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BEGIN:VEVENT
SUMMARY:Miloslav Štěpán (Masaryk University)
DTSTART:20241021T150000Z
DTEND:20241021T160000Z
DTSTAMP:20260409T093714Z
UID:SecondVDCW/1
DESCRIPTION:Title: Double categories versus factorization systems\nby Miloslav Štěp
án (Masaryk University) as part of Second Virtual Workshop on Double Cate
gories\n\n\nAbstract\nIn the first part of the talk we will recall double
categories and show that\nevery orthogonal factorization system can be tur
ned into a double category\,\nevery (nice) double category can be turned i
nto an orthogonal factorization\nsystem\, and that these processes are mut
ually inverse. We will do the same\nthing for strict factorization systems
.\n\nIn the second part of the talk we will put the above results into a w
ider\nperspective of 2-category theory: if the double category is regarded
as a\ndiagram in Cat\, producing the factorization system out of it amoun
ts to taking\na certain 2-categorial colimit of it (the codescent object).
We will mention\nthe connections to lax morphisms of algebras for a 2-mon
ad\, compositions of\npinwheels\, and possible extensions of the first par
t to weak factorization\nsystems.\n
LOCATION:https://researchseminars.org/talk/SecondVDCW/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adrian Miranda (University of Manchester)
DTSTART:20241021T160000Z
DTEND:20241021T170000Z
DTSTAMP:20260409T093714Z
UID:SecondVDCW/2
DESCRIPTION:Title: The pseudomonad perspective on double categories\, with applications to
double profunctors\nby Adrian Miranda (University of Manchester) as p
art of Second Virtual Workshop on Double Categories\n\n\nAbstract\nJust as
a category is a monad in the bicategory Span(Set)\, so a double category
is a pseudomonad in the analogous tricategory Span(Cat). Pseudo-bimodules
of between these give a notion of double profunctor\, and these in turn sp
ecialise to recover double functors. Composition of double profunctors use
s a two-dimensional colimit which\, unlike the reflexive coequalisers of t
he standard internal category setting\, are stable under pullback in Cat.
We extend Grandis and Pare’s strictification results for double categori
es and double functors to analogous strictification results for double pro
functors. We recover the tricategory PsDblCat whose morphisms are double f
unctors\, and describe work in progress\, joint with Nicola Gambino\, towa
rds a tricategory PsDblProf whose morphisms are double profunctors. Finall
y\, a dual specialisation of double profunctors will also be described giv
ing a notion of double cofunctor\, establishing the tricategory that these
maps form\, and providing a strictification result for them.\n
LOCATION:https://researchseminars.org/talk/SecondVDCW/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nima Rasekh (Universität Greifswald)
DTSTART:20241024T080000Z
DTEND:20241024T090000Z
DTSTAMP:20260409T093714Z
UID:SecondVDCW/3
DESCRIPTION:Title: Double Categories in Univalent Foundations\nby Nima Rasekh (Univers
ität Greifswald) as part of Second Virtual Workshop on Double Categories\
n\n\nAbstract\nThis is joint work with Niels van der Weide\, Benedikt Ahre
ns and Paige Randall North. In category theory\, we typically study catego
ries either up to isomorphisms (considering objects up to equality) or equ
ivalences (considering objects up to isomorphism). Fortunately\, since equ
ivalences generalize isomorphisms\, this distinction rarely causes mathema
tical difficulties. However\, double categories present a richer structure
with multiple notions of equivalence—such as isomorphisms\, horizontal
equivalences\, and gregarious equivalences—none of which subsumes all th
e others. This creates potential ambiguities\, making it necessary to spec
ify the appropriate form of equivalence in any given context. In this talk
\, I will show how moving from a set-theoretic foundation to a univalent f
oundation allows for definitions of (double) categories that inherently in
clude the desired notion of equivalence.\n
LOCATION:https://researchseminars.org/talk/SecondVDCW/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jaco Ruit (Utrecht University)
DTSTART:20241024T090000Z
DTEND:20241024T100000Z
DTSTAMP:20260409T093714Z
UID:SecondVDCW/4
DESCRIPTION:Title: A double ∞-categorical approach to formal ∞-category theory\nby
Jaco Ruit (Utrecht University) as part of Second Virtual Workshop on Doub
le Categories\n\n\nAbstract\nFormal ∞-category theory starts with the ob
servation that there are many variants of ∞-category theory\, for exampl
e\, enriched ∞-categories\, internal ∞-categories\, and monoidal ∞-c
ategories\, which come with specialized notions of adjunctions\, point-wis
e Kan extensions\, and so on. It is natural to ask whether one can give a
uniform and synthetic treatment of the foundational concepts and theorems
for these different flavors of ∞-category theory. \n \nIn this talk\, we
propose an extension of the ideas from formal (strict) category theory of
Street-Walters\, Wood\, Verity\, and Shulman\, to the ∞-categorical con
text\, and give a leisurely introduction to the theory of ∞-equipments.
