2.1 Monoidal categories and restricted decompositions

The constructions above will not cover all the structures we would like to study. For example, consider the case of a nontrivial finitely generated free Abelian group A. Its poset of subgroups is a lattice of infinite height. Therefore, we cannot talk about decompositions given the finite-dimensional requirement in Definition 2.1. Nevertheless, it is implicit here that decompositions should be sets of subgroups $A_1,\ldots , A_r$ which span a Boolean lattice inside the poset of subgroups with join $A_1\oplus \cdots \oplus A_r = \left \langle A_1,\ldots ,A_r\right \rangle = A$ . Thus, if we instead consider the subposet of complemented subgroups of A (or summands), this is a bounded poset of finite height. However, there is still a problem in this subposet: even if $\{A_1,A_2\}$ is a (weak) decomposition in the poset sense, it is not clear that $A_1\vee A_2$ coincides with the direct sum $A_1\oplus A_2$ . For example, if $A = {\mathbb {Z}}^2$ , then $A_1=\left \langle (1,2)\right \rangle $ and $A_2=\left \langle (1,0)\right \rangle $ are summands of A such that $A_1\wedge A_2 = 0$ and $A_1\vee A_2 = A$ in the poset of summands of A, but $A_1+A_2 = A_1\oplus A_2 < A$ .

A similar situation arises if we consider instead a free group of rank $\geq 2$ . By a decomposition here, we want not only the Boolean lattice condition but also ‘free’ spans. That is, if F is a free group of rank $\geq 2$ and $H,K$ are two subgroups, then $H,K$ is a weak decomposition if $H\cap K = 1$ and $H\vee K = \left \langle H,K\right \rangle =F$ . But the span $\left \langle H,K\right \rangle $ forgets the relation between the elements of H and K, and we may actually want $\left \langle H,K\right \rangle $ to be equal to the free product of H and K. An adaptation of the Abelian-case example with ${\mathbb {Z}}^2$ shows that there are free factors $H,K$ of the free group $F_2$ of rank $2$ such that $H\cap K = 1$ but $H\vee K = \left \langle H,K\right \rangle = F$ is not the free product of H and K.

To cover these and other cases, we extend our purely combinatorial setting by using concepts from category theory. We start with some notation.

For a category ${\mathcal C}$ and an object $X\in {\mathcal C}$ , write $\operatorname {\mathrm {Hom}}_{\mathcal C}(Y,X)$ for the homomorphisms from an object Y to X, and $\operatorname {\mathrm {Aut}}_{\mathcal C}(X)$ for the isomorphisms of X. Let ${\mathcal C} \downarrow X$ be the comma category with objects $(Y,f)$ , where $Y \in {\mathcal C}$ and $f \in \operatorname {\mathrm {Hom}}_{{\mathcal C}}(Y,X)$ , and morphisms between $(Y,f)$ and $(Z,g)$ being $h \in \operatorname {\mathrm {Hom}}_{{\mathcal C}}(Y,Z)$ with $f = gh$ . We consider the subcategory with objects $(Y,i)$ such that i is a monomorphism, and morphisms inherited from ${\mathcal C} \downarrow X$ . Note that a morphism $h : (Y,i) \rightarrow (Z,j)$ in comes indeed from a monomorphism $h:Y\to Z$ .

A subobject of X is an isomorphism class of objects in , where $(Y,i)$ and $(Z,j)$ are isomorphic if and only if there exists an isomorphism $\phi :Y\to Z$ such that $i = j\phi $ . We write $[(Y,i)]$ for the subobject represented by $(Y,i)$ . Let ${\mathcal {S}}(X)$ denote the collection of subobjects of X, which is a poset-category (that is, there is at most one arrow between any two objects, and this defines an anti-symmetric relation). We gather some properties in the following lemma.

Note that item 5 also implies that if $i,i':Z\to X$ are two monomorphisms and $z:=[(Z,i)]$ , $z':=[(Z,i')]$ , then ${\mathcal {S}}(X)_{\leq z}\cong {\mathcal {S}}(Z) \cong {\mathcal {S}}(X)_{\leq z'}$ . In particular, if $i:X\to X$ is a monomorphism, then ${\mathcal {S}}(X)_{\leq [(X,i)]} \cong {\mathcal {S}}(X)$ .

We want to restrict the collection of decompositions to those that have a particular shape. To specify the ‘shape’ of the decompositions, we define beforehand a ‘product of objects’, and then we select the decompositions whose join equals the product. This naturally leads us to the definition of a monoidal product in our category. However, we will require this product to have some particular properties closer to a coproduct.

The notation ‘ $\sqcup $ ’ suggests a similarity with the coproduct. Indeed, sometimes $\sqcup $ will be the coproduct of the category. However, in many cases, the category ${\mathcal C}$ may not have all coproducts, and so $\sqcup $ will serve as a remedy.

Remark 2.13. Note that, for any two objects X, Y, we have a canonical map $\iota _X:X\to X\sqcup Y$ arising from

$$\begin{align*}X \cong X \sqcup 0 \overset{\operatorname{\mathrm{id}}_X \sqcup i_0}{\longrightarrow}X\sqcup Y,\end{align*}$$

where $i_0:0\to Y$ .

To define decompositions in ${\mathcal {S}}(X)$ , we need finite height, and ${\mathcal {S}}(X)$ may have infinite dimension in general (for example, when X is a nontrivial free group in the category of groups, see Example 2.18). For that reason, we will work with the subposet ${\mathcal {S}}(X,\sqcup )$ of $\sqcup $ -complemented subobjects defined below, which we expect to have finite height and whose elements are eligible to be part of decompositions. To that end, we introduce first the notion of $\sqcup $ -compatible joins of elements.

