7,816 research outputs found
A colimit decomposition for homotopy algebras in Cat
Badzioch showed that in the category of simplicial sets each homotopy algebra
of a Lawvere theory is weakly equivalent to a strict algebra. In seeking to
extend this result to other contexts Rosicky observed a key point to be that
each homotopy colimit in simplicial sets admits a decomposition into a homotopy
sifted colimit of finite coproducts, and asked the author whether a similar
decomposition holds in the 2-category of categories Cat. Our purpose in the
present paper is to show that this is the case.Comment: Some notation changed; small amount of exposition added in intr
Noncommutativity as a colimit
Every partial algebra is the colimit of its total subalgebras. We prove this
result for partial Boolean algebras (including orthomodular lattices) and the
new notion of partial C*-algebras (including noncommutative C*-algebras), and
variations such as partial complete Boolean algebras and partial AW*-algebras.
The first two results are related by taking projections. As corollaries we find
extensions of Stone duality and Gelfand duality. Finally, we investigate the
extent to which the Bohrification construction, that works on partial
C*-algebras, is functorial.Comment: 22 pages; updated theorem 15, added propoisition 3
On the equvialence of colimits and 2-colimits
We compare the colimit and 2-colimit of strict 2-functors in the 2-category
of groupoids, over a certain type of posets. These posets are of special
importance, as they correspond to coverings of a topological space. The main
result of this paper gives conditions on the 2-functor , for
which . One can
easily see that any 2-functor can be deformed to a 2-functor
, which satisfied the conditions of the theorem
The Tits alternative for non-spherical triangles of groups
Triangles of groups have been introduced by Gersten and Stallings. They are,
roughly speaking, a generalisation of the amalgamated free product of two
groups and occur in the framework of Corson diagrams. First, we prove an
intersection theorem for Corson diagrams. Then, we focus on triangles of
groups. It has been shown by Howie and Kopteva that the colimit of a hyperbolic
triangle of groups contains a non-abelian free subgroup. We give two natural
conditions, each of which ensures that the colimit of a non-spherical triangle
of groups either contains a non-abelian free subgroup or is virtually solvable.Comment: 45 pages, 21 figures, v2: minor revision (correction of typos, new
font within figures, ...
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