2.1 Preliminaries

We begin by fixing a few notational conventions.

By a bifunctor, we mean a functor in two arguments. We make repeated use of the Leibniz construction [Reference Riehl and VerityRV14, Definition 4.4], which transforms a bifunctor into an bifunctor on arrow categories.

Definition 2.3 Given a bifunctor ${\odot } \colon \mathbf {C} \times \mathbf {D} \to \mathbf {E}$ into a category $\mathbf {E}$ with pushouts, the Leibniz construction defines a bifunctor ${\mathbin {\widehat {\odot }}} \colon \mathbf {C}^\to \times \mathbf {D}^\to \to \mathbf {E}^\to $ , with $f \mathbin {\widehat {\odot }} g$ defined for $f \colon A \to B$ and $g \colon X \to Y$ as the following induced map:

2.2 Model structures and Quillen equivalences

In the abstract, the force of our result is that a certain model category presents the $(\infty ,1)$ -category of $\infty $ -groupoids. Concretely, we work entirely in model-categorical terms, exhibiting a Quillen equivalence between this model category and another model category—simplicial sets—already known to present $\mathbf {{\infty }\text {-}Gpd}$ . We briefly fix the relevant basic definitions here but assume prior familiarity, especially with factorization systems; standard references include [Reference HoveyHov99, Reference Dwyer, Hirschhorn, Kan and SmithDHKS04].

Note that given any two of the classes $(\mathcal {C}, \mathcal {W}, \mathcal {F})$ , we can reconstruct the third: $\mathcal {C}$ is the class of maps with left lifting against $\mathcal {F} \cap \mathcal {W}$ , $\mathcal {F}$ is the class of maps with right lifting against $\mathcal {C} \cap \mathcal {W}$ , and $\mathcal {W}$ is the class of maps that can be factored as a map with left lifting against $\mathcal {F}$ followed by a map with right lifting against $\mathcal {C}$ . We will thus frequently introduce a model category by giving a description of two of its classes.

The two factorization systems are commonly generated by sets of left maps.

Now, we come to relationships between model categories.

Note that F preserves cofibrations if and only if G preserves trivial fibrations, while G preserves fibrations if and only if F preserves trivial cofibrations.

2.3 Reedy categories and elegance

The linchpin of our approach is Reedy category theory, the theory of diagrams over categories whose morphisms factor into degeneracy-like and face-like components. As our base category of interest contains non-trivial isomorphisms, we work more specifically with the generalized Reedy categories introduced by Berger and Moerdijk [Reference Berger and MoerdijkBM11].

The prototypical strict Reedy category is the simplex category $\Delta $ : the degree of an n-simplex is n, while the lowering and raising maps are the degeneracy and face maps, respectively [Reference Gabriel and ZismanGZ67, Section II.3.2].

A Reedy structure on a category $\mathbf {R}$ is essentially a tool for working with $\mathbf {R}$ -shaped diagrams. For example, a weak factorization system on any category $\mathbf {E}$ induces injective and projective Reedy weak factorization systems on the category $[\mathbf {R},\mathbf {E}]$ of $\mathbf {R}$ -shaped diagrams in $\mathbf {E}$ ; likewise for model structures. Importantly for us, any diagram of shape $\mathbf {R}$ can be regarded as built iteratively from “partial” diagrams specifying the elements at indices up to a given degree. We are specifically interested in presheaves, i.e., $\mathbf {R}^{\mathrm {op}}$ -shaped diagrams in ${\mathbf {Set}}$ . We refer to [Reference Dwyer, Hirschhorn, Kan and SmithDHKS04, Section 22; Reference Berger and MoerdijkBM11; Reference Riehl and VerityRV14; Reference ShulmanShu15] for overviews of Reedy categories and their applications.

Berger and Moerdijk’s definition of generalized Reedy category [Reference Berger and MoerdijkBM11, Definition 1.1] includes one additional axiom. Following Riehl [Reference RiehlRie17], we treat this as a property to be assumed only where necessary.

Note that any Reedy category in which all lowering maps are epic satisfies this property. The main results of this article are restricted to pre-elegant Reedy categories (Definition 5.28) for which this is always the case (Lemma 5.29); nevertheless, we try to record where only the weaker assumption is needed.

The following cancellation property will come in handy.

Proof We prove the first statement; the second follows by duality. Suppose $gf$ is a lowering map. We take Reedy factorizations $f = me$ , $g = m'e'$ , and then $e'm = m"e"$ :

This gives us a Reedy factorization $gf = (m'm")(e"e)$ . By uniqueness of factorizations, $m'm"$ must be an isomorphism; this implies $\lvert t" \rvert = \lvert t' \rvert = \lvert r \rvert $ , so $m'$ and $m"$ are also isomorphisms. Thus, $g \cong e'$ is a lowering map.

When studying ${\mathbf {Set}}$ -valued presheaves over a Reedy category, it is useful to consider the narrower class of elegant Reedy categories [Reference Berger and MoerdijkBM11, Reference Bergner and RezkBR13].

Intuitively, an elegant Reedy category is one where any pair of “degeneracies” $s {\overset {-}\leftarrow } r {\overset {-}\to } s'$ has a universal “combination” $r {\overset {-}\to } s \sqcup _r s'$ , namely, the diagonal of their pushout. The condition on the Yoneda embedding asks that any r-cell in a presheaf is degenerate along (that is, factors through) both $r {\overset {-}\to } s$ and $r {\overset {-}\to } s'$ if and only if it is degenerate along their combination. Again, the simplex category is the prototypical elegant Reedy category [Reference Gabriel and ZismanGZ67, Section II.3.2].

A presheaf $X \in \mathrm {PSh}({\mathbf {R}})$ over any Reedy category can be written as the sequential colimit of a sequence of n-skeleta containing non-degenerate cells of X only up to degree n, with the maps between successive skeleta obtained as cobase changes of certain basic cell maps. When $\mathbf {R}$ is elegant, these cell maps are moreover monic. This property gives rise to a kind of induction principle: any property closed under certain colimits can be verified for all presheaves on an elegant Reedy category by checking that it holds on basic cells. This principle is conveniently encapsulated by the following definition.

We note that when $\mathbf {E}$ is a model category with monos as cofibrations, these are all diagrams whose colimits agree with their homotopy colimits: we can compute their colimits in the $(\infty ,1)$ -category presented by $\mathbf {E}$ by simply computing their 1-categorical colimits in $\mathbf {E}$ , which is hardly the case in general. This fact is another application of Reedy category theory; see, for example, Dugger [Reference DuggerDug08, Section 14]. As a result, these colimits have homotopical properties analogous to 1-categorical properties of colimits. For example, recall that given a natural transformation $\alpha \colon F \to G$ between left adjoint functors $F,G \colon \mathbf {E} \to \mathbf {F}$ , the class of $X \in \mathbf {E}$ such that $\alpha _X$ is an isomorphism is closed under colimits. If $F,G$ are left Quillen adjoints and $\mathbf {E},\mathbf {F}$ have monomorphisms as cofibrations, then the class of X such that $\alpha _X$ is a weak equivalence is saturated by monomorphisms. This particular fact will be key in Section 7.1.

For presheaves over an elegant Reedy category, the basic cells are the quotients of representables by automorphism subgroups.

As described above, we will be studying a category $\square _{\lor }$ that is not a Reedy category. Thus, we will not use the previous proposition directly. Instead, our Section 5 establishes a generalization to categories that only embed in a Reedy category in a nice way.

2.4 Simplicial sets

To show that a given model category presents $\mathbf {{\infty }\text {-}Gpd}$ , it suffices to exhibit a Quillen equivalence to a model category already known to present $\mathbf {{\infty }\text {-}Gpd}$ . Here, our standard of comparison will be the classical Kan–Quillen model structure on simplicial sets [Reference QuillenQui67, Section II.3].

This is a strict Reedy category, in fact, an EZ category (see Remark 2.17). The raising and lowering maps are given by the face and degeneracy maps, defined as the injective and surjective maps of posets, respectively.

3.1 Model structures from premodel structures

As any two of the classes $(\mathcal {C},\mathcal {W},\mathcal {F})$ defining a model structure determines the third, any premodel structure induces a candidate class of weak equivalences.

For the remainder of this section, we fix a premodel category $\mathbf {M}$ with factorization systems $(\mathcal {C},\mathcal {F}_t)$ and $(\mathcal {C}_t,\mathcal {F})$ . The following two propositions are standard.