We now define posets of subobjects compatible with the monoidal product $\sqcup $ :

Note that by definition, ${\mathcal {S}}(X,\sqcup )$ is bounded with maximum X and minimum $0$ (the initial object), and all its elements are $\sqcup $ -complemented, and in particular complemented. Also, ${\mathcal {PD}}(X,\sqcup )$ is bounded with maximum $\{X\}$ and minimum $\emptyset $ . Moreover, ${\mathcal D}(X,\sqcup )$ is a nonempty order ideal of ${\mathcal {PD}}(X,\sqcup )$ and, in particular, $\{X\}$ is the unique maximal element of ${\mathcal D}(X,\sqcup )$ .

In view of the previous example and Example 2.10, we propose the following question:

4.1 Frames and partial bases

The following property follows immediately from the definition.

Now we use frames to recover the idea of bases. We propose the following definition for the partial basis complexes, based on inflation complexes.

If each $P_{x}$ is finite, then $(K,P)$ is an inflation of K in the sense of [Reference Björner, Wachs and WelkerBWW, Section 6]. We will see that results analogous to those from [Reference Björner, Wachs and WelkerBWW] hold in the general situation, where the sets $P_x$ are not necessarily finite or have the same size.

Intuitively, we think of $P_x$ as a set of suitable bases of the atom x. For example, if x represents a $1$ -dimensional vector space in the subspace poset, then $P_x$ can represent the set of nonzero vectors in x. If x represents a $1$ -dimensional nondegenerate subspace in the nondegenerate subspace poset of a unitary space, we can take $P_x$ to be the set of normal vectors $v\in x$ .

To simplify the notation, we will usually drop the P from the notation and leave the collection implicit. When the stated results strongly depend on the collection P, we will use the notation again.

4.2 Ordered versions

We define the ordered versions of the posets attached to ${\mathcal {S}}(X,\sqcup )$ .

Definition 4.5. Let ${\mathcal T}$ be one of the symbols ${\mathcal D}, {\mathcal {PD}}, {\mathcal F}, {\mathcal {PF}}, \mathcal {B},\mathcal {PB}$ . The ordered poset $\mathcal {O}{\mathcal T}(X,\sqcup )$ consists of distinct-element tuples $(x_1,\ldots ,x_r)$ , $r\geq 0$ , such that $\{x_1,\ldots ,x_r\}\in {\mathcal T}(X,\sqcup )$ . The order in $\mathcal {O}{\mathcal T}(X,\sqcup )$ is given by order-preserving refinement; that is,

$$\begin{align*}(x_1,\ldots,x_r)\leq (y_1,\ldots, y_s) \quad \Leftrightarrow \quad \forall \ 1\leq i\leq j\leq r, \ \exists \ k\leq l \mathrel{{\: : \: }} x_i\leq y_k \text{ and } x_j\leq y_l. \end{align*}$$

We denote by $F_{\mathcal T}:\mathcal {O}{\mathcal T}(X,\sqcup )\to {\mathcal T}(X,\sqcup )$ the forgetful poset map.

The following example gives us some idea of the structure of the ordered versions:

Example 4.7. Let $X = \mathbb {F}_{2}^2$ be the $2$ -dimensional vector space over the field with two elements. For the monoidal product, we take $\sqcup =\oplus $ , the direct sum in the category of vector spaces over $\mathbb {F}_{2}$ . There are exactly three nontrivial proper subspaces, which we may denote by $U_1, U_2, U_{3}$ . Then ${\mathcal {S}}(X,\sqcup )={\mathcal {S}}(V)$ , the poset of subspaces of V, has the following structure:

From the Hasse diagram of ${\mathcal {S}}(V)$ , we see that there are three full decompositions

$$\begin{align*}\{U_1,U_2\}, \ \{U_1, U_{3}\}, \ \{U_2, U_{3}\}.\end{align*}$$

From this, one easily determines the partial decomposition poset ${\mathcal {PD}}(X,\sqcup )={\mathcal {PD}}(V)$ and its ordered version. See Figure 5.

Figure 5

Poset of subspaces of $V = \mathbb {F}_{2}^2$ (top left), poset of partial decompositions of V (top right) and poset of ordered partial decompositions of V (bottom). Here, $U_1 = \left \langle (1,0)\right \rangle $ , $U_2 = \left \langle (1,1)\right \rangle $ , and $U_3 = \left \langle (0,1)\right \rangle $ .

Example 4.8. Let $X = \{1,\ldots , n\}$ be in the category of sets with $\sqcup = \coprod $ the disjoint union. Then ${\mathcal {S}}(X,\sqcup )$ is the Boolean lattice, ${\mathcal D}(X,\sqcup ) = \Pi (n)$ is the partition lattice, and ${\mathcal {PD}}(X,\sqcup )$ is the partial partition lattice. The ordered partial partition poset ${{\mathcal O}}{\mathcal {PD}}(X,\sqcup )$ for the case $n=3$ is given in Figure 6. The blue subposet shows the ordered partition poset ${{\mathcal O}}{\mathcal D}(X,\sqcup )$ .

Figure 6

Ordered partial and full partition poset.

Proposition 1.4 of [Reference Billera and SarangarajanBS96] states that the ordered partition lattice is the face lattice of the permutohedron $P_n$ of order n.

Since we aim to relate later the homotopy type of $\mathcal {O} {\mathcal T}(X,\sqcup )$ with that of ${\mathcal T}(X,\sqcup )$ , in the following lemmas, we take a closer look at the fibers by the canonical forgetful maps and the intervals in the ordered versions.

Proof. Since ${\mathcal D}(X,\sqcup )_{\geq \sigma }$ is the partition lattice on the set $\sigma $ by Lemma 2.5, it is clear that $F_{\mathcal D}^{-1}( {\mathcal D}(X,\sqcup )_{\geq \sigma } )$ is the ordered partition lattice on the set $\sigma $ . Now, by Proposition 1.4 of [Reference Billera and SarangarajanBS96], this is the face lattice of the permutahedron of order $|\sigma |$ . The second part of item 1 now follows since the order complex of the face lattice of a polytope is a sphere.

Item 3 follows similarly, and the shellability is implied by Theorem 6.1 of [Reference Björner and WachsBW].

