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Higher topos theory
, 2006
"... Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain com ..."
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Cited by 52 (0 self)
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Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain complex of Gvalued singular cochains on X. An alternative is to regard H n (•, G) as a representable functor on the homotopy category
THE CHU CONSTRUCTION
, 1996
"... We take another look at the Chu construction and show how to simplify it by looking at ..."
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Cited by 12 (1 self)
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We take another look at the Chu construction and show how to simplify it by looking at
The Reflectiveness of Covering Morphisms in Algebra And Geometry
, 1997
"... Each full reflective subcategory X of a finitelycomplete category C gives rise to a factorization system (E; M) on C, where E consists of the morphisms of C inverted by the reflexion I : C ! X . Under a simplifying assumption which is satisfied in many practical examples, a morphism f : A ! B lies ..."
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Cited by 7 (5 self)
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Each full reflective subcategory X of a finitelycomplete category C gives rise to a factorization system (E; M) on C, where E consists of the morphisms of C inverted by the reflexion I : C ! X . Under a simplifying assumption which is satisfied in many practical examples, a morphism f : A ! B lies in M precisely when it is the pullback along the unit jB : B ! IB of its reflexion If : IA ! IB; whereupon f is said to be a trivial covering of B. Finally, the morphism f : A ! B is said to be a covering of B if, for some effective descent morphism p : E ! B, the pullback p f of f along p is a trivial covering of E. This is the absolute notion of covering; there is also a more general relative one, where some class \Theta of morphisms of C is given, and the class Cov(B) of coverings of B is a subclass  or rather a subcategory  of the category C #B ae C=B whose objects are those f : A ! B with f 2 \Theta. Many questions in mathematics can be reduced to asking whether Cov(B) is re...
Finitary Sketches
, 1997
"... Finitary sketches, i.e., sketches with finitelimit and finitecolimit specifications, are proved to be as strong as geometric sketches, i.e., sketches with finitelimit and arbitrary colimit specifications. Categories sketchable by such sketches are fully characterized in the infinitary firstorder ..."
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Cited by 5 (0 self)
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Finitary sketches, i.e., sketches with finitelimit and finitecolimit specifications, are proved to be as strong as geometric sketches, i.e., sketches with finitelimit and arbitrary colimit specifications. Categories sketchable by such sketches are fully characterized in the infinitary firstorder logic: they are axiomatizable by oecoherent theories, i.e., basic theories using finite conjunctions, countable disjunctions, and finite quantifications. The latter result is absolute; the equivalence of geometric and finitary sketches requires (in fact, is equivalent to) the nonexistence of measurable cardinals. 1.
The Chu Construction
, 1996
"... . We take another look at the Chu construction and show how to simplify it by looking at it as a module category in a trivial Chu category. This simplifies the construction substantially, especially in the case of a nonsymmetric biclosed monoidal category. We also show that if the original category ..."
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. We take another look at the Chu construction and show how to simplify it by looking at it as a module category in a trivial Chu category. This simplifies the construction substantially, especially in the case of a nonsymmetric biclosed monoidal category. We also show that if the original category is accessible, then for any of a large class of "polynomiallike" functors, the category of coalgebras has cofree objects. 1. Introduction In a recent paper, I showed how the Chu construction, given originally in [Chu, 1979] for symmetric monoidal closed categories, could be adapted to monoidal biclosed (but not necessarily symmetric) categories. The construction, although well motivated by the necessity of providing a doubly infinite family of duals, was rather complicated with many computations involving indices. Recently I have discovered that the autonomous structure of Chu categories can be put into the familiar context of bimodules over a not necessarily commutative "algebra" objec...