# Cellular objects and Shelah s singular compactness theorem

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1 Cellular objects and Shelah s singular compactness theorem Logic Colloquium 2015 Helsinki Tibor Beke 1 Jiří Rosický 2 1 University of Massachusetts tibor 2 Masaryk University Brno Aug 7, 2015

2 how it began Theorem (Schreier, 1926): Every subgroup of a free group is free. Also true for abelian groups, abelian p-groups, and several equational varieties of k-algebras, for a field k: Lie algebras (Witt), commutative (non-associative) algebras (Shirshov), magmas (Kurosh),... Remark: there is still no complete characterization of these varieties known.

3 Conversely: Does this characterize free groups? Question Suppose G is an infinite group such that all subgroups of G of cardinality less than G, are free. (We will call such a group G almost free.) Is G free? Answer No. Eklof (1970) gave examples of non-free but almost free abelian groups (and Mekler, of non-abelian groups) of cardinality ℵ n, for any n N +. On the other hand, Higman (1951) proved that for singular κ of cofinality ω 0, an almost free group of cardinality κ is necessarily free. Hill (1970): for singular κ of cofinality ω 0, an almost free abelian group of cardinality κ is necessarily free. Hill (1974): this also holds for singular κ of cofinality ω 1.

4 Shelah: Singular Cardinal Compactness Shelah (1974) proved three related statements, each of which has the form: Let κ be singular and S a structure of size κ. If all substructures of S of size less than κ have property P, then S itself has property P. (1) structure = abelian group; P = free (2) structure = graph; P = having coloring number µ (3) structure = set of countable sets; P = having a transversal.

5 In example (2), a graph G is defined to have coloring number µ if the vertices of G can be well-ordered so that every vertex is connected to < µ vertices preceding it in the ordering. Subsequently, many other examples added, chiefly from commutative algebra: completely decomposable modules, Q-filtered modules...

6 what is Singular Cardinal Compactness really? Not an instance of compactness in any classical sense (e.g. for propositional or first-order logic, or compact cardinals) Shelah (1974) axiomatizes when his proof works Hodges (1982), building on Shelah (unpublished), gives another, elegant axiomatization Eklof (2006): SCC is about an abstract notion of free What is free about a graph with coloring number less than µ?

7 topological origin examples cellular singular cardinal compactness cells: topological origin Let D n = {x R n x 1} be the closed unit n-ball D n = {x R n x = 1} be the unit n-1-sphere D n i n Dn the inclusion. A pushout p in the category of topological spaces D n i n D n X Y p is called attaching an n-cell.

8 topological origin examples cellular singular cardinal compactness cellular maps: topological origin Let α be a well-ordered set and F : α Top a functor. Suppose for every β < α, the map F (β) F (β + 1) is attaching an n-cell (for some n N, depending on β) for limit β < α, the functor F is smooth at β. Then F (0) colim F is called a relatively cellular map. If F (0) is empty, colim F is called a cellular space. Cellular spaces are topological generalizations of the geometrically more restricted notions of simplicial complex and CW-complex.

9 topological origin examples cellular singular cardinal compactness I -cellular maps Let C be a cocomplete category and I a class of morphisms. The class of I -cellular maps is defined as the closure under isomorphisms (in the category of morphisms of C) of well-ordered smooth colimits of pushouts of elements of I. An object X is I -cellular if the map X from the initial object to X is I -cellular. This notion was isolated around 1970 by Quillen, Kan, Bousfield etc. It has proved very handy in homotopical algebra, in constructing weak factorization systems and homotopy model categories. Note that (with rare exceptions) a cellular map has many cell decompositions, without any preferred / canonical / functorial one.

10 topological origin examples cellular singular cardinal compactness I -cellular maps: example 1 Let C be an equational variety of algebras and their homomorphisms (e.g. groups, abelian groups, R-modules... ). Let A be the free algebra on the empty set, A the free algebra on a singleton and let I = {A A }. Then I -cellular objects are the same as free algebras.

11 topological origin examples cellular singular cardinal compactness I -cellular maps: example 2 Fix a ring R and let S be a set of R-modules. Let I = {0 M M S}. Then I -cellular objects are the same as I -decomposable modules. (This example would make sense in any category with coproducts.)

12 topological origin examples cellular singular cardinal compactness I -cellular maps: cartoon of example 3 consider the two families of inclusions of graphs

13 topological origin examples cellular singular cardinal compactness I -cellular maps: example 3 Work in the category of graphs and graph homomorphisms. Let µ be a cardinal and let I consist of all inclusions of graphs V, E V, E (V ) ( V ) where the set V of vertices has cardinality < µ and is a singleton (we take a representative set of these morphisms). Then I -cellular objects are precisely graphs having coloring number µ. An I -cell decomposition of a graph is the same data as a well-ordering of vertices satisfying Shelah s criterion: each vertex is connected to < µ vertices preceding it.