For item 2, we construct a homotopy between the identity and a constant map. Since $\sigma $ is a nonempty $\sqcup $ -partial decomposition (not full), there exists $x = \bigvee _{y\in \sigma } y$ , and this is different from $1$ and $0$ . Let $\tau \in F^{-1}_{{\mathcal {PD}}}( ({\mathcal {PD}}(X,\sqcup )^{*})_{\geq \sigma })$ , and write $\tau = (z_1,\ldots ,z_s)$ . Define

$$\begin{align*}z_i' = \bigvee_{y\in \sigma \mathrel{{\: : \: }} y\leq z_i} y\end{align*}$$

and let $r(\tau )$ be the tuple $(z_1',\ldots ,z_s')$ after removing the $0$ elements. We see that $F_{{\mathcal {PD}}}(r(\tau ))$ is a $\sqcup $ -partial decomposition obtained by joining elements of $\sigma $ , so $F_{{\mathcal {PD}}}(r(\tau ))\geq \sigma $ . Thus, we have well-defined order-preserving maps

$$\begin{align*}\tau \geq r(\tau) \leq (x).\end{align*}$$

This implies that $F^{-1}_{{\mathcal {PD}}}( ({\mathcal {PD}}(X,\sqcup )^{*})_{\geq \sigma })$ is contractible.

We describe now some intervals in the ordered versions. Recall property (LI) from Definition 3.2. Here, $|z|$ denotes the size of an ordered tuple z – that is, the number of (distinct) elements of the tuple.

Finally, we establish a connection between the ordered version of the poset of decompositions and the subposet of $(\sqcup ,h)$ -complemented subobjects.

Definition 4.11. Let $G:\mathcal {O}{\mathcal {PD}}(X,\sqcup )^{*} \to {(\Delta {\mathcal {S}}_h(X,\sqcup )^{*})}^{\mathrm {op}} \cup \{\emptyset \}$ be the function

$$\begin{align*}G(x_1,\ldots,x_r) = \{x_1 < x_1\vee x_2 < \cdots < x_1\vee\cdots \vee x_{r-1} \}.\end{align*}$$

It is clear that if $\{x_1,\ldots ,x_r\}$ is a $\sqcup $ -partial decomposition, then any join of its elements is $(\sqcup ,h)$ -complemented, and hence belongs to ${\mathcal {S}}_h(X,\sqcup )$ . Thus, G is a well-defined function. However, G is not order-preserving for partial decompositions: for example, if $x\leq y$ and $\{x,z\}, \{y,z\}$ are $\sqcup $ -partial decompositions, then $(x,z)\leq (y,z)$ but $G(x,z) = \{x\}\not \subseteq \{y\} = G(y,z) $ . Nevertheless, this is a poset map when restricted to decompositions:

Proof. This follows since elements of $\mathcal {O}{\mathcal D}(X,\sqcup )_{>\tau }$ are obtained by joining adjacent blocks of an ordered decomposition $\tau $ . That is, if $\sigma>\tau $ and $\tau = (x_1,\ldots ,x_r)$ , we can write

$$\begin{align*}\sigma = \bigg( {\bigvee_{i=1}^{t_1}}x_i, \bigvee_{i=t_1+1}^{t_2} x_i,\ldots, \bigvee_{i=t_{s-1}+1}^{r}x_i\bigg)\end{align*}$$

for suitable $1 \leq t_1 < t_2 < \cdots < t_s = r$ . Thus, $G(\tau )$ contains $G(\sigma )$ .

We will see later that this map turns out to be an isomorphism under unique complementation and suitable extension properties. See Lemma 5.17 and Lemma 5.18.

4.3 The augmented Bergman complex

We introduce now a (simplified) version of the augmented Bergman complex in our context. This complex was first defined for matroids, and it gained substantial attention due to the work [Reference Braden, Huh, Matherne, Proudfoot and WangBHMPW] where its Stanley-Reisner ring is a key object in the proofs of the Top-Heavy conjecture (raised by Dowling and Wilson), and the non-negativity of the coefficients of Kazhdan-Lusztig polynomials for all matroids. Its topology in the matroid case was carefully analyzed in [Reference Bullock, Kelley, Reiner, Ren, Shemy, Shen, Sun and ZhangBKR].

It is a direct consequence of the definition that the augmented Bergman complex $\Delta (X,\sqcup )$ is indeed a simplicial complex. Note that if $\sigma \in {\mathcal F}(X,\sqcup )$ is a full frame, then $\sigma \in \Delta (X,\sqcup )$ is a maximal simplex. Also note that $0 \in {\mathcal {S}}(X,\sqcup )$ is a vertex of $\Delta (X,\sqcup )$ , and a simplex containing $0$ must be a flag of ${\mathcal {S}}(X,\sqcup )^{\circ }$ .

4.4 The Charney poset

A different version of the ordered decompositions was introduced by R. Charney in [Reference CharneyC]. There she works with the poset of summands of a finitely generated free module over a Dedekind domain, and defines a poset whose elements are ordered direct sum decompositions of size two with a zigzag ordering. Another version of Charney’s poset appears in the context of split buildings for spaces with a certain form, and we interpret decompositions of size two there as direct sum decomposition pairs $(V_1,V_2)$ such that $V_1$ and the orthogonal complement of $V_2$ are totally isotropic subspaces (see [Reference Lehrer and RylandsLR]).

We adapt the definition of this poset to our context.

Definition 4.14. The Charney poset associated with X is the poset $\mathcal {G}(X,\sqcup )$ whose underlying set consists of ordered decompositions of size two:

$$\begin{align*}\mathcal{G}(X,\sqcup) = \{ z \in {\mathcal{OD}}(X,\sqcup) \mathrel{{\: : \: }} |z| = 2\},\end{align*}$$

and the ordering is given by

$$\begin{align*}(x,y)\leq (x',y') \quad \text{ if } \quad x\leq x' \text{ and } y\geq y'.\end{align*}$$

The following proposition relates the ordered decompositions to the Charney complex.

