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How to bake pi : an edible exploration of the mathematics of mathematics
\"In How to Bake Pi, math professor Eugenia Cheng provides an accessible introduction to the logic and beauty of mathematics, powered, unexpectedly, by insights from the kitchen: we learn, for example, how the béchamel in a lasagna can be a lot like the number 5, and why making a good custard proves that math is easy but life is hard.\"--Publisher description.
Book
Congruence Lattices of Ideals in Categories and (Partial) Semigroups
This monograph presents a unified framework for determining the congruences on a number of monoids and categories of transformations,
diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain
normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several
specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions;
Brauer, Temperley–Lieb and Jones partitions; linear and projective linear transformations; and partial braids. Special considerations
are needed for certain small ideals, and technically more intricate theoretical underpinnings for the linear and partial braid
categories.
eBook
Sorting with snakes
Offers lessons in sorting and similarities along with information on snakes.
Book
Infinite-Dimensional Representations of 2-Groups
A ‘2-group’ is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations
on vector spaces, 2-groups have representations on ‘2-vector spaces’, which are categories analogous to vector spaces. Unfortunately,
Lie 2-groups typically have few representations on the finite-dimensional 2-vector spaces introduced by Kapranov and Voevodsky. For this
reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional 2-vector spaces called ‘measurable categories’ (since they
are closely related to measurable fields of Hilbert spaces), and used these to study infinite-dimensional representations of certain Lie
2-groups. Here we continue this work. We begin with a detailed study of measurable categories. Then we give a geometrical description of
the measurable representations, intertwiners and 2-intertwiners for any skeletal measurable 2-group. We study tensor products and direct
sums for representations, and various concepts of subrepresentation. We describe direct sums of intertwiners, and
sub-intertwiners—features not seen in ordinary group representation theory. We study irreducible and indecomposable representations and
intertwiners. We also study ‘irretractable’ representations—another feature not seen in ordinary group representation theory. Finally,
we argue that measurable categories equipped with some extra structure deserve to be considered ‘separable 2-Hilbert spaces’, and
compare this idea to a tentative definition of 2-Hilbert spaces as representation categories of commutative von Neumann algebras.
eBook
Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity-Topoi
We develop a universal framework to study smooth higher orbifolds on the one hand and higher Deligne-Mumford stacks (as well as their
derived and spectral variants) on the other, and use this framework to obtain a completely categorical description of which stacks arise
as the functor of points of such objects. We choose to model higher orbifolds and Deligne-Mumford stacks as infinity-topoi equipped with
a structure sheaf, thus naturally generalizing the work of Lurie (2004), but our approach applies not only to different settings of
algebraic geometry such as classical algebraic geometry, derived algebraic geometry, and the algebraic geometry of commutative ring
spectra as in Lurie (2004), but also to differential topology, complex geometry, the theory of supermanifolds, derived manifolds etc.,
where it produces a theory of higher generalized orbifolds appropriate for these settings. This universal framework yields new insights
into the general theory of Deligne-Mumford stacks and orbifolds, including a representability criterion which gives a categorical
characterization of such generalized Deligne-Mumford stacks. This specializes to a new categorical description of classical
Deligne-Mumford stacks, a result sketched in Carchedi (2019), which extends to derived and spectral Deligne-Mumford stacks as well.
eBook
Higher topos theory
Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. InHigher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics.
The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.
eBook