107 research outputs found
Coequalisers of formal topology
We give a predicative construction of quotients of formal topologies. Along with earlier results on the match up between of continuous functions on real numbers (in the sense of Bishop\u27s constructive mathematics) and approximable mappings on the formal space of reals, we argue that formal topology gives an adequate foundation for constructive algebraic topology, also in the predicative sense. Predicativity is of essence when formalising the subject in logical frameworks based on Martin-Löf type theories
Lex colimits
Many kinds of categorical structure require the existence of finite limits,
of colimits of some specified type, and of "exactness" conditions between the
finite limits and the specified colimits. Some examples are the notions of
regular, or Barr-exact, or lextensive, or coherent, or adhesive category. We
introduce a general notion of exactness, of which each of the structures listed
above, and others besides, are particular instances. The notion can be
understood as a form of cocompleteness "in the lex world" -- more precisely, in
the 2-category of finitely complete categories and finite-limit preserving
functors.Comment: 38 pages; v2: final journal version, various minor changes and new
Section 5.9 dealing with filtered colimit
-typical Witt vectors with coefficients and the norm
For a profinite group we describe an abelian group of
-typical Witt vectors with coefficients in an -module (where is a
commutative ring). This simultaneously generalises the ring of Dress
and Siebeneicher and the Witt vectors with coefficients of Dotto,
Krause, Nikolaus and Patchkoria, both of which extend the usual Witt vectors of
a ring. We use this new variant of Witt vectors to give a purely algebraic
description of the zeroth equivariant stable homotopy groups of the
Hill-Hopkins-Ravenel norm of a connective spectrum , for
any finite group . Our construction is reasonably analogous to the
constructions of previous variants of Witt vectors, and as such is amenable to
fairly explicit concrete computations.Comment: 83 page
Categorical generalisations of quantum double models
We show that every involutive Hopf monoid in a complete and finitely
cocomplete symmetric monoidal category gives rise to invariants of oriented
surfaces defined in terms of ribbon graphs. For every ribbon graph this yields
an object in the category, defined up to isomorphism, that depends only on the
homeomorphism class of the associated surface. This object is constructed via
(co)equalisers and images and equipped with a mapping class group action. It
can be viewed as a categorical generalisation of the ground state of Kitaev's
quantum double model or of a representation variety for a surface. We apply the
construction to group objects in cartesian monoidal categories, in particular
to simplicial groups as group objects in SSet and to crossed modules as group
objects in Cat. The former yields a simplicial set consisting of representation
varieties, the latter a groupoid whose sets of objects and morphisms are
obtained from representation varieties.Comment: 46 page
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