Proposition 4.15. Let $\beta $ be the function

$$\begin{align*}\beta :\left\{ \begin{array}{ccc} {{\mathcal{OD}}(X,\sqcup)^{\circ}}^{\mathrm{op}} & \to & \Delta \mathcal{G}(X,\sqcup)\\ \tau=(z_0,z_1,\ldots,z_r) & \mapsto & \{ \big(\bigvee_{i=0}^j z_i, \bigvee_{k=j+1}^r z_k\big) \mathrel{{\: : \: }} 0\leq j <r \} \end{array} \right.\end{align*}$$

Then $\beta $ is a rank-preserving poset embedding whose image is downward closed.

Moreover, if properties (EX) and (CM) from Definition 3.10 hold, then $\beta $ is surjective and therefore an isomorphism.

Proof. It is clear that $\beta $ is a well-defined, order-preserving and rank-preserving map, and also injective: if $\beta (z_0,\ldots ,z_r) = \{ (x_0,y_0) < \cdots < (x_{r-1}, y_{r-1}) \}$ , then

(1) $$ \begin{align} z_j = (\bigvee_{i=0}^j z_i\big) \wedge \big( \bigvee_{k=j}^r z_k\big) = x_j \wedge y_{j-1}. \end{align} $$

Note that this meet always exists by definition of a decomposition. By convention, we set $x_{-1}=0, y_{-1}=1$ and $x_r=1,y_r=0$ .

We show now that $\beta $ is an embedding. Suppose that $\beta (\tau )\supseteq \beta (\sigma )$ with

$$\begin{align*}\beta(\tau) = \beta(z_0,\ldots,z_r) = \{ (x_0,y_0) < \cdots < (x_{r-1}, y_{r-1}) \}, \end{align*}$$

$$\begin{align*}\beta(\sigma) = \beta(w_0,\ldots,w_s) = \{ (u_0,v_0) < \cdots < (u_{s-1}, v_{s-1}) \}.\end{align*}$$

For $0\leq i\leq s-1$ , we take the unique index $0\leq j_i\leq r-1$ such that $(u_i,v_i) = (x_{j_i},y_{j_i})$ . Since $(u_{-1},v_{-1}) = (x_{-1},y_{-1})$ and $(u_s,v_s) = (x_r,y_r)$ , we set $j_{-1} = -1$ and $j_s = r$ . It is clear that $(j_i)_{i=-1}^{s}$ is a strictly monotone increasing sequence. Then, for all $0\leq i\leq s$ , we have

$$\begin{align*}w_i = u_i\wedge v_{i-1} = x_{j_i}\wedge y_{j_{i-1}} = (\bigvee_{k=0}^{j_i} z_k\big) \wedge \big( \bigvee_{k=j_{i-1}+1}^r z_k\big) = \bigvee_{k=j_{i-1}+1}^{j_i}z_k.\end{align*}$$

This implies that $\tau \leq \sigma $ , so $\beta $ is a poset embedding. It is also clear that if $\omega \subseteq \beta (z)$ , then $\omega \in \mathrm {Im}(\beta )$ .

Finally, we show that $\beta $ is surjective under properties (EX) and (CM). We argue by induction on the size r of a chain $s \in \Delta \mathcal {G}(X,\sqcup )$ and in view of (1), we show that if

$$\begin{align*}s = \{ (x_0,y_0) < \cdots < (x_r,y_r) \}, \end{align*}$$

then $y_{i-1}\wedge x_i$ exists for all $1\leq i\leq r$ and, if we set $y_{-1} = 1$ and $x_{r+1} = 1$ , then $\{x_i \wedge y_{i-1} \mathrel {{\: : \: }} 0\leq i\leq r+1\}\in {\mathcal D}(X,\sqcup )$ .

The case $r = 0$ is trivial. Suppose then that $r> 0$ and that $s = \{ (x_0,y_0) < \cdots < (x_r,y_r)\}$ . By induction, we have a $\sqcup $ -decomposition

$$\begin{align*}\sigma = \{ x_0, x_0\wedge y_1,\ldots, x_{r-1}\wedge y_{r-2}, y_{r-1}\}\in {\mathcal D}(X,\sqcup).\end{align*}$$

By property (CM) applied to $(x_{r-1},y_{r-1}) < (x_r,y_r)$ , we have $\tau := \{x_r\wedge y_{r-1}, x_{r-1}, y_{r}\}\in {\mathcal D}(X,\sqcup )$ . Hence, $\{x_r\wedge y_{r-1},y_r\}\in {\mathcal D}( {\mathcal {S}}(X,\sqcup )_{\leq y_{r-1}})$ by Remark 3.11 and it is $\sqcup $ -compatible in ${\mathcal {S}}(X,\sqcup )$ , so by property (EX), we have

$$\begin{align*}\sigma' := \left( \sigma \setminus \{y_{r-1}\} \right) \cup \tau \in {\mathcal D}(X,\sqcup).\end{align*}$$

But note that

$$\begin{align*}\sigma' = \{x_0, x_1\wedge y_0,\ldots, x_{r-1}\wedge y_{r-2}, x_r\wedge y_{r-1}, y_r\} = \{ x_i\wedge y_{i-1} \mathrel{{\: : \: }} 0\leq i\leq r+1\}.\end{align*}$$

This concludes the induction, and thus, $\beta $ is surjective since

$$\begin{align*}\beta(x_0, x_1\wedge y_0,\ldots, x_r\wedge y_{r-1}, y_r) = \{ (x_0,y_0) < \cdots < (x_r,y_r) \}.\\[-36pt] \end{align*}$$

We summarize the non-ordered constructions in Table 2.

5.1 General homotopy formulas

In this section, we first recall some known facts about homotopy types of posets and then apply them to the posets we defined in the previous sections. When writing down explicit formulas for the homotopy type, we identify the poset with its order complex. For example, ${\mathcal T} \simeq {{\mathcal S}} \vee {{\mathcal S}}$ means that the geometric realization of the order complex of ${\mathcal T}$ is homotopy equivalent to a wedge sum of two copies of the geometric realization of the order complex of ${{\mathcal S}}$ . If we need to emphasize the topological spaces behind our results and constructions, we write $|{{\mathcal S}}|$ for the geometric realization of the order complex of a poset  ${{\mathcal S}}$ .

