idempotent monad (fibrant replacement),
• $C$ is an idempotent comonad (cofibrant replacement),
• $C$ and $F$ commute up to isomorphism,
• Compatibility: If $X\leq C(Y)$ and $F(X)\leq Y$, then $F(X)\leq C(Y)$ (Salch et al., 27 Dec 2025).

Conversely, any compatible idempotent monad/comonad pair $(F,C)$ defines a model structure, yielding a bijection (for finite $A$) between strong model structures and isomorphism classes of such $(F,C)$ pairs (Theorem C+D).

4. Constructions from Topological and Matroidal Data

Topological Model Structures

Given a finite set $S$ and a topology $T$ (Moore family, closed under union), the closure operator $Cl_T: P(S)\to P(S)$ is an idempotent monad. The interior operator from another topology $T'$ provides an idempotent comonad $Int_{T'}$. Compatibility conditions are:

• $Cl_T Int_{T'} = Int_{T'} Cl_T$,
• If $U \subseteq Int_{T'}(V)$ and $Cl_T(U) \subseteq V$, then $Cl_T(U) \subseteq Int_{T'}(V)$,
• Rigidity: If $U\subseteq V$, $Cl_T(U)=Cl_T(V)$, $Int_{T'}(U)=Int_{T'}(V)$, then $U=V$.

When satisfied, these define a strong model structure on $P(S)$ where cofibrant objects are open sets of $T'$, fibrant objects are closed sets of $T$, and weak equivalences satisfy $Cl_T(U)\leq Int_{T'}(V)$ (Salch et al., 27 Dec 2025).

Matroidal Model Structures

A matroid $M$ on finite $S$ yields an idempotent monad via its closure operator $Cl_M$; a second matroid $M'$ provides the dual comonad via interior (complement of a flat). The analogous compatibility and rigidity conditions guarantee a strong matroidal model structure where fibrant objects are flats of $M$, cofibrant objects are complements of flats of $M'$ (Salch et al., 27 Dec 2025).

5. Explicit Computation and Bousfield Quiver Analysis for Small Boolean Algebras

Comprehensive enumeration of model structures can be achieved for small Boolean algebras $P(S)$:

• For $|S|=1$: Exactly three model structures arise—discrete, $C$-free, and $F$-free—lying on a linear quiver via Bousfield colocalization.
• For $|S|=2$: There are 10 retratile factorization systems, 23 model structures, including 17 strong, 9 topological, 11 matroidal, visualized on a planar Bousfield quiver with localizations/colocalizations denoted by directed edges.
• For $|S|=3$: There are 450 factorization systems, 1026 model structures (377 strong, 84 topological, 50 matroidal, 4 geometric), forming connected components and isolated nodes within the Bousfield quiver. Computational enumeration enables classification of topological/matroidal/geometric model structures and their localization relationships (Salch et al., 27 Dec 2025).

$n$	Model Structures	Strong	Topological	Matroidal	Geometric
1	3	3	1	1	0
2	23	17	9	11	1
3	1026	377	84	50	4

6. Extension to Type Spaces and Applications

Model structures on preorders also extend to the context of type spaces in model theory (Baglini et al., 11 Sep 2025). Given an ordered structure $(M,\le,\dots)$ and type spaces $S_k(A)$:

• The induced preorder $\precapprox$ on types is defined by the existence of realizations $\alpha\models p$, $\beta\models q$ with $\alpha\leq\beta$ in a monster model.
• For definably complete linear orders, the classification of types by cuts in $\dcl(A)$ yields an isomorphism $$(S_1(A)/\approx,\precapprox)\cong(\mathrm{CC}(A),\le)$$, providing precise control over the preorder extension from $M$ to $S_k(M)$ (Baglini et al., 11 Sep 2025).
• Applications include the analysis of divisibility orders on ultrafilters, where the preorder encodes arithmetic divisibility and admits independence results connected to CH and cofinality properties of ultrapowers.

7. Significance and Broader Connections

The systematic development of model structures on preorders opens avenues for studying categorical homotopy-theoretic phenomena within purely order-theoretic or combinatorial data. The framework unifies factorization systems, monad-comonad duality, and localization/colocalization structures. Explicit computational results on finite Boolean algebras demonstrate the feasibility of exhaustive classification, while connections to type space preorders show the utility in model theory and ultrafilter arithmetic. This program develops a language-independent, order-theoretic foundation for model category theory and extends classical results to new algebraic and logical domains (Salch et al., 27 Dec 2025, Baglini et al., 11 Sep 2025).