14 topological origin examples cellular singular cardinal compactness I -cellular maps: example 4 Work in the category of directed bipartite graphs, i.e. triples U, V, E where U, V are sets and E is a relation from A to B. Let I consist of the three morphisms where is the empty set, a singleton. Pushouts by these create a disjoint edge, create a vertex in the second partition, create an edge between existing vertices respectively. I -cellular objects are precisely directed bipartite graphs U, V, E that posses at least one transversal U V : a cell decomposition of a graph provides data for a transversal, and vice versa.

15 topological origin examples cellular singular cardinal compactness cellular SCC: cleanest version Let B be a locally finitely presentable category and I a set of morphisms. Let X B be an object whose size X is singular. If all subobjects of X of size less than X are I -cellular, then X itself is I -cellular.

16 topological origin examples cellular singular cardinal compactness cellular SCC: clarifying remarks The size of an object X of a locally presentable category is the cardinal predecessor of the least cardinal κ such that X is κ-presentable. (The least such κ will indeed be a successor cardinal.) This notion of size is intrinsic to the category, and coincides with the naive notion of cardinality of the underlying set in all cases of interest. It suffices to have enough subobjects of X to be cellular for the conclusion to hold. Could have a proper class of generating maps as long as the size of their domains is bounded from above. The size of X should be big enough (above the Löwenheim Skolem number).

17 topological origin examples cellular singular cardinal compactness Singular Cardinal Compactness: Cellular version Theorem (B. Rosický, 2014) Let B be a locally finitely presentable category, µ a regular uncountable cardinal and I a class of morphisms with µ-presentable domains. Let X B be an object with max{µ, card(fp B)} < X. Assume (i) X is a singular cardinal (ii) there exists φ < X such that for all successor cardinals κ + with φ < κ + < X, there exists a dense κ + -filter of Sub(X ) consisting of I -cellular objects. Then X is I -cellular.

18 topological origin examples cellular singular cardinal compactness the one that got away Theorem (Hodges, 1982) Let k be a field and K/k a field extension. Suppose K is κ-generated over k for some singular cardinal κ, and for all intermediate extensions L between k and K, if L is λ-generated over k for some regular cardinal λ, then L is a purely transcendental extension of k. Then K itself is a purely transcendental extension of k. This is a corollary of Hodges s axiomatization of Shelah s proof. It is not a case of the cellular version of singular compactness.

19 the main theorem about the proof questions Singular Cardinal Compactness: Functorial version Theorem (B. Rosický, 2014) Let A be an accessible category with filtered colimits, B a finitely accessible category and F : A B a functor preserving filtered colimits. Let X B be an object with max{µ F, card(fp B)} < X. Assume (i) X is a singular cardinal (ii) there exists φ < X such that for all successor cardinals κ + with φ < κ + < X, the image of F contains a dense κ + -filter of Sub(X ) (iii) F -structures extend along morphisms. Then X is in the image of F.

20 the main theorem about the proof questions functorial SCC: clarifying remarks Let F : A B be a functor. We say that F -structures extend along morphisms if, given any morphism g : X Y and object U of A, together with an isomorphism i : F (U) F (X ) in B, there exists a morphism f : U V and isomorphism j : F (V ) F (Y ) such that commutes. F (U) F (f ) F (V ) i F (X ) F (g) F (Y ) An object of B is in the image of the functor F if it is isomorphic to F (X ) for some X in A. j

21 the main theorem about the proof questions about the proof Assume first the target category is Pre(C), the category of presheaves on a small category C. Adapt the set-based proof of Hodges (1982). Fairly easy since the poset of subobjects of any object of Pre(C) is a complete distributive lattice. Characterize the categories B that posses a nice enough embedding B Pre(C) into a presheaf category so that one can deduce the conclusion for F : A B, given that it holds for the composite F : A B Pre(C). The functorial version implies both the cellular version and Hodges s version.

22 the main theorem about the proof questions the big picture

23 the main theorem about the proof questions questions Both the cellular and functorial versions of SCC are machines for churning out conditionals of the following type: If X is almost free and its size is singular, then it is free. Such a statement could be rather worthless for two reasons: There exists no almost free X whose size is singular. If X is almost free then it is free (regardless of size).

24 the main theorem about the proof questions bother! Theorem (Hodges, 1982) Let k be a field and K/k a field extension. Suppose K is κ-generated over k for some singular cardinal κ, and for all intermediate extensions L between k and K, if L is λ-generated over k for some regular cardinal λ, then L is a purely transcendental extension of k. Then K itself is a purely transcendental extension of k. Is this a statement about the empty set? That is, are there any such K/k?

25 the main theorem about the proof questions the combinatorics of (counter)examples Concerning groups, abelian groups, µ-colorable graphs, set transversals: We ve uniformized the proofs in singular characteristics (where almost free objects are free). Is there a way to unify the construction of paradoxical i.e. almost free but non-free objects in regular characteristics? Note that these constructions are fairly few and scattered. They seem to work best for ℵ n for finite n, or under V = L or similar set-theoretic assumptions.

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