75 research outputs found
Local Presentability Of Certain Comma Categories
It follows from standard results that if A and C are locally ^presentable categories and F:A^C is a ^-accessible functor, then the comma category IdC^F is locally ^-presentable. We show that, under the same hypotheses, F^IdC is also locally ^-presentable
On epimorphisms and monomorphisms of Hopf algebras
We provide examples of non-surjective epimorphisms in the category
of Hopf algebras over a field, even with the additional requirement that
have bijective antipode, by showing that the universal map from a Hopf algebra
to its enveloping Hopf algebra with bijective antipode is an epimorphism in
\halg, although it is known that it need not be surjective. Dual results are
obtained for the problem of whether monomorphisms in the category of Hopf
algebras are necessarily injective. We also notice that these are automatically
examples of non-faithfully flat and respectively non-faithfully coflat maps of
Hopf algebras.Comment: 17 pages; changed the abstract, revised introduction, shortened some
proofs; to appear in J. Algebr
Presentably symmetric monoidal infinity-categories are represented by symmetric monoidal model categories
We prove the theorem stated in the title. More precisely, we show the
stronger statement that every symmetric monoidal left adjoint functor between
presentably symmetric monoidal infinity-categories is represented by a strong
symmetric monoidal left Quillen functor between simplicial, combinatorial and
left proper symmetric monoidal model categories.Comment: v3: 17 pages, references updated and exposition improved, accepted
for publication in Algebraic and Geometric Topolog
Lifting accessible model structures
A Quillen model structure is presented by an interacting pair of weak
factorization systems. We prove that in the world of locally presentable
categories, any weak factorization system with accessible functorial
factorizations can be lifted along either a left or a right adjoint. It follows
that accessible model structures on locally presentable categories - ones
admitting accessible functorial factorizations, a class that includes all
combinatorial model structures but others besides - can be lifted along either
a left or a right adjoint if and only if an essential "acyclicity" condition
holds. A similar result was claimed in a paper of Hess-Kedziorek-Riehl-Shipley,
but the proof given there was incorrect. In this note, we explain this error
and give a correction, and also provide a new statement and a different proof
of the theorem which is more tractable for homotopy-theoretic applications.Comment: This paper corrects an error in the proof of Corollary 3.3.4 of "A
necessary and sufficient condition for induced model structures"
arXiv:1509.0815
The categorical basis of dynamical entropy
Many different branches of theoretical and applied mathematics require a
quantifiable notion of complexity. One such circumstance is a topological
dynamical system - which involves a continuous self-map on a metric space.
There are many notions of complexity one can assign to the repeated iterations
of the map. One of the foundational discoveries of dynamical systems theory is
that these have a common limit, known as the topological entropy of the system.
We present a category-theoretic view of topological dynamical entropy, which
reveals that the common limit is a consequence of the structural assumptions on
these notions. One of the key tools developed is that of a qualifying pair of
functors, which ensure a limit preserving property in a manner similar to the
sandwiching theorem from Real Analysis. It is shown that the diameter and
Lebesgue number of open covers of a compact space, form a qualifying pair of
functors. The various notions of complexity are expressed as functors, and
natural transformations between these functors lead to their joint convergence
to the common limit
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