In light of the above, we use the arrows and to denote trivial cofibrations and fibrations also in a premodel structure.

We now reduce the problem of checking 2-out-of-3 for $\mathcal {W}(\mathcal {C},\mathcal {F})$ to a reduced collection of special cases of 2-out-of-3 where some or all maps belong to $\mathcal {C}$ or $\mathcal {F}$ .

Proof Conditions A–C all follow by straightforward applications of 2-out-of-3 for $\mathcal {W}(\mathcal {C},\mathcal {F})$ . Suppose conversely that we have A–C and let maps $g \colon Y \to Z$ and $f \colon X \to Y$ be given. Then, using the two factorization systems and condition C, we have the following diagram:

Suppose first that g and f are weak equivalences. Then, we may choose the factorizations of f and g such that the map is a trivial cofibration and the map is a trivial fibration. Thus, $gf$ factors as a trivial cofibration followed by a trivial fibration, i.e., is a weak equivalence.

Now suppose that f and $gf$ are weak equivalences. We may choose the factorization of f such that the map is a trivial cofibration. The composite is then a trivial cofibration, so the composite is a trivial fibration by condition B. Then, the map is a trivial fibration by condition A. Hence, g is a weak equivalence. By the dual argument, if g and $gf$ are weak equivalences then so is f.

3.2 Cylindrical premodel structures

Now, we derive a simpler set of criteria for premodel structures equipped with a compatible adjoint functorial cylinder.

Definition 3.9 A functorial cylinder on a category $\mathbf {E}$ is a functor $\mathbb {I} \otimes (-) \colon \mathbf {E} \to \mathbf {E}$ equipped with endpoint and contraction transformations fitting in a diagram as shown:

An adjoint functorial cylinder is a cylinder such that $\mathbb {I} \otimes (-)$ is a left adjoint.

There is a dual notion of functorial path object consisting of a functor $\mathbb {I} \oslash (-)$ and natural transformations $\delta _k \oslash (-) \colon \mathbb {I} \otimes (-) \to \mathrm {Id}$ and $\varepsilon \oslash (-) \colon \mathrm {Id} \to \mathbb {I} \otimes (-)$ . By transposition, each adjoint functorial cylinder corresponds to an adjoint functorial path object.

Definition 3.14 Given $f \colon A \to B$ in a finitely cocomplete category with a functorial cylinder and , we write $\mathrm {M}_{k}(f)$ for its k-sided mapping cylinder, defined as the pushout

The k-sided mapping cylinder factorization of f is the factorization

Fix once more a premodel category $\mathbf {M}$ with factorization systems $(\mathcal {C},\mathcal {F}_t)$ and $(\mathcal {C}_t,\mathcal {F})$ . We show that condition C is reducible to condition A when $\mathbf {M}$ is cylindrical by relating trivial fibrations with dual strong deformation retracts.

Definition 3.18 In a category with a functorial cylinder, we say $f \colon Y \to X$ is a dual strong k-oriented deformation retract for some $k \in \{0,1\}$ when we have a map $s \colon X \to Y$ such that $fs = \mathrm {id}$ and a homotopy $h \colon \mathbb {I} \otimes {Y} \to Y$ such that $h(\delta _k \otimes Y) = sf$ , $h(\delta _{1-k} \otimes Y) = \mathrm {id}$ , and $fh$ is a constant homotopy. Equivalently (if the category is finitely cocomplete), f is a dual strong k-oriented deformation retract when we have a diagonal filler

The following is a standard construction (see, e.g., [Reference QuillenQui67, Lemma I.5.1]).

Proof Let $f \colon Y \to X$ be an $\mathcal {R}$ -map between $\mathcal {L}$ -objects. We solve two lifting problems in turn:

The maps s and h exhibit f as a dual strong $0$ -oriented deformation retract; we may similarly construct a $1$ -oriented equivalent.

Corollary 3.20 Let $(\mathcal {L},\mathcal {R})$ be a cylindrical weak factorization system on a category with a functorial cylinder. Then, in any diagram of the form

the horizontal map is a dual strong k-oriented deformation retract for any $k \in \{0,1\}$ .

Proof Let $s \colon X \to Y$ and $h \colon \mathbb {I} \otimes {Y} \to Y$ be as in the definition of dual strong k-oriented deformation retract. Then, the diagram

exhibits f as a retract of a trivial fibration.

Proof Suppose we have a weak equivalence $X \to Y$ factoring as a trivial cofibration followed by a fibration, thus a diagram of the following form:

We first take a pullback and factorize the induced gap map as a trivial cofibration followed by a fibration.

By Corollary 3.20, the composites and are dual strong deformation retracts, thus trivial fibrations by Lemma 3.21. Then, the composite is a trivial fibration by composition, so is a trivial fibration by right cancellation.

Finally, we prove for reference below that the cancellation properties opposite of condition A always hold in a cylindrical premodel structure, though we will not need this fact.

Proof The first factor $A \to \mathrm {M}_{k}(f)$ in the factorization of $f \colon A \to B$ decomposes as the composite

The first map is a cobase change of $0 \to B$ , thus an $\mathcal {L}$ -map. The last map is a cobase change of $\partial \otimes A \cong \partial \mathbin {\widehat {\otimes }} (0 \to A)$ , thus also an $\mathcal {L}$ -map.

Proof Let be a cofibration between trivially cofibrant objects. Consider the commutative square

The top horizontal map is a trivial cofibration by Lemma 3.24, while the right vertical map is a trivial cofibration by cylindricality. The bottom map is split monic, so m is a retract of a trivial cofibration and thus a trivial cofibration itself.

Proof Given a diagram

we apply Lemma 3.25 to f in the coslice under A via Remark 3.17.

3.3 Model structures from the fibration extension property

We now narrow our attention to premodel structures satisfying properties common to cubical-type model structures: first, that all objects are cofibrant, and second, that fibrations extend along trivial cofibrations, the latter of which follows in particular from the existence of enough fibrant universes classifying fibrations. Note that our conditions cease to be self-dual at this point; moreover, the result is a criterion sufficient but not necessary to obtain a model structure.

Proof Suppose trivial fibrations have right cancellation in $\mathbf {M}$ and let $p \colon Y \to X$ be a map lifting against cofibrations between cofibrant objects. We take a cofibrant replacement of Y, obtaining maps . By cancellation, it suffices to show the composite $p' \colon Y' \to X$ is a trivial fibration. We appeal to the retract argument: $p'$ has the lifting property against the left part of its (cofibration, trivial fibration) factorization—this being a cofibration between cofibrant objects—so is a retract of the right part of its factorization. It is thus itself a trivial fibration.

The converse is an elementary exercise in lifting. Suppose the (cofibration, trivial fibration) factorization system is generated by cofibrations between cofibrant objects, let and $g \colon Y \to X$ be such that $gf$ is a trivial fibration. Given a cofibration between cofibrant objects and a lifting problem

we solve lifting problems first against f and then against $gf$ :

The composite $fv$ is a lift for the original square.

In particular, Lemma 3.27 tells us that trivial fibrations have right cancellation in any premodel structure where all objects are cofibrant. If the premodel structure is additionally cylindrical, then condition C is also always satisfied.

Proof Suppose we have and . We take their composite’s (trivial cofibration, fibration) factorization:

We intend to show q is a trivial fibration. By Corollary 3.20 and the assumption that all objects are cofibrant, p has the structure of a dual strong 0-oriented deformation retract. Thus, we have a diagonal lift

Using that q is a fibration, we show that q is a dual strong deformation retract by solving a lifting problem of the form

The map is the following composite:

The first map is a cobase change of the trivial cofibration m, while the final map is a cobase change of the trivial cofibration $\partial \mathbin {\widehat {\otimes }} n$ ; thus the composite is indeed a trivial cofibration. The diagonal lift exhibits q as a dual strong deformation retract, thus, a trivial fibration by Lemma 3.21.

Thus, in a cylindrical premodel structure where all objects are cofibrant, the only non-trivial property necessary to apply Theorem 3.23 is left cancellation for trivial cofibrations among cofibrations. This we can further reduce to the following condition.

Definition 3.29 (FEP)

We say a premodel category $\mathbf {M}$ has the fibration extension property when for any fibration and trivial cofibration , there exists a fibration whose base change along m is f:

The fibration extension property can, in particular, be obtained from the existence of fibrant classifiers for fibrations, i.e., fibrant universes of fibrations. We do not generally expect to have a single classifier for all fibrations, only those below a certain size. Thus, we now consider a setup where a premodel category sits inside a larger category containing classifiers for its fibrations.

