This paper is an extended version of our proceedings paper announced at LICS'16; in order to complement it, this version is written from a different viewpoint including topos-theoretic aspect on our work. Technically, this paper introduces and studies the class of semi-galois categories, which extend galois categories and are dual to profinite monoids in the same way as galois categories are dual to profinite groups; the study on this class of categories is aimed at providing an axiomatic reformulation of Eilenberg's theory of varieties of regular languages--- a branch in formal language theory that has been developed since the mid 1960's and particularly concerns systematic classification of regular languages, finite monoids, and deterministic finite automata. In this paper, detailed proofs of our central results announced at LICS'16 are presented, together with topos-theoretic considerations. The main results include (I) a proof of the duality theorem between profinite monoids and semi-galois categories, extending the duality theorem between profinite groups and galois categories; based on this results on semi-galois categories, we then discuss (II) a reinterpretation of Eilenberg's theory from a viewpoint of duality theorem; in relation with this reinterpretation of the theory, (III) we also give a purely topos-theoretic characterization of classifying topoi BM of profinite monoids M among general coherent topoi, which is a topos-theoretic application of (I). This characterization states that a topos E is equivalent to the classifying topos BM of some profinite monoid M if and only if E is (i) coherent, (ii) noetherian, and (iii) has a surjective coherent point. This topos-theoretic consideration is related to the logical and geometric problems concerning Eilenberg's theory that we addressed at LICS'16, which remain open in this paper.
Provided here is a proof of the duality theorem between profinite monoids and semi-galois categories in the full form of a contravariant equivalence between suitable categories. Our proof is given intentionally as a natural extension of an elementary proof of the duality between profinite groups and galois categories. This elementary proof will be valuable in its own right mostly for those who are particularly concerned with profinite monoids and with comparison with the case of profinite groups: The structure of profinite monoids is still mysterious in general, while their analysis plays a fundamental role in Eilenberg's theory @cite_17 . We shall not proceed to generalize this duality theorem itself in this paper, since we instead proceed to another direction in : We consider another dual (coherent topoi) of semi-galois categories and see that they are axiomatized exactly as coherent noetherian topoi with coherent surjective points. This consideration serves yet another (topos-theoretic) viewpoint on the classical Eilenberg variety theory.
@cite_17
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We study Kan extensions in three weakenings of the Eilenberg-Moore dou- ble category associated to a double monad, that was introduced by Grandis and Par e. To be precise, given a normal oplax double monad T on a double categoryK, we consider the double categories consisting of pseudo T -algebras, 'weak' vertical T -morphisms, hor- izontal T -morphisms and T -cells, where 'weak' means either 'lax', 'colax' or 'pseudo'. Denoting these double categories by Algw(T ), where w = l, c or ps accordingly, our main result gives, in each of these cases, conditions ensuring that (pointwise) Kan extensions can be lifted along the forgetful double functor Algw(T ) ! K. As an application we recover and generalise a result by Getzler, on the lifting of pointwise left Kan extensions along symmetric monoidal enriched functors. As an application of Getzler's result we prove, in suitable symmetric monoidal categories, the existence of bicommutative Hopf monoids that are freely generated by cocommutative comonoids. Author(s): Johnson, William Andrew | Advisor(s): Scanlon, Thomas | Abstract: This dissertation is a collection of results in model theory, related in one way or another to fields, NIP theories, and elimination of imaginaries. The most important result is a classification of dp-minimal fields, presented in Chapter 9. We construct in a canonical fashion a non-trivial Hausdorff definable field topology on any unstable dp-minimal field. Usingthis we classify the dp-minimal pure fields and valued fields up to elementary equivalence. Furthermore we prove that every VC-minimal field is real closed or algebraically closed.In Chapter 11, we analyze the theories of existentially closed fields with several valuations and orderings, as studied by van den Dries. We show that these model complete theories are NTP2, and