These ∞-equipments are certain double ∞-categories in which many conce
pts of category theory may be developed and expressed (using only the doub
le categorical structure). We will present an overview and highlight some
of these aspects. Now\, since this approach yields category theories for t
he objects of these ∞-equipments\, developing a category theory for a fl
avor of ∞-categories is a question of constructing the right suitable am
bient ∞-equipment. Throughout the talk\, we discuss some of these exampl
es of ∞-equipments.\n
LOCATION:https://researchseminars.org/talk/SecondVDCW/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lander Hermans (University of Antwerp)
DTSTART:20241024T100000Z
DTEND:20241024T110000Z
DTSTAMP:20260409T093714Z
UID:SecondVDCW/5
DESCRIPTION:Title: Virtual double categories as coloured box operads\nby Lander Herman
s (University of Antwerp) as part of Second Virtual Workshop on Double Cat
egories\n\n\nAbstract\nVirtual double categories are a 2-categorification
of multicategories and compare to double categories as multicategories com
pare to monoidal categories. In algebraic topology\, multicategories are a
lso known as coloured operads and are extensively used to encode algebraic
operations\, thus generalizing operads.\n\nBy viewing virtual double cate
gories as coloured versions of box operads\, we shift our point of view: f
rom objects of study to algebraic gadgets encoding higher operations. This
is exemplified by our main application: we present a box operad Lax encod
ing lax functors U->Cat(k) into the category of k-linear categories. They
appear in algebraic geometry as prestacks generalizing structure sheaves a
nd (noncommutative) deformations thereof.\n\nIn the second part of the tal
k\, I will sketch key components of our main result: a Koszul duality for
box operads. For example\, to every box operad we can associate a canonica
l L_\\infty-algebra. A salient feature is that these results can be explai
ned purely in terms of (virtual) double categorical diagrams\, or in our t
erms\, stackings of boxes.\n\nIf time permits\, I will explain how it appl
ies to Lax in order to tackle their deformation and homotopy theory.\n
LOCATION:https://researchseminars.org/talk/SecondVDCW/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Keisuke Hoshino (Kyoto University)
DTSTART:20241029T235900Z
DTEND:20241030T010000Z
DTSTAMP:20260409T093714Z
UID:SecondVDCW/6
DESCRIPTION:Title: Double categories of relations relative to factorisation systems\nb
y Keisuke Hoshino (Kyoto University) as part of Second Virtual Workshop on
Double Categories\n\n\nAbstract\nThe double category of relations and the
double category of spans have historically been key examples in the study
of double categories. More generally\, given a class $\\mathcal{M}$ of mo
rphisms in a finitely complete category $\\mathbf{C}$\, one can define an
$\\mathcal{M}$-relation $A \\nrightarrow B$ as a span that belongs to $\\m
athcal{M}$ as a morphism into $A\\times B$. If $\\mathcal{M}$ is the right
class of a stable factorisation system\, the $\\mathcal{M}$-relations and
morphisms in $\\mathbf{C}$ form a double category.\n\nIn this talk\, I wi
ll first give a characterisation of double categories that arise from stab
le factorisation systems in this manner.\n\nI will then explore how differ
ent classes of stable factorisation systems can be characterised in terms
of their corresponding double categories. This include the (regular epi\,
mono) factorisation system on a regular category\, which has been studied