4.1 The semilattice cube category

We now introduce this article’s main character: the (join-)semilattice cube category $\square _{\lor }$ generated by an interval object, finite Cartesian products, and a binary connection operator. Like other Cartesian cube categories, it is a (single-sorted) Lawvere theory [Reference William LawvereLaw63]: a finite product category in which every object is a finite power of some distinguished object.

Definition 4.1 The theory of (join-)semilattices consists of an associative and commutative binary operator $\lor $ for which all elements are idempotent, which we call the join. This means the following laws:

$$ \begin{align*} (x \lor y) \lor z = x \lor (y \lor z), \qquad \quad x \lor y = y \lor x, \qquad \quad x \lor x = x. \end{align*} $$

The theory of 01-bounded (join-)semilattices consists, in addition to the above, of two constants $0$ , $1$ and the following laws:

$$ \begin{align*} 0 \lor x = x, \qquad \quad 1 \lor x = 1. \end{align*} $$

The (join-)semilattice cube category $\square _{\lor }$ is the Lawvere theory of 01-bounded semilattices. Concretely, the objects of $\square _{\lor }$ are of the form $T^n$ for $n \in \mathbb {N}$ , and the morphisms $T^m \to T^n$ are n-ary tuples of expressions over $0,1,\lor $ in m variables modulo the equations above. We write $\mathbf {T}_\lor $ for the Lawvere theory of semilattices.

Recall that the category of algebras of a Lawvere theory $\mathbf {T}$ is the category of finite-product-preserving functors from $\mathbf {T}$ to ${\mathbf {Set}}$ , which supports an “underlying set” functor $U \colon [\mathbf {T},{\mathbf {Set}}]_{\mathrm {fp}} \to {\mathbf {Set}}$ given by evaluation at the distinguished object $T^1$ . This functor has a left adjoint $F \colon {\mathbf {Set}} \to \mathrm {Alg}({\mathbf {T}})$ which produces the free $\mathbf {T}$ -algebra on a set, and the covariant Yoneda embedding restricts to an embedding $\mathbf {T}^{\mathrm {op}} \to \mathrm {Alg}({\mathbf {T}})$ sending $T^n$ to the free algebra on n elements. We write $\mathbf {SLat}$ and $\mathbf {01}\mathbf {SLat}$ for the categories of algebras of $\mathbf {T}_\lor $ and $\square _{\lor ,}$ respectively. Concretely, these are the categories of sets equipped with the operations described in Definition 4.1 and operation-preserving morphisms between them.

It can also be useful to take an order-theoretic perspective on $\mathbf {SLat}$ and $\mathbf {01}\mathbf {SLat}$ , identifying them as subcategories of the category $\mathbf {Pos}$ of posets and monotone maps. Recall that the operator $\lor $ induces a poset structure on any semilattice, with $x \le y$ when $x \lor y = y$ .

In particular, the interval $[1] \in \mathbf {Pos}$ is a 01-bounded semilattice.

Proposition 4.5 The interval is a dualizing object for a duality between the categories of finite semilattices and finite 01-bounded semilattices, which is to say that we have the following categorical equivalence:

Given a semilattice A, the 01-bounded semilattice structure on ${\mathbf {SLat}}(A,[1])$ is defined pointwise from that on $[1]$ ; likewise ${\mathbf {01}\mathbf {SLat}}(B,[1])$ has a pointwise semilattice structure for any $B \in \mathbf {01}\mathbf {SLat}$ . This extends the duality between the augmented simplex category and the category of finite intervals (i.e., finite bounded linear orders and bound-preserving monotone maps) observed by Joyal [Reference JoyalJoy97, Section 1.1; Reference WraithWra93].

By way of this duality, we have in particular an embedding of $\square _{\lor }$ in the category of finite semilattices, induced by the embedding of its opposite in its category of models:

Here, we use that the free semilattice on a finite set of generators is a finite semilattice. Unpacking, this embedding sends $T^n$ to ${\mathbf {01}\mathbf {SLat}}(F(n),[1]) \cong {{\mathbf {Set}}}(n,U[1]) \cong [1]^n$ .

We can also describe the cubes in $\mathbf {SLat}$ as free semilattices on posets. Given a poset A, write $1 \mathbin {\star } A$ for the poset obtained by adjoining a minimum element $\bot $ to A. For any set S, we have a monotone map $\eta _n \colon 1 \mathbin {\star } S \to [1]^S$ sending $\bot $ to $\bot $ and $i \in S$ to the element of $[1]^S$ with $1$ at its ith component and $0$ elsewhere.

4.2 Cubical-type model structure on semilattice cubical sets

We now define our model structure on $\mathrm {PSh}{(\square _{\lor })}$ using Corollary 3.33. That our case satisfies the corollary’s hypotheses is essentially an application of existing work, namely, [Reference Cavallo, Mörtberg and SwanCMS19] or [Reference AwodeyAwo23], so we do not give many proofs, only enough of an outline to guide an unfamiliar reader through the appropriate references. We point to [Reference Gambino and SattlerGS17; Reference SattlerSat17; Reference Awodey, Gambino and HazratpourAGH24, §8) for further details on constructing model structures of this kind and to [Reference Licata, Orton, Pitts and SpittersLOPS18] for the definition of the universe in particular.

4.2.2 Unbiased fibrations

In order to apply Corollary 3.33, we must check that we have a fibration between fibrant objects in $\mathrm {PSh}{(\square _{\lor })}$ classifying fibrations in $\mathrm {PSh}_{\kappa }({\square _{\lor }})$ . This follows from work on cubical models of type theory, specifically the interpretation of universes. Our cube category falls within the ambit of [Reference Angiuli, Brunerie, Coquand, Harper, Hou (Favonia) and LicataABCHFL21], which describes a universe $p_{\mathrm {fib}} \colon \widetilde U_{\mathrm {fib}} \to U_{\mathrm {fib}}$ with fibration structures on $p_{\mathrm {fib}}$ and $U_{\mathrm {fib}}$ in type-theoretic terms; Awodey gives a construction of the same in categorical language [Reference AwodeyAwo23, Sections 6–8].

However, the fibrations used in these models are not a priori the fibrations we defined in the previous section: they are what Awodey [Reference AwodeyAwo23] calls unbiased fibrations, which lift not only against (pushout products with) endpoint inclusions $\delta _k \colon 1 \to \mathbb {I}$ but against generalized points on the interval. To see that $\widehat {\square }_{\lor }^{\mathrm {ty}}$ is compatible with this model of type theory, we check here that biased (i.e., ordinary) and unbiased fibrations coincide in the presence of a connection.

Definition 4.20 Given $r \colon B \to \mathbb {I}$ and $f \colon A \to B$ , their unbiased mapping cylinder is the following pushout:

Note that $\mathrm {M}_{\delta _k !_B}(f)$ is the ordinary k-sided mapping cylinder (Definition 3.14). We write $r \mathbin {\widehat {\times }}_B m\colon \mathrm {M}_{r}(m) \to \mathbb {I} \times B$ for the unique map fitting in the diagram

This is the pushout product in the slice over B of and , hence the notation. Note that $(\delta _k !_B) \mathbin {\widehat {\times }}_B f$ is the ordinary pushout product $\delta _k \mathbin {\widehat {\times }} f$ .

Definition 4.21 We say $f \colon Y \to X$ is an unbiased fibration when it has the right lifting property against $r \mathbin {\widehat {\times }}_B m$ for all $r \colon B \to \mathbb {I}$ and .

Lemma 4.22 $r \mathbin {\widehat {\times }}_B m$ is a trivial cofibration for any $r \colon B \to \mathbb {I}$ and .

Proof Define . Take a pushout of $\delta _0 \mathbin {\widehat {\times }} m$ :

Define a map $v \colon \mathrm {M}_{1}(r \mathbin {\widehat {\times }}_B m) \to C$ like so:

Take the pushout of $\delta _1 \mathbin {\widehat {\times }} (r \mathbin {\widehat {\times }}_B m)$ by this map:

Then, we can exhibit $r \mathbin {\widehat {\times }}_B m$ as a retract of $n'n$ :

As a retract of a trivial cofibration, $r \mathbin {\widehat {\times }}_B m$ is thus a trivial cofibration.

Corollary 4.23 A map is a fibration in $\widehat {\square }_{\lor }^{\mathrm {ty}}$ if and only if it is an unbiased fibration.

