29 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 approaches ontology and metaontology through mathematics,
and more precisely 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 E and
thus of Ω E yield nonclassical, many-valued logics.
Framed this way, ontology suddenly becomes more mathematical: a solid corpus of tech-
niques 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