Assume that ${{\mathcal S}}$ is a poset of finite height n. We say that ${{\mathcal S}}$ is spherical if it is $(n-1)$ -connected or, equivalently, homotopy equivalent to a wedge of n-spheres. The poset ${{\mathcal S}}$ is Cohen-Macaulay if ${{\mathcal S}}$ is spherical and for all $x,y\in {{\mathcal S}}$ , the intervals ${{\mathcal S}}_{>x}$ , ${{\mathcal S}}_{<y}$ and ${{\mathcal S}}_{> x} \cap {{\mathcal S}}_{< y}$ (if $x<y$ ) are spherical of height $n-h(x)-1$ , $h(y)-1$ and $h(y)-h(x)-2$ , respectively. This is the classical notion of homotopically Cohen-Macaulayness of a poset or simplicial complex. This implies Cohen-Macaulayness over any field, which is defined by the corresponding high homological connectedness.

Recall that if $f,g:{\mathcal T}\to {{\mathcal S}}$ are order-preserving maps such that $f(x)\leq g(x)$ for all $x\in {\mathcal T}$ , then f and g are homotopy equivalent. We will repeatedly use this fact throughout this paper to obtain homotopy equivalences and establish the contractibility of some posets.

The first tool we will need is a consequence of Propositions 3.1, 3.5 and 3.7 from [Reference Welker, Ziegler and ŽivaljevićWZZ]. Note that in [Reference Welker, Ziegler and ŽivaljevićWZZ], Proposition 3.1 was formulated only for diagrams over finite posets, but the proof only relies on paracompactness of the spaces involved (see, for example, [Reference SegalSe]). As a consequence, the results from [Reference Welker, Ziegler and ŽivaljevićWZZ] can be applied to finite height posets.

Proposition 5.1. Let ${{\mathcal S}},{\mathcal T}$ be posets of finite height and $f:{\mathcal T}\to {{\mathcal S}}$ a poset map such that for all $x\in {{\mathcal S}}$ , the inclusion $f^{-1}\big (\,{{\mathcal S}}_{<x}\,\big ) \hookrightarrow f^{-1}\big (\,{{\mathcal S}}_{\leq x}\,\big )$ is homotopy equivalent to the constant map with image $c_x\in f^{-1}({{\mathcal S}}_{\leq x})$ . Then there is a homotopy equivalence

$$\begin{align*}{\mathcal T} \simeq {{\mathcal S}} \vee \bigvee_{x\in {{\mathcal S}}} f^{-1}\big(\,{{\mathcal S}}_{\leq x}\,\big) * {{\mathcal S}}_{>x},\end{align*}$$

where the wedge identifies $x\in {{\mathcal S}}$ with $c_x\in f^{-1}({{\mathcal S}}_{\leq x})$ .

The following well-known fact will be useful for verifying the requirements of Proposition 5.1.

Then from Proposition 5.1 and Lemma 5.2, we can derive the following version of Quillen’s fiber theorem, which we will use repeatedly.

We will also use the non-Hausdorff mapping cylinder (see [Reference Barmak and MinianBM]) of a map $f:{\mathcal T}\to {{\mathcal S}}$ between posets. This is the poset $M_f = {\mathcal T}\coprod {{\mathcal S}}$ such that its ordering $\leq $ restricts to the given ordering in ${\mathcal T}$ and ${{\mathcal S}}$ , and for $x\in {\mathcal T}$ and $y\in {{\mathcal S}}$ , we have $x < y$ if $f(x)\leq y$ . Note that if ${{\mathcal S}}$ has a unique maximal element $1_{{{\mathcal S}}}$ , then this is also the unique maximal element of $M_f$ , and we can write $M_f^{\circ } = M_f \setminus \{1_{{{\mathcal S}}}\}$ .

Example 5.4. Let ${\mathcal T}$ be the poset with points $(0,0),(0,1),(1,0),(1,1)$ , with $(a,b)\leq (c,d)$ if and only if $a\leq c$ . Let ${{\mathcal S}}$ be the poset $\{0 < 1\}$ , and consider the projection $f:{\mathcal T}\to {{\mathcal S}}$ defined by $f(a,b)=a$ . Then the non-Hausdorff mapping cylinder of f is the poset displayed in Figure 7.

Figure 7

Hasse diagram of the non-Hausdorff mapping cylinder of the poset map $f:{\mathcal T}\to {{\mathcal S}}$ .

The adjective non-Hausdorff is motivated by the non-Hausdorffness of the construction when considering finite posets as finite topological spaces. We refer the reader to [Reference Barmak and MinianBM] for a more detailed discussion.

Lemma 5.5. Let $f:{\mathcal T}\to {{\mathcal S}}$ be a map of posets of finite height, where ${{\mathcal S}}$ is bounded. If $M := \{ x \in {\mathcal T}~|~f(x) = 1_{{{\mathcal S}}}\}$ is an antichain, then $M_f^{\circ } = M_f \setminus \{1_{{{\mathcal S}}}\}$ is homotopy equivalent to

$$ \begin{align*}\bigvee_{x \in M} \Sigma {\mathcal T}_{< x}.\end{align*} $$

Let us apply Lemma 5.5 to the map f from Example 5.4.

Example 5.6. Let ${\mathcal T}$ , ${{\mathcal S}}$ and f be as in Example 5.4. Note that ${{\mathcal S}}$ is bounded. Following the notation of Lemma 5.5, we have that $M = \{ (1,0), (1,1)\}$ is an antichain. Thus, $M_f^{\circ }$ is the poset obtained from ${\mathcal T}$ after adding an extra element $0$ above $(0,0)$ and $(0,1)$ . The order complex of this poset is depicted in Figure 8, which clearly has the homotopy type of $S^1\vee S^1$ . This coincides with the description given in Lemma 5.5.

Figure 8

Geometric realization of the poset $M_f^{\circ }$ from Example 5.4.