Proof If $f \colon Y \to X$ is an unbiased fibration, then lifting against any $\delta _k \mathbin {\widehat {\times }} m$ is obtained as lifting against $(\delta _k{!}_B) \mathbin {\widehat {\times }}_B m$ . The converse is Lemma 4.22.

Remark 4.24 For the reader more comfortable with cubical type theories, we give the type-theoretic analog to the proof of Corollary 4.23. The ABCHFL type theory equips types with a composition operator of the following form:

In the presence of a connection, we can derive a term satisfying the equations required of $\mathsf {com}_{i.A}^{{r} \to {s}}\,\left [{\varphi } \mapsto {i.M}\right ]\,{M_0}$ using only composition $\varepsilon \to s,$ where $\varepsilon \in \{0,1\}$ , namely, the term Q below.

Remark 4.25 We can also use $\lor $ to show that any fibration is an equivariant fibration in the sense of the ACCRS model structure [Reference Awodey, Cavallo, Coquand, Riehl and SattlerACCRS24]. For simplicity, let us restrict attention to lifting along $\delta ^n_1 \colon 1 \to \mathbb {I}^n$ , which is the simplest case; we leave it as an exercise to formulate and derive unbiased equivariant lifting by combining the proof of Lemma 4.22 with the following sketch. A more complete proof (for simplicial sets rather than semilattice cubical sets, but with the same argument) is in [Reference Awodey, Cavallo, Coquand, Riehl and SattlerACCRS24, Proposition 6.1.7].

Write for the wide subcategory of isomorphisms of $\square _{\lor }$ . We have a functor $\delta \colon \Sigma \to \mathrm {PSh}{(\square _{\lor })}^\to $ sending $[1]^n$ to $\delta _1^n \colon 1 \to \mathbb {I}^n$ and $\sigma \colon [1]^n \cong [1]^n$ to $(\mathrm {id},\sigma ) \colon \delta _1^n \to \delta _1^n$ . Take $u_{\delta \Sigma }$ to be the composite

A uniform equivariant 1-fibration is a right $u_{\delta \Sigma }$ -map.

Suppose $f \colon Y \to X$ is a uniform fibration and let $m \colon A {\rightarrowtail } B$ and a lifting problem $(y,x) \colon \delta _1^n \mathbin {\widehat {\times }} m \to f$ be given. We have a map ${\uparrow _n} \colon {[1]} \times [1]^n \to [1]^n$ sending $(t, i_1, \ldots , i_n) \mapsto (t \lor i_1, \ldots , t \lor i_n)$ which we use to form a lifting problem against $\delta _1 \mathbin {\widehat {\times }} (\mathbb {I}^n \times m)$ :

The composite tikz69 is our desired lift, while the uniformity conditions follow from those on f and the fact that ${\uparrow _n} \circ (\mathbb {I} \times \sigma ) \cong \sigma \circ {\uparrow _n}$ for any $\sigma \colon [1]^n \cong [1]^n$ .

4.2.3 Universe

To define a universe classifying fibrations, we use a theorem of Licata et al. [Reference Licata, Orton, Pitts and SpittersLOPS18]. The cardinal $\kappa $ provides a Grothendieck universe in ${\mathbf {Set}}$ , from which Hofmann and Streicher’s construction produces a universe $p_U \colon \widetilde U \to U$ in $\mathrm {PSh}{(\square _{\lor })}$ classifying $\kappa $ -small maps [Reference Hofmann and StreicherHS97, Reference Streicher, Crosilla and SchusterStr05, Reference AwodeyAwo24]. Our classifier for $\kappa $ -small fibrations shall be a subuniverse of $p_U$ . The key property of $\mathrm {PSh}{(\square _{\lor })}$ is that the cocylinder $(-)^{\mathbb {I}}$ has a right adjoint, i.e., that $\mathbb {I}$ is internally tiny: we have $(-)^{\mathbb {I}} \cong ((-) \times [1])^*$ and therefore . This property is common to cube categories but fails, for example, in simplicial sets. We refer to Swan [Reference SwanSwa22] for a deeper analysis.

Given a $\kappa $ -small map $f \colon Y \to X$ with characteristic map $A \colon X \to U$ , we define a family $X^{\mathbb {I}} \to U$ whose sections correspond to fibration structures on A. To do so, it is convenient to work in the internal extensional type theory of the universe $p_U$ in the style of Orton and Pitts [Reference Orton and PittsOP18].Footnote ⁵ Writing $\top \colon 1 \to \Omega $ for the subobject classifier in $\mathrm {PSh}{(\square _{\lor })}$ , the maps ${!}_\Omega \colon \Omega \to 1$ and $\top $ are both classified by $p_U$ ,Footnote ⁶ so appear as a closed type $\cdot \vdash \Omega : U$ and type family $\varphi \colon \Omega \vdash {[\varphi ]} : U,$ respectively. The interval likewise appears as a closed type $\cdot \vdash \mathbb {I} : U$ with inhabitants $\cdot \vdash 0,1 : \mathbb {I}$ .

Definition 4.26 Given a type $A \colon U$ , define its type of trivial fibration structures $\operatorname {\mathrm {TFib}} A \colon U$ as follows:

Definition 4.27 Given $k \in \{0,1\}$ and $A \colon X \to U$ , define the pullback exponential $(\delta _k \mathbin {\widehat {\to }} A) : (\Sigma p \colon X^{\mathbb {I}}.\ A(p(k))) \to U$ internally as follows:

Definition 4.28 Given $A \colon X \to U$ , define $\operatorname {\mathrm {Fib}}_kA \colon X^{\mathbb {I}} \to U$ for $k \in \{0,1\}$ and then $\operatorname {\mathrm {Fib}} A \colon X^{\mathbb {I}} \to U$ as follows:

Proposition 4.29 Let $f \colon Y \to X$ be given with classifying map $A \colon X \to U$ . Then, f is a uniform fibration if and only if the type $\Pi p \colon X^{\mathbb {I}}.\ (\operatorname {\mathrm {Fib}} A)(p)$ is inhabited.

Proof See [Reference Awodey, Gambino and HazratpourAGH24, Corollary 8.7].

Using the right adjoint to $(-)^{\mathbb {I}}$ , we carve out the subuniverse of $p_U$ corresponding to families $A \colon X \to U$ for which $\Pi p \colon X^{\mathbb {I}}.\ (\operatorname {\mathrm {Fib}} A)(p)$ is inhabited. For this step, we return to working externally, as $\sqrt [\mathbb {I}]{-}$ does not straightforwardly internalize; Licata et al. [Reference Licata, Orton, Pitts and SpittersLOPS18] use a global sections modality to axiomatize $\sqrt [\mathbb {I}]{-}$ internally, while Riley [Reference RileyRil24] has recently proposed a type theory which directly represents $\sqrt [\mathbb {I}]{-}$ as a modality. The following definition and proposition constitute Theorem 5.2 of [Reference Licata, Orton, Pitts and SpittersLOPS18].

Definition 4.30 Define $p_{\mathrm {fib}} \colon \widetilde U_{\mathrm {fib}} \to U_{\mathrm {fib}}$ by pullback as follows:

Proposition 4.31 [Reference Licata, Orton, Pitts and SpittersLOPS18, Theorem 5.2]

If $f \colon Y \to X$ is the pullback of $p_U$ along some $A \colon X \to U$ , then f is a uniform fibration if and only if A factors through $\pi _1 \colon U_{\mathrm {fib}} \to U$ .

Corollary 4.32 The map $p_{\mathrm {fib}}$ is a uniform fibration.

Proof $p_{\mathrm {fib}}$ is the pullback of $p_U$ along $\pi _1$ , which of course factors through itself.

Finally, we need a fibrancy structure on the universe U itself. This is the most technically involved argument; we defer to prior work.

Proposition 4.33 The object $U_{\mathrm {fib}}$ is uniform fibrant.

Proof A fibrancy structure on $U_{\mathrm {fib}}$ is described in type-theoretic language in [Reference Angiuli, Brunerie, Coquand, Harper, Hou (Favonia) and LicataABCHFL21, Section 2.12], while Awodey [Reference AwodeyAwo23, Section 8] gives an external categorical construction.

Theorem 4.34 (Cubical-type model structure on semilattice cubical sets)

There is a model structure on $\mathrm {PSh}_{\kappa }({\square _{\lor }})$ in which:

• • The cofibrations are the monomorphisms;

• • The fibrations are those maps with the right lifting property against all pushout products $\delta _k \mathbin {\widehat {\times }} m$ of an endpoint inclusion with a monomorphism.

