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Constructive algebraic integration theory without choice. Dagstuhl proceedings
 Mathematics, Algorithms, Proofs, number 05021 in Dagstuhl Seminar Proceedings. Internationales Begegnungs und Forschungszentrum (IBFI), Schloss Dagstuhl
, 2005
"... Abstract. We present a constructive algebraic integration theory. The theory is constructive in the sense of Bishop, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. Since we avoid impredicative methods the results may also be interpret ..."
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Abstract. We present a constructive algebraic integration theory. The theory is constructive in the sense of Bishop, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. Since we avoid impredicative methods the results may also be interpreted in MartinL type theory or in a predicative topos in the sense of Moerdijk and Palmgren. We outline how to develop most of Bishop’s theorems on integration theory that do not mention points explicitly. Coquand’s constructive version of the Stone representation theorem is an important tool in this process. It is also used to give a new proof of Bishop’s spectral theorem.
Topological and smooth stacks
"... Abstract. We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a “stack over a stack ” and the distinction between small stacks (which are algebraic objects) and large stacks (which are ..."
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Cited by 8 (0 self)
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Abstract. We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a “stack over a stack ” and the distinction between small stacks (which are algebraic objects) and large stacks (which are generalized spaces). 1.
EXACT CATEGORIES
, 2008
"... We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3×3lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We th ..."
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Cited by 3 (0 self)
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We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3×3lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then
SET THEORY FOR CATEGORY THEORY
, 810
"... Abstract. Questions of settheoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical co ..."
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Abstract. Questions of settheoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical constructions are permissible. In this expository paper we summarize and compare a number
Contents
, 810
"... and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C ∞rings that is obtained by patching together homotopy zerosets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable nor ..."
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and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C ∞rings that is obtained by patching together homotopy zerosets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable normal bundle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a PontrjaginThom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived manifolds. In particular, the intersection A ∩ B of submanifolds A, B ⊂ X exists on the categorical level in our theory, and a cup product formula [A] ⌣ [B] = [A ∩ B] holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a categorification of intersection theory.
CONSTRUCTIVE POINTFREE TOPOLOGY ELIMINATES NONCONSTRUCTIVE REPRESENTATION THEOREMS FROM RIESZ SPACE THEORY
, 807
"... Abstract. In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that almost falgebras are commutative. The proof is obtained relat ..."
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Abstract. In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that almost falgebras are commutative. The proof is obtained relatively straightforward from the proof by Buskes and van Rooij by using the pointfree StoneYosida representation theorem by Coquand and Spitters. The StoneYosida representation theorem for Riesz spaces [LZ71, Zaa83] shows how to embed every Riesz space into the Riesz space of continuous functions on its spectrum. Theorem 1. [StoneYosida] Let R be an Archimedean Riesz space (vector lattice) with unit. Let Σ be its (compact Hausdorff) space of representations. Define the continuous function ˆr(σ): = σ(r) on Σ. Then r ↦ → ˆr is a Riesz embedding of R into C(Σ,). The theorem is very convenient, but sometimes better avoided, since it leads out of the theory of Riesz spaces. To quote Zaanen [Zaa97]:
Realizability Categories
"... This thesis contains a collection of results of my Ph.D. research in the area of realizability and category theory. My research was an exploration of the intersection of these areas focused on gaining a deeper understanding rather than on answering a specific question. This gave us some theorems tha ..."
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This thesis contains a collection of results of my Ph.D. research in the area of realizability and category theory. My research was an exploration of the intersection of these areas focused on gaining a deeper understanding rather than on answering a specific question. This gave us some theorems that help to define what realizability
3) The family {1: U → U} is covering for all
"... with a topology T. A Grothendieck topology T consists of a collection of subfunctors R ⊂ hom ( , U), U ∈ C, called covering sieves, such that the following axioms hold: 1) (base change) If R ⊂ hom ( , U) is covering and φ: V → U is a morphism of C, then the subfunctor φ −1 (R) = {γ: W → V  φ · γ ∈ ..."
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with a topology T. A Grothendieck topology T consists of a collection of subfunctors R ⊂ hom ( , U), U ∈ C, called covering sieves, such that the following axioms hold: 1) (base change) If R ⊂ hom ( , U) is covering and φ: V → U is a morphism of C, then the subfunctor φ −1 (R) = {γ: W → V  φ · γ ∈ R} is covering for V. 2) (local character) Suppose that R, R ′ ⊂ hom ( , U) are subfunctors and R is covering. If φ −1 (R ′) is covering for all φ: V → U in R, then R ′ is covering. 3) hom ( , U) is covering for all U ∈ C 1 Typically Grothendieck topologies arise from covering families in sites C having pullbacks. Covering families are sets of functors which generate covering sieves. Suppose that C has pullbacks. A topology T on C consists of families of sets of morphisms {φα: Uα → U}, U ∈ C, called covering families, such that the following axioms hold: 1) Suppose that φα: Uα → U is a covering family and that ψ: V → U is a morphism of C. Then the collection V ×U Uα → V is a covering family for V. 2) If {φα: Uα → V} is covering, and {γα,β: Wα,β → Uα is covering for all α, then the family of composites is covering.