by Carboni and Walters in terms of cartesian bicategories\, and by Lambert
in terms of cartesian double categories. By considering the (isomorphism\
, all) factorisation system\, we also recover the double category of spans
\, a structure that has been examined bicategorically by Lack\, Walters\,
and Wood\, and double-categorically by Aleiferi. I will explain how these
known results are related to our general theorem.\n\nThis talk is based on
joint work with Hayato Nasu\, and the results can be found in our paper a
vailable at https://arxiv.org/abs/2310.19428.\n
LOCATION:https://researchseminars.org/talk/SecondVDCW/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maru Sarazola (University of Minnesota)
DTSTART:20241029T220000Z
DTEND:20241029T230000Z
DTSTAMP:20260409T093714Z
UID:SecondVDCW/7
DESCRIPTION:Title: Double-categorical frameworks for algebraic K-theory\nby Maru Saraz
ola (University of Minnesota) as part of Second Virtual Workshop on Double
Categories\n\n\nAbstract\nIn the last few years\, double categories have
made an appearance in the world of algebraic K-theory. The goal of this ta
lk is to introduce the main ideas in some of these new K-theoretical frame
works\, highlighting their different features and possible uses. No prior
background on K-theory will be assumed.\n
LOCATION:https://researchseminars.org/talk/SecondVDCW/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Evan Patterson (Topos Institute)
DTSTART:20241029T230000Z
DTEND:20241029T235900Z
DTSTAMP:20260409T093714Z
UID:SecondVDCW/8
DESCRIPTION:Title: Double-categorical logic in theory and practice\nby Evan Patterson
(Topos Institute) as part of Second Virtual Workshop on Double Categories\
n\n\nAbstract\nCategory theory contains a broad array of gadgets out of wh
ich to build specialized logics\, as explored in categorical logic and pro
gramming language theory. Double category theory can organize and systemat
ize the use of these categorical gadgets. In the first part of the talk\,
I review joint work with Michael Lambert on double theories and their mode
ls. Double theories are categorified theories whose models are categories
equipped with extra structure. In the second part\, I show how this work c
an be put to practical effect. I describe ongoing work with collaborators
at Topos Institute to build CatColab\, an interactive environment for form
al\, interoperable\, conceptual modeling. In technical terms\, CatColab is
a structure editor for models of domain-specific logics\, as defined by d
ouble theories.\n
LOCATION:https://researchseminars.org/talk/SecondVDCW/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rui Prezado (Universidade de Aveiro)
DTSTART:20241030T160000Z
DTEND:20241030T170000Z
DTSTAMP:20260409T093714Z
UID:SecondVDCW/9
DESCRIPTION:Title: Change-of-base for generalized multicategories\nby Rui Prezado (Uni
versidade de Aveiro) as part of Second Virtual Workshop on Double Categori
es\n\n\nAbstract\nA *multicategory* is a categorical structure that consis
ts of objects\nand *multimorphisms*\, whose domain consists of a finite (p
ossibly empty)\nstring of objects\, and whose codomain consists of a singl
e object. Their\ncomposition and identity\, as well as the unity and assoc
iativity\nproperties\, are well modeled by the extension of the *free mono
id monad*\non $\\mathsf{Set}$ to the (proarrow) equipment\n$\\mathsf{Span}
(\\mathsf{Set})$ of *spans* in $\\mathsf{Set}$.\nMulticategories\, togethe
r with their respective functors\, can be\nobtained by considering a suita