We write $\widehat {\square }_{\lor }^{\mathrm {ty}}$ for this model category.

Proof By Corollary 3.33 applied with $\mathrm {PSh}_{\kappa }({\square _{\lor }})$ inside $\mathrm {PSh}{(\square _{\lor })}$ and the factorization systems $(\mathcal {M},\mathcal {F}_t)$ and $(\mathcal {C}_t,\mathcal {F})$ defined in this section. Clearly all objects are cofibrant, and every fibration in $\mathrm {PSh}_{\kappa }({\square _{\lor }})$ is classified by $p_{\mathrm {fib}} \colon \widetilde U_{\mathrm {fib}} \to U_{\mathrm {fib}}$ , which is a fibration (Corollary 4.32) between fibrant objects (Proposition 4.33).

Our question now is whether $\widehat {\square }_{\lor }^{\mathrm {ty}}$ presents $\mathbf {{\infty }\text {-}Gpd}$ . More narrowly, we can ask whether the following comparison adjunction evinces a Quillen equivalence between $\widehat {\square }_{\lor }^{\mathrm {ty}}$ and $\widehat {\Delta }^{\mathrm {kq}}$ .

Definition 4.35 (Triangulation)

Define to be the functor sending the n-cube $[1]^n$ to the n-fold product $(\Delta ^{1})^n$ of the $1$ -simplex, with the evident functorial action. The triangulation functor $\mathrm {T} \colon \mathrm {PSh}{(\square _{\lor })} \to \mathrm {PSh}({\Delta })$ is the left Kan extension of :

Triangulation has a right adjoint, the nerve functor defined by .

4.3 Idempotent completion

Although the triangulation adjunction is the most immediate means of comparing $\widehat {\square }_{\lor }^{\mathrm {ty}}$ and $\widehat {\Delta }^{\mathrm {kq}}$ , it is not the most convenient. Ideally, we would like to have a comparison on the level of the base categories, some functor $i \colon \Delta \to \square _{\lor }$ or vice versa, in which case we would obtain an adjoint triple ${i}_! \dashv i^* \dashv {i}_*$ on their presheaf categories. This is too much to hope for, but we can define an embedding from $\Delta $ into the idempotent completion of $\square _{\lor }$ , following the strategy used by Sattler [Reference SattlerSat19] and Streicher and Weinberger [Reference Streicher and WeinbergerSW21] to relate $\Delta $ and $\square _{\land \!\lor }$ . The category of presheaves on any category $\mathbf {C}$ is equivalent to the category of presheaves on its idempotent completion $\overline {\mathbf {C}}$ , the closure of $\mathbf {C}$ under splitting of idempotents [Reference Borceux and DejeanBD86]. We shall exhibit an embedding $\blacktriangle \colon \Delta \to \overline {\square }_\lor $ ; by composing the triple ${\blacktriangle }_! \dashv \blacktriangle \!^* \dashv {\blacktriangle }_*$ with the adjoint equivalence , we obtain a triple relating $\mathrm {PSh}({\Delta })$ and $\mathrm {PSh}{(\square _{\lor })}$ .

We then observe that (Lemma 4.48); thus, the upshot of this detour is that $\mathrm {T}$ is also a right adjoint. It will, however, be easier to study the adjunction ${\blacktriangle }_! \dashv \blacktriangle \!^*$ than , in particular because both ${\blacktriangle }_!$ and $\blacktriangle \!^*$ are left Quillen adjoints (Corollary 4.53 and Lemma 4.54). We will first show in Section 7.1 that ${\blacktriangle }_! \dashv \blacktriangle \!^*$ is a Quillen equivalence, then deduce formally that $\blacktriangle \!^* \dashv {\blacktriangle }_*$ and are also Quillen equivalences.

The splitting of an idempotent is unique up to isomorphism if it exists: s is the equalizer of the pair $f,\mathrm {id} \colon A \to A$ , while r is the coequalizer of the same. We say that $\mathbf {C}$ is idempotent complete if every idempotent splits.

Equivalently, an idempotent completion is a universal (in a bicategorical sense) fully faithful functor $\mathbf {C} \to \overline {\mathbf {C}}$ into an idempotent complete category. We shall only need the following consequence of this characterization.

We can describe the idempotent completion of $\square _{\lor }$ concretely as a full subcategory of $\mathbf {SLat}$ .

We show that is an idempotent completion using the following observations of Horn and Kimura.

Recall from Remark 4.4 that we have an embedding $\Delta \to \mathbf {SLat}$ . The induced lattice structure on a simplex is distributive, so this embedding factors through $\overline {\square }_\lor $ .

We can now decompose the triangulation functor.

4.4 Two Quillen adjunctions

In light of the equivalence $\mathrm {PSh}({\overline {\square }_\lor }) \simeq \mathrm {PSh}{(\square _{\lor })}$ , it now suffices to compare $\widehat {\Delta }^{\mathrm {kq}}$ with the induced model structure on $\mathrm {PSh}({\overline {\square }_\lor })$ , which again has monomorphisms for cofibrations and fibrations generated by pushout products $\delta _k \mathbin {\widehat {\times }} m$ . We begin by observing that both ${\blacktriangle }_!$ and $\blacktriangle \!^*$ are left Quillen adjoints.

Proof Write $\Delta _{\mathrm {a}}$ for the augmented simplex category, the full subcategory of $\mathbf {Pos}$ consisting of the objects $[n]$ for $n \in \mathbb {N}$ as well as . Write $\overline {\square }_{\lor {\mathrm {a}}}$ for the category of finite distributive semilattices, which similarly freely extends $\overline {\square }_\lor $ with an initial object (the empty semilattice). The inclusion $\blacktriangle $ extends to an inclusion between the augmented categories:

The square is a pullback and the vertical maps are (discrete) Grothendieck opfibrations, so the square is exact in the sense that the canonical map $\upsilon ^* {(\blacktriangle _{\mathrm {a}})}_! \to {\blacktriangle }_! \iota ^*$ is invertible [nLa24, Proposition 5.2 and Corollary 3.1]. This is also straightforward to check directly: the functors are cocontinuous, so it suffices to check on representables, and $\iota ^*$ and $\upsilon ^*$ preserve all representables except for the initial representable, which they send to an initial object. Since $\iota $ is fully faithful, this gives ${\blacktriangle }_! \cong {\blacktriangle }_! \iota ^* {\iota }_* \cong \upsilon ^* {(\blacktriangle _{\mathrm {a}})}_! {\iota }_*$ . Therefore, it suffices to prove that ${(\blacktriangle _{\mathrm {a}})}_!$ preserves monomorphisms.

Just like in simplicial sets, the monomorphisms in augmented simplicial sets form the left class of a weak factorization system generated by boundary inclusions (of augmented simplices). As ${(\blacktriangle _{\mathrm {a}})}_!$ is a left adjoint, it therefore suffices to show that it sends boundary inclusions to monomorphisms. The boundary inclusion is the joint image of the non-identity face maps $\Delta ^{k} {\overset {+}\to } \Delta ^{n}$ . The joint image of a set of maps $(f_i : A_i \to B)_{i \in I}$ in any pretopos is computed as the coequalizer of

It, therefore, suffices to check that ${(\blacktriangle _{\mathrm {a}})}_!$ sends face maps to monomorphisms and preserves pullbacks of cospans whose legs are face maps. As face maps are monic, the latter condition implies the former. For the latter condition, as face maps go between representables and $\Delta _{\mathrm {a}}$ has these pullbacks, it suffices to check that $\blacktriangle _{\mathrm {a}}$ preserves pullbacks of cospans whose legs are face maps. In fact, $\blacktriangle _{\mathrm {a}}$ creates such pullbacks, as any subposet of a linear poset is again linear.

The following statements can be phrased more generally at the level of cylinder objects in a model category. They also have evident dual version in terms of path objects with fibrancy assumptions instead.

Proof The top maps in the following diagram are trivial cofibrations because A is cofibrant:

The claim follows using 2-out-of-3.

In a cylindrical model category, a homotopy retract is a pair of maps $s \colon X \to Y$ , $r \colon Y \to X$ equipped with a homotopy $h \colon \mathbb {I} \otimes {X} \to X$ from $rs$ to $\mathrm {id}_X$ .

