445 research outputs found
Categorical notions of fibration
Fibrations over a category , introduced to category theory by
Grothendieck, encode pseudo-functors , while
the special case of discrete fibrations encode presheaves . A two-sided discrete variation encodes functors , which are also known as profunctors from to . By work of
Street, all of these fibration notions can be defined internally to an
arbitrary 2-category or bicategory. While the two-sided discrete fibrations
model profunctors internally to , unexpectedly, the dual two-sided
codiscrete cofibrations are necessary to model -profunctors internally
to -.Comment: These notes were initially written by the second-named author to
accompany a talk given in the Algebraic Topology and Category Theory
Proseminar in the fall of 2010 at the University of Chicago. A few years
later, the now first-named author joined to expand and improve in minor ways
the exposition. To appear on "Expositiones Mathematicae
t-structures are normal torsion theories
We characterize -structures in stable -categories as suitable
quasicategorical factorization systems. More precisely we show that a
-structure on a stable -category is
equivalent to a normal torsion theory on , i.e. to a
factorization system where both classes
satisfy the 3-for-2 cancellation property, and a certain compatibility with
pullbacks/pushouts.Comment: Minor typographical corrections from v1; 25 pages; to appear in
"Applied Categorical Structures
Coend calculus
The book formerly known as "This is the (co)end, my only (co)friend".Comment: This is the version ready for submissio
Categorical Ontology I - Existence
The present paper is the first piece of a series whose aim is to develop an approach to ontology and metaontology through category theory. We exploit the theory of elementary toposes to claim that a satisfying ``theory of existence'', and more at large ontology itself, can both be obtained through category theory. In this perspective, an ontology is a mathematical object: it is a category, the universe of discourse in which our mathematics (intended at large, as a theory of knowledge) can be deployed. The internal language that all categories possess prescribes the modes of existence for the objects of a fixed ontology/category.
This approach resembles, but is more general than, fuzzy logics, as most choices of \clE and thus of \Omega_\clE yield nonclassical, many-valued logics.
Framed this way, ontology suddenly becomes more mathematical: a solid corpus of techniques can be used to backup philosophical intuition with a useful, modular language, suitable for a practical foundation. As both a test-bench for our theory, and a literary divertissement, we propose a possible category-theoretic solution of Borges' famous paradoxes of Tlön's ``nine copper coins'', and of other seemingly paradoxical construction in his literary work. We then delve into the topic with some vistas on our future works
- …