1,337 research outputs found
Modalities in homotopy type theory
Univalent homotopy type theory (HoTT) may be seen as a language for the
category of -groupoids. It is being developed as a new foundation for
mathematics and as an internal language for (elementary) higher toposes. We
develop the theory of factorization systems, reflective subuniverses, and
modalities in homotopy type theory, including their construction using a
"localization" higher inductive type. This produces in particular the
(-connected, -truncated) factorization system as well as internal
presentations of subtoposes, through lex modalities. We also develop the
semantics of these constructions
Fejer and Suffridge polynomials in the delayed feedback control theory
A remarkable connection between optimal delayed feedback control (DFC) and
complex polynomial mappings of the unit disc is established. The explicit form
of extremal polynomials turns out to be related with the Fejer polynomials. The
constructed DFC can be used to stabilize cycles of one-dimensional non-linear
discrete systems
Symmetric monoidal noncommutative spectra, strongly self-absorbing -algebras, and bivariant homology
Continuing our project on noncommutative (stable) homotopy we construct
symmetric monoidal -categorical models for separable -algebras
and noncommutative spectra using the
framework of Higher Algebra due to Lurie. We study smashing (co)localizations
of and with respect to strongly
self-absorbing -algebras. We analyse the homotopy categories of the
localizations of and give universal characterizations
thereof. We construct a stable -categorical model for bivariant
connective E-theory and compute the connective E-theory groups of
-stable -algebras. We also introduce and study the
nonconnective version of Quillen's nonunital K'-theory in the framework of
stable -categories. This is done in order to promote our earlier result
relating topological -duality to noncommutative motives to the
-categorical setup. Finally, we carry out some computations in the case
of stable and -stable -algebras.Comment: 26 pages; v2 revised in accordance with arXiv:1412.8370, corrections
in Sections 3 and 4; v3 minor changes, to appear in J. Noncommut. Geo
Goodwillie's Calculus of Functors and Higher Topos Theory
We develop an approach to Goodwillie's calculus of functors using the
techniques of higher topos theory. Central to our method is the introduction of
the notion of fiberwise orthogonality, a strengthening of ordinary
orthogonality which allows us to give a number of useful characterizations of
the class of -excisive maps. We use these results to show that the pushout
product of a -equivalence with a -equivalence is a
-equivalence. Then, building on our previous work, we prove a
Blakers-Massey type theorem for the Goodwillie tower. We show how to use the
resulting techniques to rederive some foundational theorems in the subject,
such as delooping of homogeneous functors.Comment: 40 pages, (a slightly modified version of) this paper is accepted for
publication by the Journal of Topolog
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