Proof For the non-trivial direction, assume that L sends representables to weakly contractible objects. Given $n \ge 1$ and $I \subseteq [n]$ , write $\Lambda ^{n}_{I}$ for the union of the subobjects $d_i \colon \Delta ^{n-1} {\rightarrowtail } \Delta ^{n}$ over $i \in I$ . We check by induction that L sends to a trivial cofibration for $n \in \mathbb {N}$ and $\emptyset \subsetneq I \subsetneq [n]$ . When $\lvert I \rvert = 1$ , $\Lambda ^{n}_{I}$ is the representable $\Delta ^{n-1}$ , so the claim holds by assumption and 2-out-of-3. Otherwise, choose some $i \in I$ . We have the following pushout square, which is preserved by L:

By induction hypothesis, L sends the left vertical map to a trivial cofibration. As trivial cofibrations are closed under cobase change, L then also sends the right vertical map to a trivial cofibration. By induction hypothesis, L sends to a trivial cofibration. By 2-out-of-3, we conclude that L sends to a trivial cofibration. For $I = [n]\setminus k$ , we obtain that L sends the horn inclusion $\Lambda ^{n}_{k} \to \Delta ^{n}$ to a trivial cofibration. This makes L a left Quillen adjoint.

The combinatorics of the above proof have a conceptual explanation in terms of the pushout join in augmented simplicial sets, which produces boundary inclusions and horn inclusions starting from the maps $\emptyset \to 1$ and $\Delta ^{-1} \to 1$ .

We quickly see that ${\blacktriangle }_! \dashv \blacktriangle \!^*$ is a Quillen coreflection in the following sense.

5.1 Cellular presentations and Reedy monomorphisms

For the theory of cellular presentations of diagrams over Reedy categories, we follow Riehl and Verity [Reference Riehl and VerityRV14, Reference RiehlRie17]. Almost none of the content in this section is novel. For simplicity, we restrict our attention throughout to presheaves, though much of the theory generalizes to functors from a Reedy category into any category.

5.1.1 Weighted colimits

Riehl and Verity observe that many arguments in Reedy category theory are naturally phrased in terms of weighted (co)limits. While more fundamental to enriched category theory, these can have a clarifying role even in ordinary (i.e., ${\mathbf {Set}}$ -enriched) category theory.

Definition 5.1 Let $\mathbf {E}$ be a category. Let a functor $W \colon \mathbf {C}^{\mathrm {op}} \to {\mathbf {Set}}$ (the weight) and a diagram $F \colon \mathbf {C} \to \mathbf {E}$ be given. A weighted colimit for this data is an object , equipped with a natural transformation , such that for any $X \in \mathbf {E}$ the induced map

of sets is an isomorphism.

Informally, the weight W specifies how many “copies” of each object in the diagram F to include in the weighted colimit .

Example 5.2 The ordinary colimit of a diagram $F \colon \mathbf {C} \to \mathbf {E}$ can be described as , a colimit weighted by the terminal presheaf $1 \in \mathrm {PSh}({\mathbf {C}})$ . Conversely, any weighted colimit admits a characterization as an ordinary colimit over the category of elements of W:

In particular, any cocomplete category has weighted colimits.

Example 5.3 Recall that a tensor of a set $S \in {\mathbf {Set}}$ and object $X \in \mathbf {E}$ is an object $S \mathbin {\ast } X$ such that morphisms $S \mathbin {\ast } X \to Y$ correspond to objects ${{\mathbf {Set}}}(S,{\mathbf {E}}(X,Y))$ , i.e., families of morphisms $f_s \colon X \to Y$ for $s \in S$ . In ordinary category theory, this is simply the S-ary coproduct $\coprod _{s \in S} X$ , so can be expressed as the weighted colimit of the constant diagram $\Delta X \colon S \to \mathbf {E}$ . Alternatively, we can encode the tensor as the S-weighted colimit of the diagram $X \colon \mathbf {1} \to \mathbf {E}$ over the terminal category. We can characterize any weighted colimit as a coend of tensors:

We will always be working in cocomplete categories. For a given $\mathbf {C}$ , weighted colimits over $\mathbf {C}$ are then functorial in both the weight and the diagram, giving a bifunctor . This functoriality will be an essential tool. In particular, we will often take a family of weighted colimits over a family of weights.

Notation 5.4 Given a family of weights $W \colon \mathbf {D} \times \mathbf {C}^{\mathrm {op}} \to {\mathbf {Set}}$ and $F \colon \mathbf {C} \to \mathbf {E}$ , we write for the result of calculating the weighted colimit pointwise, that is, .

Remark 5.5 From the characterization in terms of ordinary colimits, it follows that weighted colimits in presheaf categories are computed pointwise. Thus, for $W \colon \mathbf {C}^{\mathrm {op}} \to {\mathbf {Set}}$ and $F \colon \mathbf {C} \times \mathbf {D}^{\mathrm {op}} \to {\mathbf {Set}}$ , we have , where on the left we regard F as a functor $\mathbf {C} \to \mathrm {PSh}({\mathbf {D}})$ .

It follows quickly from the universal property defining weighted colimits that the bifunctor preserves colimits in both arguments. It is therefore determined by its behavior on representable weights, which is simply characterized.

Proposition 5.6 Naturally in $c \in \mathbf {C}$ and $X \colon \mathbf {C} \to \mathbf {E}$ , we have .

Corollary 5.7 Naturally in $W \colon \mathbf {D}^{\mathrm {op}} \to {\mathbf {Set}}$ , $V \colon \mathbf {D} \times \mathbf {C}^{\mathrm {op}} \to {\mathbf {Set}}$ , and $F \colon \mathbf {C} \to \mathbf {E}$ , we have .

Proof By cocontinuity, it suffices to check the case where W is representable.

Notation 5.8 In this section, we use the notation for the hom-bifunctor ${\mathbf {C}}(-,-)$ . Thus, the representable functor for $c \in \mathbf {C}$ , written in our usual notation, may now be written as , while we also have the co-representable . With our notation for parameterized weighted colimits, Proposition 5.6 then tells us that for any $X \in \mathrm {PSh}({\mathbf {C}})$ . We have an analogous equation in the second argument: .

5.1.2 Cellular presentations of presheaves

A central theorem of Reedy theory is the existence of cellular presentations: when $\mathbf {R}$ is a Reedy category, any $\mathbf {R}$ -indexed diagram is a sequential colimit of maps that successively attach cells of increasing degree. Likewise, any natural transformation between $\mathbf {R}$ -indexed diagrams decomposes as a transfinite composite of such maps. In the Riehl–Verity style, the intermediate objects and maps are obtained by taking (Leibniz) weighted colimits of the input diagram. As for any diagram X, one can exhibit a cellular presentation for X by constructing a cellular presentation for and then applying the cocontinuous functor .

For the remainder of this section, we fix a Reedy category $\mathbf {R}$ .

Definition 5.9 For each $n \in \mathbb {N}$ , define to be the subfunctor of arrows of degree less than n.

Definition 5.10 For any $n \in \mathbb {N}$ , write ${\mathbf {R}}[n]$ for the subcategory of $\mathbf {R}$ consisting of objects of degree n and isomorphisms between them. We introduce the following notation for restrictions of where one argument or the other is required to have a given degree:

We similarly introduce notation for the corresponding restrictions of the skeleton bifunctor $\mathrm {sk}_{{<}n}{\mathbf {R}} \colon \mathbf {R}^{\mathrm {op}} \times \mathbf {R} \to {\mathbf {Set}}$ :

Finally, we write and for the restrictions of the inclusion .

Notation 5.11 For $r \in \mathbf {R}$ of degree n, we abbreviate and . Likewise, we write and .

Definition 5.12 For any $f \colon X \to Y$ in $\mathrm {PSh}({\mathbf {R}})$ and $n \in \mathbb {N}$ , the ${<} n$ -skeleton map for f is the Leibniz weighted colimit

We write $\mathrm {sk}_{{<}n}{f} \in \mathrm {PSh}({\mathbf {R}})$ for the domain of this map, which we call the ${<}n$ -skeleton of f; its codomain is Y. For $Y \in \mathrm {PSh}({\mathbf {R}})$ , we write $\mathrm {sk}_{{<}n}{Y}$ for the n-skeleton of the map $0 {\rightarrowtail } Y$ .