analyze forking, dividing, and burden in these theories. The theory of algebraically closed fields with n independent valuation rings turns out to be an example of such a theory. This provides a new and natural example of an NTP2 theory which is neithersimple nor NIP, nor even a conceptual hybrid of something simple and something NIP.In Chapter 8, we exhibit a bad failure of elimination of imaginaries in a dense o-minimal structure. We produce an exotic interpretable set which cannot be put in definable bijection with a definable set, after naming any amount of parameters. However, we show that these exotic interpretable sets are still amenable to some of the tools of tame topology: they mustadmit nice definable topologies locally homeomorphic to definable sets.Chapter 12 proves the existence of Z nZ-valued definable strong Euler characteristics on pseudofinite fields, which measure the non-standard “size” of definable sets, mod n. The non-trivial result is that these “sizes” are definable in families of definable sets. This could probably be proven e cohomology, but we give a more elementary proof relying heavily on the theory of abelian varieties.We also present simplified and new proofs of several model-theoretic facts, including the definability of irreducibility and Zariski closure in ACF (Chapter 10), and elimination of imaginaries in ACVF (Chapter 6). This latter fact was originally proven by Haskell, Hrushovski, and Macpherson. We give a proof that is drastically simpler, inspired by Poizat’s proofs of elimination of imaginaries in ACF and DCF. Whereas formal category theory is classically considered within a @math -category, in this paper a double-dimensional approach is taken. More precisely we develop such theory within the setting of hypervirtual double categories, a notion extending that of virtual double category by adding cells with nullary target. [...] After this the notion of weak' Kan extension within a hypervirtual double category is considered, together with three strengthenings. [...] The notion of yoneda embedding is then considered in a hypervirtual double category, and compared to that of a good yoneda structure on a @math -category; the latter in the sense of Street-Walters and Weber. Conditions are given ensuring that a yoneda embedding @math defines @math as the free small cocompletion of @math , in a suitable sense. In the second half we consider formal category theory in the presence of algebraic structures. In detail: to a monad @math on a hypervirtual double category @math several hypervirtual double categories @math of @math -algebras are associated, [...]. This is followed by the study of the creation of, amongst others, left Kan extensions by the forgetful functors @math . The main motivation of this paper is the description of conditions ensuring that yoneda embeddings in @math lift along these forgetful functors, as well as ensuring that such lifted algebraic yoneda embeddings again define free small cocompletions, now in @math . As a first example we apply the previous to monoidal structures on categories, hence recovering Day convolution of presheaves and Im-Kelly's result on free monoidal cocompletion, as well as obtaining a "monoidal Yoneda lemma". Our main theorems provide a single geometric setting in which polynomial representatives for Schubert classes in the integral cohomology ring of the flag manifold are determined uniquely, and have positive coefficients for geometric reasons. This results in a geometric explanation for the naturality of Schubert polynomials and their associated combinatorics. Given a permutation w in S_n, we consider a determinantal ideal I_w whose generators are certain minors in the generic n x n matrix (filled with independent variables). Using multidegrees' as simple algebraic substitutes for torus-equivariant cohomology classes on vector spaces, our main theorems describe, for each ideal I_w: - variously graded multidegrees and Hilbert series in terms of ordinary and double Schubert and Grothendieck polynomials; - a Grobner basis consisting of minors in the generic n x n matrix; - the Stanley-Reisner complex of the initial ideal in terms of known combinatorial diagrams associated to permutations in S_n; and - a procedure inductive on weak Bruhat order for listing the facets of this complex, thereby generating the coefficients of Schubert polynomials by a positive recursion on combinatorial diagrams. We show that the initial ideal is Cohen-Macaulay, by identifying the Stanley-Reisner complex as a special kind of subword complex in S_n'', which we define generally for arbitrary Coxeter groups, and prove to be shellable by giving an explicit vertex decomposition. We also prove geometrically a general positivity statement for multidegrees of subschemes.