Example 5.25. Let C be the category from Figure 5.3.
x
f1
xx
f2
h11
h12
&&
y
h1
h2
&&
y1
f21
f11
xxz
Figure 5.3: An acyclic category
The unlabeled morphisms are given by
γ11h1f1f11 h11 h2f1f11h12γ12 γ22h2f2f21 h12 h1f2f21h11γ21.
Figure 5.4 shows the realizations ofC with respect to Gmin andGmax. On the left hand side we have the realization with respect toGmin, where we have only labeled 0– and1–cells. On the right hand side we have the realization with respect toGmax, where the morphisms in the picture represent their respective closures. The reader may convince himself that these yield indeed all objects inDpC,Gmaxq.
f1
h11 h12
f1
f2
h12 h11
f2
h1
f11
h2
f21 γ11 γ12
γ22 γ21
idy
idx idy1
idx
idy
idx idy1
idx
f1
h11 h12
f1
f2
h12 h11
f2
h1
f11
h2
f21 idz
idy
idx idy1
idx
idy
idx idy1
idx
xγ11y xγ12y
xγ22y
xγ21y
Figure 5.4: The realizations of the category from Figure 5.3 with respect toGmin
(left) andGmax (right)
Proof. Elementary calculations show that pairs of factorizations, for which the pairs of first and second coordinates satisfy the properties of Definition 5.8, do satisfy those properties for the product.
Since trivial factorizations of the product category correspond to pairs of trivial factorizations of the factor categories, products of decomposition sets contain all trivial decompositions of the product.
Remark 5.27. The product of maximal decomposition sets is the maximal decom-position set of the product category.
Proposition 5.28. The product of minimal decomposition sets is the minimal decomposition set of the product category if and only if the factors are discrete categories.
Proof. Pairs of trivial decompositions correspond to trivial decompositions of the product if and only if either both first or both second coordinates are identities.
If and only if both factor categories have non-identity morphisms this is not the case for any such pair.
Theorem 5.29. For decomposition sets G1 of C1 and G2 of C2 and a chain C inC1C2
xCyG1G2 xπ1pCqyG1 xπ2pCqyG2
holds, whereπ1, π2 denote the projection functors of the product category.
Proof. Let us show the first inclusion by induction on the size ofG1 andG2. For the start we considerGminpC1qGminpC2q. Decompositions corresponding to pairs of trivial decompositions indeed generate new morphisms, but they change neither π1pCqnor π2pCq. Thus even xCyGminpC1qGminpC2q π1pCq π2pCqis true. For the induction itself set G11 : G1Y tpf1, h1qu. If xCyG11G2 xCyG1G2, by the induction hypothesis, this equals xπ1pCqyG1 xπ2pCqyG2, which is a subcategory ofxπ1pCqyG11 xπ2pCqyG2. So w.l.o.g. assume, for some morphismf2 ofxπ2pCqyG2
the pair pf1, f2q is a morphism of xCyG11G2 which does not exist in xCyG1G2. So there has to be a decomposition (for which w.l.o.g. f1, f2 are both the front morphisms)ppf1, f2q,ph1, h2qq PG11G2zG1G2such thatph1f1, h2f2qdoes exist inxCyG1G2 xπ1pCqyG1xπ2pCqyG2. Thus, in particular,h1f1is a morphism of xπ1pCqyG1. This gives the existence off1 inxπ1pCqyG11. Sopf1, f2qis a morphism ofxπ1pCqyG11 xπ2pCqyG2 which proves the first inclusion.
For the second inclusion note that for a decomposition pf1, h1qof G1, the pair ppf1,idq,ph1,idqqis a decomposition of the product G1G2. Thus any morphism f1 of xπ1pCqyG1 gives rise to a morphism pf1,idq of xCyG1G2, no matter which identity is meant. Similarly, we get morphisms pid, f2q of xCyG1G2 from mor-phismsf2ofxπ2pCqyG2. Therefore arbitrary morphismspf1, f2q pf1,idq pid, f2q ofxπ1pCqyG1 xπ2pCqyG2 belong toxCyG1G2.
So the set of morphisms of the two subcategories is identical.
Corollary 5.30. Let G1,G2 be decomposition sets of C1 resp. C2, then
DpC1C2,G1G2q DpC1,G1q DpC2,G2q
via the canonical isomorphisms of taking products and projecting to coordinates.
Proof. Clearly the functors are inverse to each other. So it is enough to check that they are well defined. For chains pf1, . . . , fnq of C1 and ph1, . . . , hmq of C2, Theorem 5.29 applied to the chainC ppf1, h1q, . . . ,pfn, h1q, . . . ,pfn, hmqq imme-diately shows that taking products is well defined. By Theorem 5.29 we obtain π1pxCyG1G2q xπ1pCqyG1 andπ2pxCyG1G2q xπ2pCqyG2 So the projection func-tors are well defined, too.
Example 5.31. Consider Example 5.25. The category shown in Figure 5.4 is isomorphic to a product of the (up to ismorphism unique) acyclic category with two objects and two non-identity morphisms with itself. Since for this category there are no proper decompositions, there is only one decomposition set and real-izations of the decomposition complex are subdivided spheresS1. Thus, indeed, the decomposition complex of the product is a subdivided torusS1S1, see Figure 5.4.
Example 5.32. Consider Example 5.14. Realizations of decomposition complexes of the factors are subdivided spheresS1resp. the interval D1. Thus the decompo-sition complexes of the product is a subdivided cylinderS1D1. Figure 5.5 shows realizations of the decomposition complexes with respect to the maximal (left) and the minimal (right) decomposition set.
Theorem 5.33. LetG1be a decomposition set ofC1 and letG2be a decomposition set ofC2. Then
DpC1
ºC2,G1YG2q DpC1,G1q º
DpC2,G2q.
h1 f11
h f21
f2
f1
idx
idy1 idz
idy
h1 f11
h f21
f2 γ2
f1 γ1
idx
idy1 idz
idy
Figure 5.5: Realizations of decomposition complexes of Example 5.14 with respect to the maximal(left) and the minimal(right) decomposition set.
Proof. Since decompositions of the coproduct correspond to decompositions of the summands and chains are either contained in C1 or in C2, the statement is true, like in the case of nerves.
Remark 5.34. The properties of symmetry and downwards closedness are pre-served under taking products. The same is true for realizations. Coordinate-wise products of G– andG1–realizations are GG1-realizations.
We end with a theorem from Kozlov [23], which becomes a corollary of our theory.
Corollary 5.35. [23, Thm. 10.21] For arbitrary acyclic categoriesC andD
|DpCD,GminpCDqq| subdivides |DpC,GminpCqq| |DpD,GminpDqq|.
Proof. Since GminpCDq GminpCq GminpDq, we can apply Corollary 5.24.
Then it is just left to note that as sets
|DpC,GminpCqq| |DpD,GminpDqq| |DpCD,GminpCq GminpDqq|
holds by Corollary 5.30.
6 Summary
We understand how different subdivisions of Bergman complexes are related to each other. The unification of the different languages of chains, nested sets and matroid types enables us to generalize these concepts to arbitrary posets and even acyclic categories. Since all comes down to product decompositions of intervals, the main advantage of decomposition complexes over order complexes is their closedness under products.
Whenever chains of specific posets have combinatorial interpretations and/or there are suitable product structures, it is reasonable to consider decomposition complexes to gain deeper insights. Potentially, chains that are glued together can be characterized in terms of these combinatorial interpretations - like combinatorial types of phylogenetic trees.
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