Note that the ${<}0$ -skeleton map is . For each $m \le n \in \mathbb {N}$ , the inclusion $\mathrm {sk}_{{<}m}{\mathbf {R}} {\rightarrowtail } \mathrm {sk}_{{<}n}{\mathbf {R}}$ induces a morphism $\mathrm {sk}_{{<}m}{f} \to \mathrm {sk}_{{<}n}{f}$ by functoriality of weighted colimits, and the fact that is the union of the subfunctors $\mathrm {sk}_{{<}n}{\mathbf {R}}$ implies that $Y \cong \operatorname *{\mathrm {colim}}_{n \in \mathbb {N}} \mathrm {sk}_{{<}n}{f}$ . Thus, we have a natural decomposition of f as the transfinite composite $\mathrm {sk}_{{<}0}{f} \to \mathrm {sk}_{{<}1}{f} \to \mathrm {sk}_{{<}2}{f} \to \cdots $ where we may compute $\mathrm {sk}_{{<}n}{f} \cong X \sqcup _{\mathrm {sk}_{{<}n}{X}} \mathrm {sk}_{{<}n}{Y}$ . The chain of skeleta may be further decomposed in terms of latching maps.

Definition 5.13 Given $f \colon X \to Y$ in $\mathrm {PSh}({\mathbf {R}})$ and $r \in \mathbf {R}$ , define the latching map $\widehat {\ell }_{r}{f} \in {\mathbf {Set}}^\to $ for f at r by the Leibniz weighted colimit

The codomain of this map is $Y_r$ ; we write $L_{r}{f}$ for its domain and call this the latching object for f at r.

We write $\widehat {\ell }_{r}{Y}$ and $L_{r}{Y}$ for the latching map and object of $0 {\rightarrowtail } Y$ at r. For general $f \colon X \to Y$ , we can calculate that $L_{r}{f} \cong X_r \sqcup _{L_{r}{X}} L_{r}{Y}$ and $\widehat {\ell }_{r}{f} \cong [ f_r, L_{r}{f} ]$ . It is convenient to have notation for the collected ${\mathbf {R}}[n]$ -sets of latching maps at a given degree.

Definition 5.14 Given $f \colon X \to Y$ and $n \in \mathbb {N}$ , we define the nth latching map of f by . We write $L_{n}{f} \in \mathrm {PSh}({{\mathbf {R}}[n]})$ for its domain and ${f}_{n} \in \mathrm {PSh}({{\mathbf {R}}[n]})$ for its codomain.

These maps are assembled from the latching maps at the individual objects of degree n: we have $(\widehat {\ell }_{n}{f})_r \cong \widehat {\ell }_{r}{f}$ for each $r \in {\mathbf {R}}[n]$ .

We can now exhibit the maps between successive ${<}n$ -skeleta as pushouts of Leibniz weighted colimits of boundary inclusions and latching maps. The induced decomposition of a map f into a sequential colimit of pushouts of basic maps is what we mean by a cellular presentation of f.

Proposition 5.15 [Reference RiehlRie17, Corollary 4.21]

For any $f \colon X \to Y$ and $n \in \mathbb {N}$ , we have a pushout square of the following form:

We refer to the maps as cell maps.

Proof By applying to a pushout square in $\mathbf {R}^{\mathrm {op}} \times \mathbf {R} \to {\mathbf {Set}}$ ; see [Reference RiehlRie17, Theorem 4.15].

Corollary 5.16 Every $f \colon X \to Y$ in $\mathrm {PSh}({\mathbf {R}})$ has a cellular presentation by maps of the form .

For our purposes, namely, working with properties saturated by monomorphisms, it is important to know when the cell maps are monic.

Definition 5.17 A map $f \colon X \to Y$ in $\mathrm {PSh}({\mathbf {R}})$ is a Reedy monomorphism when $\widehat {\ell }_{r}{f}$ is monic in ${\mathbf {Set}}$ for all $r \in \mathbf {R}$ .

Here and in the following, we are specializing the theory of Reedy cofibrations to the (mono, epi) weak factorization system on ${\mathbf {Set}}$ . To see when Reedy monomorphisms have monic cell maps, we use the following lemma. Recall that a map is epi-projective if it has the left lifting property against all epimorphisms.

Proposition 5.18 Let $\mathbf {C}$ be a small category, $f \in [\mathbf {C}^{\mathrm {op}}, {\mathbf {Set}}]^\to $ , and $g \in [\mathbf {C}, {\mathbf {Set}}]^\to $ . If f is epi-projective and g is monic, then is monic.

Proof By [Reference RiehlRie17, Lemma 3.13 and Corollary 3.17] applied to the (mono, epi) weak factorization system on ${\mathbf {Set}}$ .

Lemma 5.19 If isos act freely on lowering maps in $\mathbf {R}$ , then is epi-projective in ${\mathbf {R}}[n] \to {\mathbf {Set}}$ .

Proof A given morphism from r to an object of degree n is either a lowering map or has degree less than n. This induces the following coproduct decomposition in ${\mathbf {R}}[n] \to {\mathbf {Set}}$ :

Since epi-projective is the left class in a weak factorization system, it is stable under cobase change. It thus suffices to show that $\mathbf {R}^-(r, -)$ is epi-projective. Since isos act freely on $\mathbf {R}^-(r, -)$ , it is the left Kan extension along some functor $A \to {\mathbf {R}}[n]$ of some $F \colon A \to {\mathbf {Set}}$ with A a set. Recall that epimorphisms are characterized levelwise in ${\mathbf {Set}}$ -valued functors. By adjoint transposition, it thus suffices to show that F is epi-projective. Since A is a set, this just means that F is levelwise epi-projective. And in ${\mathbf {Set}}$ , every object is epi-projective.

Corollary 5.20 Suppose that isos act freely on lowering maps in $\mathbf {R}$ . Given a Reedy monic $f \in \mathrm {PSh}({\mathbf {R}})^\to $ , the map is monic for all $n \in \mathbb {N}$ .

Proof We have for every $r \in \mathbf {R}$ . We know is epi-projective by Lemma 5.19, and $\widehat {\ell }_{n}{f}$ is monic by assumption, so their Leibniz weighted colimit is monic by Proposition 5.18.

5.1.3 EZ decompositions

The Reedy monomorphisms with initial domain can be characterized more simply: an object X is Reedy monic exactly if every element of X writes uniquely up to isomorphism as a degeneracy of a non-degenerate element of X. We are not aware of a proof of this precise statement (Lemma 5.24) in the literature, though we would be surprised if it were unknown. We use Cisinski’s term “EZ decomposition” [Reference CisinskiCis06, Proposition 8.1.13] for what Berger and Moerdijk call standard decompositions.

Definition 5.21 Let $X \in \mathrm {PSh}({\mathbf {R}})$ . We say that $x \in X_r$ is non-degenerate when every lowering map $e \colon r {\overset {-}\to } s$ admitting an $x' \in X_s$ with $x'e = x$ is an isomorphism. An EZ decomposition of $x \in X_r$ is a pair $(e,x'),$ where $x' \in X_s$ is non-degenerate, $e \colon r \to s$ is a lowering map, and $x = x'e$ . We regard two EZ decompositions $(e_0,x_0')$ and $(e_1,x_1')$ of x as isomorphic when there exists an isomorphism $\theta \colon s_0 \cong s_1$ in $\mathbf {R}$ such that $x_0'\theta = x_1'$ and $e_0 = e_1\theta $ . We say X has unique EZ decompositions when any two EZ decompositions of any element of X are isomorphic.

Remark 5.22 Every element of a presheaf admits at least one EZ decomposition: for any $x \in X_r,$ there exists a minimal $n \in \mathbb {N}$ such that x factors though a lowering map to an object of degree n, and any such factorization is an EZ decomposition.

Proposition 5.23 [Reference Riehl and VerityRV14, Observation 3.23]

Given $X \in \mathrm {PSh}({\mathbf {R}})$ and $r \in \mathbf {R}$ , we have an isomorphism

where $X_- \in \mathrm {PSh}({\mathbf {R}^-})$ is the restriction of X along the Reedy category inclusion $\mathbf {R}^- \to \mathbf {R}$ .

Lemma 5.24 A presheaf $X \in \mathrm {PSh}({\mathbf {R}})$ is Reedy monic if and only if it has unique EZ decompositions.