In general, we also have the following formula relating the Euler characteristic of posets related by an order-preserving map (see, for example, Corollary 3.2 of [Reference WalkerWal81]).

Lemma 5.7. Let $f:{\mathcal T}\to {{\mathcal S}}$ be a map of finite posets. Then

$$\begin{align*}\tilde{\chi}({\mathcal T}) = \tilde{\chi}({{\mathcal S}}) - \sum_{x\in {{\mathcal S}}} \tilde{\chi}(f^{-1}({{\mathcal S}}_{\leq x}))\tilde{\chi}({{\mathcal S}}_{>x}).\end{align*}$$

5.2 Homotopy formulas and results for posets from Section 2 - Section 4

Our first homotopy equivalence relates the strictly partial decomposition with the subposet of $(\sqcup ,h)$ -complemented subobjects. We refer the reader to Definition 3.7 for the different notions of complementation employed here, as well as for the definition of the subposet ${\mathcal {S}}_h(X,\sqcup )$ . Recall that $\Phi (\sigma ) = \bigvee _{x\in \sigma } x$ for a partial decomposition $\sigma \in {\mathcal {PD}}(X,\sqcup )$ .

The main applications of the results in this subsection can be found in Section 6. Here, we will give just some small examples.

Now we apply the wedge-decomposition result to relate the ordered and unordered versions of the posets ${\mathcal D},{\mathcal {PD}},{\mathcal F},{\mathcal {PF}}, \mathcal {B}, \mathcal {PB}$ . Recall the definition of $\mathcal {O} K$ for a simplicial complex K from Remark 4.6. We will sometimes write CM for Cohen-Macaulay (do not confuse with property (CM) from Definition 3.10).

We study now natural maps between our posets that might be useful to compute the homotopy type relations between them. We begin with the relation between the complex of partial bases $\mathcal {B}(X,\sqcup ,P)$ and the frame complex. Since these are inflation complexes in the sense of [Reference Björner, Wachs and WelkerBWW] (see Definition 4.3), by their Theorem 6.2 adapted to the infinite situation, we can prove:

Proposition 5.10. Let K be a finite-dimensional simplicial complex and $P = (P_x)_{x\in \mathcal {A}}$ a collection of nonempty sets $P_x$ indexed by the set of vertices $\mathcal {A}$ of K. Let $p:(K,P)\to K$ be the deflation map. Then we have the following homotopy equivalence:

$$\begin{align*}(K,P) \simeq K \vee \bigvee_{\sigma\in K} \bigg( \bigvee_{\prod_{x\in \sigma} |P_x|-1} S^{|\sigma|-1} \bigg) * \operatorname{\mathrm{Lk}}_K(\sigma).\end{align*}$$

Furthermore, $p^{-1}(K_{\leq \sigma }) = P_{x_1} * \cdots * P_{x_r}$ for $\sigma = \{x_1,\ldots , x_r\}\in K$ (where the $P_x$ are regarded as discrete complexes), and K is homotopically CM if and only if $(K,P)$ is homotopically CM. The same conclusions hold for Cohen-Macaulayness over a ring.

In particular, this applies to $K = {\mathcal {PF}}(X,\sqcup )$ or ${\mathcal F}(X,\sqcup )$ with $(K,P) = \mathcal {PB}(X,\sqcup ,P)$ or $\mathcal {B}(X,\sqcup ,P)$ , respectively.

We refer to Section 6 for concrete applications of these results. For instance, in the case of (finite) matroids (see Subsection 6.4), it is well-known that the independence complex is shellable and hence homotopically Cohen-Macaulay. In our language, this is the partial basis complex associated with the lattice of flats. Therefore, as a consequence of the previous result, the corresponding frame complex is homotopically Cohen-Macaulay.

We derive some results on the augmented Bergman complex using the non-Hausdorff mapping cylinder.

Proof. Consider the map $\varphi : \Delta (X,\sqcup ) \to M_\Phi ^{\circ }$ given by

$$\begin{align*}\varphi(\sigma \cup \{x_1,\ldots,x_r\}) = \begin{cases} \sigma & r = 0,\\ x_r & r> 0. \end{cases}\end{align*}$$

Clearly, $\varphi $ is a well-defined order-preserving map. To prove it is a homotopy equivalence, we show that the lower homotopy fibers are contractible. Indeed, for $\sigma \in {\mathcal F}(X,\sqcup )$ , we have

$$\begin{align*}\varphi^{-1} \big( {M_\Phi^{\circ}}_{\leq \sigma} \big) = \Delta(X,\sqcup)_{\leq \sigma}\simeq *.\end{align*}$$

On the other hand, for $x\in {\mathcal {S}}(X,\sqcup )^{\circ }$ , we get

$$\begin{align*}\sigma \cup \{x_1,\ldots,x_r\} \in \varphi^{-1}\big( {M_\Phi^{\circ}}_{\leq x} \big) \, \Longleftrightarrow \, \Phi(\sigma) \leq x \text{ and } x_r\leq x.\end{align*}$$

When $r>0$ , the condition $\Phi (\sigma )\leq x$ follows immediately from $x_r\leq x$ . In this way, we can construct a contraction homotopy inside this fiber:

$$\begin{align*}\sigma \cup \{x_1,\ldots,x_r\}\leq \sigma\cup \{x_1,\ldots,x_r,x\} \geq \{x\}.\end{align*}$$

Note that all these terms lie in $\varphi ^{-1} \big ( {M_\Phi ^{\circ }}_{\leq \sigma } \big )$ .

Since the lower homotopy fibers are contractible, by Quillen’s fiber theorem (see Theorem 5.3), we conclude that $\varphi $ is a homotopy equivalence.

Since the elements of the poset ${\mathcal F}(X,\sqcup )$ whose join is the unique maximal element of ${\mathcal {S}}(X,\sqcup )$ consist of exactly the set of full frames, they form an antichain. In conjunction with Lemma 5.5, this immediately implies the following result, which is well-known in the matroid case (see [Reference Bullock, Kelley, Reiner, Ren, Shemy, Shen, Sun and ZhangBKR]).

