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TRIPLES, ALGEBRAS AND COHOMOLOGY
 REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES
, 2003
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Adjointness in foundations
 Dialectica
, 1969
"... Author’s commentary In this article we see how already in 1967 category theory had made explicit a number of conceptual advances that were entering into the everyday practice of mathematics. For example, local Galois connections (in algebraic geometry, model theory, linear algebra, etc.) are globali ..."
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Author’s commentary In this article we see how already in 1967 category theory had made explicit a number of conceptual advances that were entering into the everyday practice of mathematics. For example, local Galois connections (in algebraic geometry, model theory, linear algebra, etc.) are globalized into functors, such as Spec, carrying much more information. Also, “theories ” (even when presented symbolically) are viewed explicitly as categories; so are the background universes of sets that serve as the recipients for models. (Models themselves are functors, hence preserve the fundamental operation of substitution/composition in terms of which the other logical operations can be characterized as local adjoints.) My 1963 observation (referred to by Eilenberg and Kelly in La Jolla, 1965), that cartesian closed categories serve as a common abstraction of type theory and propositional logic, permits an invariant algebraic treatment of the essential problem of proof theory, though most of the later work by proof theorists still relies on presentationdependent formulations. This article sums up a stage of the development of the relationship between
Higher fundamental functors for simplicial sets, Cahiers Topologie Géom
 Diff. Catég
"... Abstract. An intrinsic, combinatorial homotopy theory has been developed in [G3] for simplicial complexes; these form the cartesian closed subcategory of simple presheaves in!Smp, the topos of symmetric simplicial sets, or presheaves on the category!å of finite, positive cardinals. We show here how ..."
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Abstract. An intrinsic, combinatorial homotopy theory has been developed in [G3] for simplicial complexes; these form the cartesian closed subcategory of simple presheaves in!Smp, the topos of symmetric simplicial sets, or presheaves on the category!å of finite, positive cardinals. We show here how this homotopy theory can be extended to the topos itself,!Smp. As a crucial advantage, the fundamental groupoid Π1:!Smp = Gpd is left adjoint to a natural functor M1: Gpd =!Smp, the symmetric nerve of a groupoid, and preserves all colimits – a strong van Kampen property. Similar results hold in all higher dimensions. Analogously, a notion of (nonreversible) directed homotopy can be developed in the ordinary simplicial topos Smp, with applications to image analysis as in [G3]. We have now a homotopy ncategory functor ↑Πn: Smp = nCat, left adjoint to a nerve Nn = nCat(↑Πn(∆[n]), –). This construction can be applied to various presheaf categories; the basic requirements seem to be: finite products of representables are finitely presentable and there is a representable 'standard interval'.
Universal properties of Span
 in The Carboni Festschrift, Theory and Applications of Categories 13 (2005
"... Abstract. We give two related universal properties of the span construction. The first involves sinister morphisms out of the base category and sinister transformations. The second involves oplax morphisms out of the bicategory of spans having an extra property; we call these “jointed ” oplax morphi ..."
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Abstract. We give two related universal properties of the span construction. The first involves sinister morphisms out of the base category and sinister transformations. The second involves oplax morphisms out of the bicategory of spans having an extra property; we call these “jointed ” oplax morphisms.
A Cellular Nerve for Higher Order Categories
, 1999
"... Introduction The following text arose from the desire to establish a firm relationship between higher order categories and topological spaces. Our approach combines the algebraic features of Michael Batanin's !operads [1] with the geometric features of Andr'e Joyal's cellular sets [15] and tries t ..."
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Introduction The following text arose from the desire to establish a firm relationship between higher order categories and topological spaces. Our approach combines the algebraic features of Michael Batanin's !operads [1] with the geometric features of Andr'e Joyal's cellular sets [15] and tries to mimick as far as possible the classical construction of the simplicial nerve of a category. Higher order categories have attracted much attention in the last decade, due to their appearance in several mathematical areas. The ultimate goal is perhaps a faithful algebraic description of homotopy systems [13]. Since a homotopy between homotopies has the shape of a disk, the next higher homotopy the shape of a ball, and so on, we chose "ball compexes", i.e. globular sets [26], as the primitive combinatorial objects. The globular structure is precisely what underlies an !category [26], once its mu
The Incommunicability of Content
 Mind
, 1966
"... 1. Setting up the foundations 3 2. The EilenbergSteenrod axioms 4 3. Stable and unstable homotopy groups 5 4. Spectral sequences and calculations in homology and homotopy 6 ..."
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1. Setting up the foundations 3 2. The EilenbergSteenrod axioms 4 3. Stable and unstable homotopy groups 5 4. Spectral sequences and calculations in homology and homotopy 6
Stable Algebraic Topology, 19451966
 Mind
, 1966
"... this paper appeared four years before Milnor's discovery of exotic dierential structures on spheres [Mil56a]. For an embedding f , he went further and showed that the homotopy type of a tubular neighborhood of f is independent of the dierentiable structure on the ambient manifold. He then introduced ..."
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this paper appeared four years before Milnor's discovery of exotic dierential structures on spheres [Mil56a]. For an embedding f , he went further and showed that the homotopy type of a tubular neighborhood of f is independent of the dierentiable structure on the ambient manifold. He then introduced the notion of ber homotopy equivalence and proved that the ber homotopy type of the tangent bundle of a manifold is independent of its dierentiable structure. He observed that the 10 J. P. MAY
A Theory of Adjoint Functors —with some Thoughts about their Philosophical Significance
, 2005
"... The question “What is category theory ” is approached by focusing on universal mapping properties and adjoint functors. Category theory organizes mathematics using morphisms that transmit structure and determination. Structures of mathematical interest are usually characterized by some universal map ..."
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The question “What is category theory ” is approached by focusing on universal mapping properties and adjoint functors. Category theory organizes mathematics using morphisms that transmit structure and determination. Structures of mathematical interest are usually characterized by some universal mapping property so the general thesis is that category theory is about determination through universals. In recent decades, the notion of adjoint functors has moved to centerstage as category theory’s primary tool to characterize what is important and universal in mathematics. Hence our focus here is to present a theory of adjoint functors, a theory which shows that all adjunctions arise from the birepresentations of “chimeras ” or “heteromorphisms ” between the objects of different categories. Since representations provide universal mapping properties, this theory places adjoints within the framework of determination through universals. The conclusion considers some unreasonably effective analogies between these mathematical
A History of Duality in Algebraic Topology
"... This paper became the starting point of investigations of homology for more general spaces than merely finite complexes or open subsets of R ..."
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This paper became the starting point of investigations of homology for more general spaces than merely finite complexes or open subsets of R