Proof Suppose that X is Reedy monic. We show that any two EZ decompositions of any $x \in X_r$ are isomorphic by induction on $\lvert r \rvert $ . Let two such factorizations $(e_0,x_0)$ , $(e_1,x_1)$ be given. If either of $e_0$ or $e_1$ is an isomorphism, then the other must be as well, in which case the factorizations are trivially isomorphic; thus, we can assume that each $e_k$ strictly decreases degree. Then, $(e_0,x_0)$ and $(e_1,x_1)$ belong to $L_{r}{X_-}$ ; because X is Reedy monic, they are moreover equal therein. By the concrete characterization of colimits in ${\mathbf {Set}}$ , we have a finite sequence of lowering spans $s_i \overset {f_i}\leftarrow t_i \overset {f^{\prime }_i} \rightarrow s_{i+1}$ for $0 \le i < n$ , always with $\lvert s_i \rvert ,\lvert t_i \rvert < \lvert r \rvert $ , together with elements for each $i \le n$ , such that $y_0 = x_0$ , $y_n = x_1$ , and $y_if_i = y_{i+1}f^{\prime }_i$ :

By taking an EZ decomposition of each $y_i$ and absorbing the lowering map into $f^{\prime }_i,f_{i+1} $ , we can assume without loss of generality that each $y_i$ is non-degenerate. Then, for each i, the equation $y_if_i = y_{i+1}f^{\prime }_i$ makes $(y_i,f_i)$ and $(y_{i+1},f^{\prime }_i)$ EZ decompositions of the same element of $X_{t_i}$ . As $\lvert t_i \rvert < \lvert r \rvert $ , it follows by induction hypothesis that they are isomorphic. Chaining these isomorphisms, we conclude that $(e_0,x_0)$ and $(e_1,x_1)$ are isomorphic.

Now, suppose conversely that X has unique EZ decompositions. By Proposition 5.23, it suffices to show the map $L_{r}{X_-} \to X_r$ is monic. The elements of $L_{r}{X_-}$ are pairs $(e \colon r {\overset {-}\to } s, x \in X_s),$ where e is a strictly lowering map, quotiented by the relation $(fe,x) = (e,xf)$ for any $f \in \mathbf {R}^-$ ; the latching map sends $(e,x)$ to $xe \in X_r$ . Let $(e_0,x_0),(e_1,x_1) \in L_rX_-$ be given such that $x_0e_0 = x_1e_1$ . Without loss of generality, we may assume that these are EZ decompositions, in which case they are isomorphic and thus equal as elements of $L_{r}{X_-}$ .

5.1.4 Saturation by monomorphisms

Now, we check that the class of Reedy monic presheaves is contained in the saturation by monos of the set of automorphism quotients of representables, assuming isos act freely on lowering maps in $\mathbf {R}$ .

Lemma 5.25 For any $X \in \mathrm {PSh}({{\mathbf {R}}[n]})$ , the presheaf is a coproduct of automorphism quotients of representables.

Proof Write ${\mathbf {R}}[n]$ as a coproduct of groups ${\mathbf {R}}[n] \cong \coprod _{i} G_i$ . Using the characterization of orbits as quotients by stabilizer groups, we may decompose X as a coproduct of orbits , where $r_i \in \mathbf {R}$ is the point of $G_i$ . By cocontinuity of , we then have

as desired.

Lemma 5.26 Any colimit of a groupoid of representables in $\mathrm {PSh}({\mathbf {R}})$ is Reedy monic.

Proof Let a groupoid $\mathbf {G}$ and $d \colon \mathbf {G} \to \mathbf {R}$ be given. Set . We show that C has unique EZ decompositions. Let two EZ decompositions $(e_0,x_0)$ and $(e_1,x_1)$ of the same element of C be given. As colimits are computed pointwise, each $x_k$ factors as $x_k= \iota _km_k$ through some leg of the coproduct and we have an arrow $g \colon i_0 \cong i_1$ in $\mathbf {G}$ making the following diagram commute:

Each $m_k$ must be a raising map because $x_k$ is non-degenerate. By uniqueness of Reedy factorizations, we have an isomorphism $\theta \colon s_0 \cong s_1$ fitting in the diagram above.

Theorem 5.27 Let $\mathbf {R}$ be a Reedy category in which isos act freely on lowering maps. Let $\mathcal {P} \subseteq \mathrm {PSh}({\mathbf {R}})$ be a class of objects such that

• • for any $r \in \mathbf {R}$ and $H \le \mathrm {Aut}_{\mathbf {R}}{(r)}$ , we have ;

• • $\mathcal {P}$ is saturated by monomorphisms.

Then, $\mathcal {P}$ contains every Reedy monic presheaf.

Proof First, we show by induction on n that $\mathrm {sk}_{{<}n}{X} \in \mathcal {P}$ for any Reedy monic presheaf X. It then follows that $X \cong \operatorname *{\mathrm {colim}}_{n \in \mathbb {N}} \mathrm {sk}_{{<}n}{X} \in \mathcal {P}$ by saturation.

In the base case, $\mathrm {sk}_{{<}0}{X}$ is the empty coproduct and thus belongs to $\mathcal {P}$ by saturation. For any $n \in \mathbb {N}$ , we have the following pushout square by Proposition 5.15:

The upper horizontal map is monic by Corollary 5.20, the lower by closure of monos in $\mathrm {PSh}({\mathbf {R}})$ under cobase change. We have $\mathrm {sk}_{{<}n}{X} \in \mathcal {P}$ by induction hypothesis. The upper-right corner is , which belongs to $\mathcal {P}$ by Lemma 5.25. Finally, the upper-left corner is by definition the following pushout object:

The upper horizontal map is monic by Proposition 5.18 and Lemma 5.19, as we can write it as the pushout product . The object is in $\mathcal {P}$ by Lemma 5.25. Using Corollary 5.7, we have

for any F. The objects and thus belong to $\mathcal {P}$ by Lemmas 5.25 and 5.26 and the induction hypothesis. By saturation, the upper-left corner of our original pushout diagram now belongs to $\mathcal {P}$ . For the same reason, we conclude that $\mathrm {sk}_{{<}n+1}{X}$ belongs to $\mathcal {P}$ .

5.2 Pre-elegant Reedy categories

We next consider the subclass of Reedy categories in which any span of lowering maps has a pushout. This restriction has some simplifying consequences (e.g., that all lowering maps are epic), and we can characterize the Reedy monic presheaves over such categories as those preserving lowering pushouts.

Intuitively, this means that any pair of lowering maps from the same object has a universal combination, the diagonal of their pushout. Of course, any elegant Reedy category is pre-elegant, so $\Delta $ is one example. Our motivating example is the (surjective, mono) Reedy structure on the category of finite inhabited semilattices, which is pre-elegant but not elegant. In Section 6, we see this is an instance of a general class of examples: the (surjective, mono) Reedy structure on the category $\mathrm {Alg}({\mathbf {T}})_{\mathrm {fin}}$ of finite algebras for a Lawvere theory $\mathbf {T}$ is always pre-elegant, but not necessarily elegant.

The following lemma generalizes the fact that any lowering map in an elegant Reedy category is split epic, with essentially the same proof as Bergner and Rezk’s Proposition 3.8(3) [Reference Bergner and RezkBR13].

Proof Consider a lowering map $e \colon r {\overset {-}\to } s$ . We take the pushout of e with itself, then use its universal property to see that the legs of the pushout are split monic:

Any split mono is a raising map (Corollary 2.15), so $f_0,f_1$ are isomorphisms. Thus, e is epic.

Proof Let a pushout square of lowering maps be given like so:

Suppose we have $x_0 \in X_{s_0}$ and $x_1 \in X_{s_1}$ such that $x_0e_0 = x_1e_1$ ; we show this data determines a unique element of $X_t$ restricting to $x_0$ and $x_1$ . For each $k \in \{0,1\}$ , take an EZ decomposition $(g_k,y_k)$ of $x_k$ . Then, $(g_0e_0,y_0)$ and $(g_1e_1,y_1)$ are EZ decompositions of the same map, so by Lemma 5.24, they are isomorphic via some $\theta \colon u_0 \cong u_1$ . The universal property of the pushout in $\mathbf {R}$ then provides a map $h_1 \colon t \to u_1$ like so:

This gives our desired element $y_1h_1 \in X_t$ restricting to $x_k$ along each $f_k$ . Note that $h_1$ is a lowering map by Lemma 2.14.

To see that this element is unique, suppose we have $x \in X_t$ such that $xf_k = x_k$ for $k \in \{0,1\}$ . Take an EZ decomposition $(h,y)$ of X, say through $u \in \mathbf {R}$ . By uniqueness of EZ decompositions, we have isomorphisms $\psi _k$ as shown:

Because isos act freely on lowering maps, we have $\psi _1^{-1}\psi _0 = \theta $ . It follows from the universal property of the pushout in $\mathbf {R}$ that $\psi _1h = h_1$ , thus that $yh = y_1h_1$ as desired.