160 research outputs found
Foundations of Algebraic Theories and Higher Dimensional Categories
Universal algebra uniformly captures various algebraic structures, by
expressing them as equational theories or abstract clones. The ubiquity of
algebraic structures in mathematics and related fields has given rise to
several variants of universal algebra, such as symmetric operads, non-symmetric
operads, generalised operads, and monads. These variants of universal algebra
are called notions of algebraic theory. In the first part of this thesis, we
develop a unified framework for notions of algebraic theory which includes all
of the above examples. Our key observation is that each notion of algebraic
theory can be identified with a monoidal category, in such a way that theories
correspond to monoid objects therein. We introduce a categorical structure
called metamodel, which underlies the definition of models of theories. We also
consider morphisms between notions of algebraic theory, which are a monoidal
version of profunctors. Every strong monoidal functor gives rise to an adjoint
pair of such morphisms, and provides a uniform way to establish isomorphisms
between categories of models in different notions of algebraic theory. A
general structure-semantics adjointness result and a double categorical
universal property of categories of models are also shown.
In the second part of this thesis, we shift from the general study of
algebraic structures, and focus on a particular algebraic structure: higher
dimensional categories. Among several existing definitions of higher
dimensional categories, we choose to look at the one proposed by Batanin and
later refined by Leinster. We show that the notion of extensive category plays
a central role in Batanin and Leinster's definition. Using this, we generalise
their definition by allowing enrichment over any locally presentable extensive
category.Comment: 134 pages, PhD thesi
The oplax limit of an enriched category
We show that 2-categories of the form \mathscr{B}\mbox{-}\mathbf{Cat} are
closed under slicing, provided that we allow to range over
bicategories (rather than, say, monoidal categories). That is, for any
-category , we define a bicategory
such that
\mathscr{B}\mbox{-}\mathbf{Cat}/\mathbb{X}\cong
(\mathscr{B}/\mathbb{X})\mbox{-}\mathbf{Cat}. The bicategory
is characterized as the oplax limit of ,
regarded as a lax functor from a chaotic category to , in the
2-category of bicategories, lax functors and icons. We prove
this conceptually, through limit-preservation properties of the 2-functor
\mathbf{BICAT}\to 2\mbox{-}\mathbf{CAT} which maps each bicategory
to the 2-category \mathscr{B}\mbox{-}\mathbf{Cat}. When
satisfies a mild local completeness condition, we also show that
the isomorphism \mathscr{B}\mbox{-}\mathbf{Cat}/\mathbb{X}\cong
(\mathscr{B}/\mathbb{X})\mbox{-}\mathbf{Cat} restricts to a correspondence
between fibrations in \mathscr{B}\mbox{-}\mathbf{Cat} over on
the one hand, and -categories admitting certain powers
on the other.Comment: 20 page
Shape transformations of lipid vesicles by insertion of bulky-head lipids
Lipid vesicles, in particular Giant Unilamellar Vesicles (GUVs), have been increasingly
important as compartments of artificial cells to reconstruct living cell-like systems in a
bottom-up fashion. Here, we report shape transformations of lipid vesicles induced by
polyethylene glycol-lipid conjugate (PEG lipids). Statistical analysis of deformed vesicle
shapes revealed that shapes vesicles tend to deform into depended on the concentration
of the PEG lipids. When compared with theoretically simulated vesicle shapes, those
shapes were found to be more energetically favorable, with lower membrane bending
energies than other shapes. This result suggests that the vesicle shape transformations
can be controlled by externally added membrane molecules, which can serve as a
potential method to control the replications of artificial cells
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