18,360 research outputs found
The third cohomology group classifies crossed module extensions
We give an elementary proof of the well-known fact that the third cohomology
group H^3(G, M) of a group G with coefficients in an abelian G-module M is in
bijection to the set Ext^2(G, M) of equivalence classes of crossed module
extensions of G with M.Comment: Further references adde
On the second cohomology group of a simplicial group
We give an algebraic proof for the result of Eilenberg and Mac Lane that the
second cohomology group of a simplicial group G can be computed as a quotient
of a fibre product involving the first two homotopy groups and the first
Postnikov invariant of G. Our main tool is the theory of crossed module
extensions of groups.Comment: Further references added. This article is an extension of the
published versio
The functors Wbar and Diag o Nerve are simplicially homotopy equivalent
Given a simplicial group G, there are two known classifying simplicial set
constructions, the Kan classifying simplicial set Wbar G and Diag N G, where N
denotes the dimensionwise nerve. They are known to be weakly homotopy
equivalent. We will show that Wbar G is a strong simplicial deformation retract
of Diag N G. In particular, Wbar G and Diag N G are simplicially homotopy
equivalent
On the 3-arrow calculus for homotopy categories
We develop a localisation theory for certain categories, yielding a 3-arrow
calculus: Every morphism in the localisation is represented by a diagram of
length 3, and two such diagrams represent the same morphism if and only if they
can be embedded in a 3-by-3 diagram in an appropriate way. The methods to
construct this localisation are similar to the Ore localisation for a 2-arrow
calculus; in particular, we do not have to use zigzags of arbitrary length.
Applications include the localisation of an arbitrary model category with
respect to its weak equivalences as well as the localisation of its full
subcategories of cofibrant, fibrant and bifibrant objects, giving the homotopy
category in all four cases. In contrast to the approach of Dwyer, Hirschhorn,
Kan and Smith, the model category under consideration does not need to admit
functorial factorisations. Moreover, our method shows that the derived category
of any abelian (or idempotent splitting exact) category admits a 3-arrow
calculus if we localise the category of complexes instead of its homotopy
category.Comment: Applications added. Minor changes. This article is an extension of
the published versio
The Quantum Query Complexity of Algebraic Properties
We present quantum query complexity bounds for testing algebraic properties.
For a set S and a binary operation on S, we consider the decision problem
whether is a semigroup or has an identity element. If S is a monoid, we
want to decide whether S is a group.
We present quantum algorithms for these problems that improve the best known
classical complexity bounds. In particular, we give the first application of
the new quantum random walk technique by Magniez, Nayak, Roland, and Santha
that improves the previous bounds by Ambainis and Szegedy. We also present
several lower bounds for testing algebraic properties.Comment: 13 pages, 0 figure
Comparing maximal mean values on different scales
When computing the average speed of a car over different time periods from
given GPS data, it is conventional wisdom that the maximal average speed over
all time intervals of fixed length decreases if the interval length increases.
However, this intuition is wrong. We investigate this phenomenon and make
rigorous in which sense this intuition is still true
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