1,720,973 research outputs found
t-structures on stable (infinity,1)-categories
No full textThe present work re-enacts the classical theory of t-structures reducing the classical definition coming from Algebraic Geometry to a rather primitive categorical gadget: suitable reflective factorization systems (defined in the work of Rosický, Tholen, and Cassidy-Hébert-Kelly), which we call "normal torsion theories" following. A relation between these two objects has previously been noticed by other authors, on the level of the triangulated homotopy categories of stable (infinity,1)-categories. The main achievement of the present thesis is to observe and prove that this relation exists genuinely when the definition is lifted to the higher-dimensional world where the notion of triangulated category comes from
Hearts and towers in stable infinity-categories ∞ -categories
Get PDFWe exploit the equivalence between t-structures and normal torsion theories on a stable ∞-category to show how a few classical topics in the theory of triangulated categories, i.e., the characterization of bounded t-structures in terms of their hearts, their associated cohomology functors, semiorthogonal decompositions, and the theory of tiltings, as well as the more recent notion of Bridgeland’s slicings, are all particular instances of a single construction, namely, the tower of a morphism associated with a J-slicing of a stable ∞-category Open image in new window , where J is a totally ordered set equipped with a monotone Z-action
Coend calculus
Get PDFThe book formerly known as "This is the (co)end, my only (co)friend".Comment: This is the final version a moment before book printing; slightly
slimmer version, lots of mistakes removed. On May 2023, a small edit has been
done, maintaining all typos that CUP removed during the copyediting phase
--and that I am not allowed to remove. Good luck finding those
Automata and Coalgebras in Categories of Species
No full textInternational audienceWe study generalized automata (in the sense of Adámek-Trnková) in Joyal’s category of (set-valued) combinatorial species, and as an important preliminary step, we study coalgebras for its derivative endofunctor ∂ and for the ‘Euler homogeneity operator’ L∘∂ arising from the adjunction L⊣∂⊣R
Differential 2-rigs
Get PDFWe study the notion of a "differential 2-rig", a category R with coproducts
and a monoidal structure distributing over them, also equipped with an
endofunctor D : R -> R that satisfies a categorified analogue of the Leibniz
rule. This is intended as a tool to unify various applications of such
categories to computer science, algebraic topology, and enumerative
combinatorics. The theory of differential 2-rigs has a geometric flavour but
boils down to a specialization of the theory of tensorial strengths on
endofunctors; this builds a surprising connection between apparently
disconnected fields. We build "free 2-rigs" on a signature, and we prove
various initiality results: for example, a certain category of colored species
is the free differential 2-rig on a single generator.Comment: In Proceedings ACT 2022, arXiv:2307.1551
Categorical Ontology I - Existence
Get PDFThe 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
Functorial Erkennen
Get PDFWe outline a ‘formal theory of scientific theories’ rooted in the theory of profunctors; the category-theoretic asset stresses the fact that the scope of scientific knowledge is to build ‘meaningful connections’ (i.e. well-behaved adjunctions) between a linguistic object (a ‘theoretical category’ T ) and the world W said language ought to describe. Such a world is often unfathomable, and thus we can only resort to a smaller fragment of it in our analysis: this is the ‘observational category’ O ⊆ W. From this we build the category [O op , Set] of all possible displacements of observational terms O. The self-duality of the bicategory of profunctors accounts for the fact that theoretical and observational terms can exchange their rôle without substantial changes in the resulting predictive-descriptive theory; this provides evidence for the idea that their separation is a mere linguistic convention; to every profunctor R linking T and O one can associate an object O ] R T obtained glueing together the two categories and accounting for the mutual relations subsumed by R. Under mild assumptions, such an arrangement of functors, profunctors, and gluings provides a categorical interpretation for the ‘Ramseyfication’ operation, in a very explicit sense: in a scientific theory, if a computation entails a certain behaviour for the system the theory describes, then saturating its theoretical variables with actual observed terms, we obtain the entailment in the world
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Accessibility and presentability in 2-categories
Get PDFWe outline a definition of accessible and presentable objects in a 2-category
endowed with a "KZ context", that is to say a pair of
lax-idempotent monads interacting in a prescribed way; this perspective
suggests a unified treatment of many "Gabriel-Ulmer like" theorems, asserting
how presentable objects arise as reflections of generating ones. We outline the
notion of "(Gabriel-Ulmer) envelope" for a KZ context, sufficient to concoct
Gabriel-Ulmer duality. We end the paper with a roundup of examples, involving
classical (set-based and enriched), low dimensional category theory, and a
perspective for future work, rooted in higher category theory and homotopy
theory.Comment: Final version, accepted for publication on JPA
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