Proposition 5.12. Let M be the set of full frames of ${\mathcal {S}}(X,\sqcup )$ . Then $\Delta (X,\sqcup )$ is homotopy equivalent to

$$ \begin{align*}\bigvee_{\sigma \in M} S^{|\sigma|-1}.\end{align*} $$

In particular, $\Delta (X,\sqcup )$ is spherical, and if M is empty (i.e., there are no full frames), then $\Delta (X,\sqcup )$ is contractible.

We have the following partial result on the connectivity of ${\mathcal {PD}}(X,\sqcup )^{*}$ if we have information on the lower intervals of both ${\mathcal {S}}_h(X,\sqcup )$ and the partial decompositions. We will use property (LI) from Definition 3.2 to prove the following theorem.

Proof. Note that ${\mathcal {S}}_h(X,\sqcup )^{*}$ is spherical of dimension $n-2$ by hypothesis, and that the inclusion ${\mathcal {S}}_h(X,\sqcup )^{*}\hookrightarrow {\mathcal {PD}}(X,\sqcup )^{*}\setminus {\mathcal D}(X,\sqcup )$ is a homotopy equivalence by Proposition 5.8. Therefore, it is enough to show that the inclusion $i:{\mathcal {PD}}(X,\sqcup )^{*}\setminus {\mathcal D}(X,\sqcup ) \hookrightarrow {\mathcal {PD}}(X,\sqcup )^{*}$ is an $(n-2)$ -equivalence.

We analyze the lower fibers of full decompositions for the map i and apply Quillen’s fiber theorem later. Let $\sigma _0\in {\mathcal D}(X,\sqcup )^{\circ }$ . By property (LI), the lower fiber $i^{-1}({\mathcal D}(X,\sqcup )^{\circ })$ can be identified with the poset

$$\begin{align*}F_{\sigma_0} := \big(\prod_{y\in \sigma} {\mathcal{PD}}(Y,\sqcup) \big)^{*} \setminus \prod_{y\in \sigma} {\mathcal D}(Y,\sqcup).\end{align*}$$

Write $\sigma _0 = \{y_1,\ldots ,y_r\}$ , and ${\mathcal {PD}}(y_i) := {\mathcal {PD}}(Y_i,\sqcup )$ , ${\mathcal D}(y_i) := {\mathcal D}(Y_i,\sqcup )$ and ${\mathcal {S}}_h(y_i) := {\mathcal {S}}_h(Y_i,\sqcup )$ . We claim that $F_{\sigma _0}$ is homotopy equivalent to ${\mathcal T} := \big ( \prod _{i=1}^r {\mathcal {S}}_h(y_i) \big )^{*}$ .

Let $\Psi :F_{\sigma _0}\to {\mathcal T}$ be the map such that

$$\begin{align*}\Psi(\tau_1,\ldots,\tau_r) = (\Phi(\tau_1),\ldots,\Phi(\tau_r)). \end{align*}$$

Clearly, $\Psi $ is well-defined and order-preserving.

On the other hand, we have a map $\Gamma : {\mathcal T} \to F_{\sigma _0}$ which is just the canonical inclusion $\Gamma (x_1,\ldots ,x_r) = ( \{x_1\} \setminus \{0\}, \ldots , \{x_r\}\setminus \{0\})$ . Then $\Psi \circ \Gamma $ is the identity map on ${\mathcal T}$ , and

$$\begin{align*}\Gamma( \Psi(\tau_1,\ldots,\tau_r) )= (\{\Phi(\tau_1)\}\setminus\{0\},\ldots,\{\Phi(\tau_r)\}\setminus\{0\}) \geq (\tau_1,\ldots,\tau_r).\end{align*}$$

Now, ${\mathcal T}$ is the proper part of the direct product of bounded posets, and hence, it is a standard fact that the geometric realization of ${\mathcal T}$ is isomorphic to an iterated suspension of a join of the spaces $|{\mathcal {S}}_h(y_i)^{*}|$ :

$$\begin{align*}|{\mathcal T}| \cong \Sigma( \cdots \Sigma\big(\Sigma\big(|{\mathcal{S}}_h(y_1)^{*}|*|{\mathcal{S}}_h(y_2)^{*}|\big)*|{\mathcal{S}}_h(y_3)^{*}|\big) \cdots * |{\mathcal{S}}_h(y_r)^{*}|\big).\end{align*}$$

See [Reference WalkerWal88]. Since ${\mathcal {S}}_h(y_i)^{*}$ is $(h(y_i)-3)$ -connected by hypothesis, we conclude that ${\mathcal T}$ is

$$\begin{align*}3(r-1) + \sum_i (h(y_i) -3) = 3(r - 1) + n - 3r = n - 3\end{align*}$$

connected. However, we know that ${\mathcal D}(X,\sqcup )^{\circ }_{>\sigma _0}$ is the proper part of the partition lattice on $\sigma _0$ , so it is spherical of dimension $r-3$ , and hence $(r-4)$ -connected (the empty space is $(-2)$ -connected by convention). Thus, $F_{\sigma _0} * {\mathcal D}(X,\sqcup )^{\circ }_{>\sigma _0}$ is

$$\begin{align*}(n-3) + (r-4) + 2 = n+r-5\end{align*}$$

connected. In particular, since $r\geq 2$ , these are $(n-3)$ -connected. By Theorem 5.3, we conclude that i is an $(n-2)$ -equivalence, and therefore, ${\mathcal {PD}}(X,\sqcup )^{*}$ is $(n-3)$ -connected.

