230 research outputs found
The Glueing Construction and Double Categories
We introduce Artin-Wraith glueing and locally closed inclusions in double
categories. Examples include locales, toposes, topological spaces, categories,
and posets. With appropriate assumptions, we show that locally closed
inclusions are exponentiable, and the exponentials are constructed via
Artin-Wraith glueing. Thus, we obtain a single theorem establishing the
exponentiability of locally closed inclusions in these five cases.Comment: 19 pages, presented at CT201
Span, Cospan, and Other Double Categories
Given a double category D such that D_0 has pushouts, we characterize
oplax/lax adjunctions between D and Cospan(D_0) such that the right adjoint is
normal and restricts to the identity on D_0, where Cospan(D_0) denotes the
double category on D_0 whose vertical morphisms are cospans. We show that such
a pair exists if and only if D has companions, conjoints, and 1-cotabulators.
The right adjoints are induced by the companions and conjoints, and the left
adjoints by the 1-cotabulators. The notion of a 1-cotabulator is a common
generalization of the symmetric algebra of a module and Artin-Wraith glueing of
toposes, locales, and topological spaces. Along the way, we obtain a
characterization of double categories with companions and conjoints as those
for which the identity functor on D_0 extends to a normal lax functor from
Cospan(D_0) to D.Comment: 16 page
Glueing and Orthogonality for Models of Linear Logic
We present the general theory of the method of glueing and associated technique of orthogonality for constructing categorical models of all the structure of linear logic: in particular we treat the exponentials in detail. We indicate simple applications of the methods and show that they cover familiar examples.
Motives of Deligne-Mumford Stacks
For every smooth and separated Deligne-Mumford stack , we associate a
motive in Voevodsky's category of mixed motives with rational
coefficients \mathbf{DM}^{\eff}(k,\mathbb{Q}). When is proper over a
field of characteristic 0, we compare with the Chow motive associated to
by Toen (\cite{t}). Without the properness condition we show that is
a direct summand of the motive of a smooth quasi-projective variety.Comment: to appear in Advances in Mathematic
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