Example 5.14. Let L be the lattice of subsets of $\{1,2,3,4\}$ of size different from three. In Subsection 6.4, we will see that this is the lattice of flats of the uniform matroid $U_{4,3}$ . Then, by Example 2.19, $L = {\mathcal {S}}(X,\sqcup )$ if L is regarded as a category with monoidal product $\sqcup =\vee $ , and $X=\{1,2,3,4\}$ the maximal element. Note L has height three, and ${\mathcal {PD}}(L) = {\mathcal {PD}}(X,\sqcup )$ . Also, $L = {\mathcal {S}}_h(X,\sqcup )$ , and by the previous theorem, we conclude that the inclusion $i:L^{*} \hookrightarrow {\mathcal {PD}}(L)^{*}$ is a $1$ -equivalence. Since $L^{*}$ is connected, we see that ${\mathcal {PD}}(L)^{*}$ is connected. Now we show that ${\mathcal {PD}}(L)^{*}$ is simply connected, so the inclusion i induces the trivial map in the fundamental groups.

If we regard $L^{*}$ as a $1$ -dimensional simplicial complex, it is not hard to see that $L^{*}$ becomes simply connected after attaching $2$ -cells to the cycles of the form

(2) $$ \begin{align} \{1\}<\{1,2\}>\{2\}<\{2,3\}>\{3\}<\{1,3\}>\{1\}. \end{align} $$

Thus, ${\mathcal {PD}}(L)^{*}$ is simply connected if each such cycle is null-homotopic when included in ${\mathcal {PD}}(L)^{*}$ . Let C denote the cycle from (2), seen as a subposet of $L^{*}$ . Then $i:C\to {\mathcal {PD}}(L)^{*}$ is null-homotopic via the following equivalences of maps:

$$\begin{align*}i(x) = \{x\} \geq \{\{k\} \mathrel{{\: : \: }} k\in \{1,2,3\}\cap x\} \leq \{\{1\},\{2\},\{3\}\}.\end{align*}$$

From this, we can conclude that ${\mathcal {PD}}(L)^{*}$ is at least $1$ -connected. Moreover, it is not hard to show that ${\mathcal {PD}}(L)^{*}$ collapses to the subposet whose elements are the nonempty subsets of size at most three of $\{1,2,3,4\}$ (which is the proper part of the Boolean lattice of rank four). Since this poset has dimension two and it is $1$ -connected, we conclude that ${\mathcal {PD}}(L)^{*}$ has the homotopy type of a wedge of $2$ -spheres.

In the following two results, we will see that the situation in the previous example actually generalizes to a more general setting.

Proof. There is an $(n-2)$ -dimensional sphere embedded in the order complex of ${\mathcal {S}}_h(X,\sqcup )$ which is obtained by taking the usual apartment of a basis, here regarded as a frame. Concretely, this sphere is a Coxeter complex obtained as the order complex of the subposet

$$\begin{align*}S_{\sigma} := \big\{\bigvee_{x\in \tau} x \mathrel{{\: : \: }} \emptyset\neq \tau\subsetneq \sigma \big\}.\end{align*}$$

Since $S_{\sigma }$ is an $(n-2)$ -dimensional sphere, its class gives a nonzero cycle for the $(n-2)$ -homology group of ${\mathcal {S}}_h(X,\sqcup )^{*}$ , which is naturally included in $\widetilde {H}_{n-2}( {\mathcal {S}}(X,\sqcup )^{*},{\mathbb {Z}})$ .

On the other hand, note that the image of $S_{\sigma }$ under the canonical map ${\mathcal {S}}_h(X,\sqcup )^{*}\to {\mathcal {PD}}(X,\sqcup )^{*}$ is contained in the subposet

$$\begin{align*}T_{\sigma} = \{ \tau \mathrel{{\: : \: }} \emptyset\neq \tau\subseteq \tau'\in{\mathcal D}(X,\sqcup)^{\circ}_{\geq \sigma} \}.\end{align*}$$

Then $T_{\sigma }$ is the proper part of the partial decomposition poset of the Boolean lattice on $\sigma $ . In particular, by Corollary 5.19, the poset $T_{\sigma }$ is contractible. Hence, the inclusion of the sphere $S_{\sigma } \hookrightarrow {\mathcal {PD}}(X,\sqcup )^{*}$ is null-homotopic (and thus trivial in homology).

5.3 Unique complementation

Now we explore some consequences of the relations between the different posets we defined when we additionally have unique downward $(\sqcup ,h)$ -complementation and property (EX) holds. Recall that property (EX) states that if we have a partial decomposition $\sigma $ of ${\mathcal {S}}(X,\sqcup )$ and we pick a full decomposition $\tau \in {\mathcal D}({\mathcal {S}}(X,\sqcup )_{\leq y})$ of the lower interval of one of its elements $y\in \sigma $ which is $\sqcup $ -compatible in ${\mathcal {S}}(X,\sqcup )$ , then $( \sigma \setminus \{y\})\cup \tau $ is a partial decomposition of X. See Definition 3.10.

Proof. Since G is a surjective poset map by hypothesis, it is enough to prove that $G(\tau ) \supseteq G(\sigma )$ implies that $\tau \leq \sigma $ .

Assume then that $G(\tau )\supseteq G(\sigma )$ . Write $\tau =(z_1,\ldots ,z_r)$ and $\sigma =(w_1,\ldots ,w_s)$ . For $1\leq j\leq s-1$ , let $1\leq \beta (j) \leq r$ be such that

$$\begin{align*}y_j:=w_1\vee\cdots\vee w_j = z_1\vee\cdots\vee z_{\beta(j)}.\end{align*}$$

We claim that

$$\begin{align*}w_{j+1} = z_{\beta(j)+1}\vee \cdots\vee z_{\beta(j+1)}.\end{align*}$$

Indeed, $w_{j+1}$ is the unique $(\sqcup ,h)$ -complement of

$$\begin{align*}y_j = z_1\vee\cdots\vee z_{\beta(j)}\end{align*}$$

in

$$\begin{align*}y_j \vee w_{j+1} = y_j \vee z_{\beta(j)+1} \vee \cdots\vee z_{\beta(j)}.\end{align*}$$

But also $z_{\beta (j)+1} \vee \cdots \vee z_{\beta (j)}$ is the unique $(\sqcup ,h)$ -complement of $y_j$ in $y_j \vee w_{j+1}$ . Thus, they must coincide. This shows that $\tau \leq \sigma $ .