ble kind of algebra with respect to the\nextended free monoid monad on $\\
mathsf{Span}(\\mathsf{Set})$.\n\nThis abstraction can be carried out with
any *cartesian* monad $T$ on a\ncategory $\\mathcal A$ with pullbacks. We
have an extension of $T$ to the\nequipment $\\mathsf{Span}(\\mathcal{A})$
of spans in $\\mathcal A$\, and the\nrespective "$T$-algebras" are the so-
called *$T$-categories internal to\n$\\mathcal A$*\, whose category we den
ote by $\\mathsf{Cat}(T\,\\mathcal A)$.\nMost importantly\, if we are prov
ided with another cartesian monad $S$ on\na category $\\mathcal B$\, and a
suitable monad morphism\n$(F\,\\phi) \\colon (\\mathcal A\, T) \\to (\\ma
thcal\nB\, S)$\, it was shown in [3] that $(F\,\\phi)$ induces a *change-o
f-base*\nfunctor $\\mathsf{Cat}(T\,\\mathcal A) \\to\n\\mathsf{Cat}(S\,\\m
athcal B)$.\n\nThe enriched counterpart of such generalized multicategorie
s were first\nconsidered in [1]\; in essence\, *enriched $(T\,\\mathcal V)
$-categories*\ncan be obtained as the "$T$-algebras" for a suitable monad
$T$ on the\nequipment $\\mathcal V$-$\\mathsf{Mat}$\, where the enriching
category\n$\\mathcal V$ is a suitable monoidal category. Likewise\, these
also have\na notion of change-of-base functors.\n\nIn general\, we can con
sider *horizontal lax $T$-algebras* [2] for a\nlax monad $T$ on a pseudodo
uble category $\\mathbb D$\, special cases of\nwhich are enriched and inte
rnal generalized multicategories. This talk\naims to present notions of *c
hange-of-base functors* between such\nhorizontal lax algebras\, the study
of which was motivated by\nunderstanding the relationship between enriched
and internal\nmulticategorical structures\, the main topic of study of [4
]\, joint\nwork with F. Lucatelli Nunes.\n\nWe assume the basics of double
category theory\, and some familiarity\nwith multicategories will prove t
o be worthwhile.\n\n1. M. M. Clementino\, W. Tholen. Metric\, topology a
nd multicategory -- a\n common approach. *J. Pure Appl. Algebra*\, (179
):13--47\, 2003.\n\n2. G. Cruttwell\, M. Shulman. A unified framework fo
r generalized\n multicategories. *Theory Appl. Categ.*\, 24(21):580--65
5\, 2010.\n\n3. T. Leinster. *Higher Operads\, Higher Categories*\, volu
me 298 of\n London Mathematical Society Lecture Note Series. Cambridge\
n University Press\, 2004.\n\n4. R. Prezado\, F. Lucatelli Nunes. Gen
eralized multicategories:\n change-of-base\, embedding and descent. To
appear in *Appl. Categ.\n Structures.*\n
LOCATION:https://researchseminars.org/talk/SecondVDCW/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dylan McDermott (University of Oxford)
DTSTART:20241030T150000Z
DTEND:20241030T160000Z
DTSTAMP:20260409T093714Z
UID:SecondVDCW/10
DESCRIPTION:Title: The formal theories of presheaves and cocompletions\nby Dylan McDe
rmott (University of Oxford) as part of Second Virtual Workshop on Double
Categories\n\n\nAbstract\nMany categories can usefully be seen as completi
ons of other categories under classes of colimits. For instance\, if we fr
eely add small colimits to a small category A then we get presheaves on A\
, and every finitely accessible category arises by completing a small cate
gory under filtered colimits. The theory of cocompletions of a V-category
A is well-developed\, at least for V a symmetric monoidal closed category\
, in particular\, it is known that every such cocompletion is given by tak
ing a class of presheaves on A. I will talk about some ongoing work to dev
elop a formal categorical analogue of this theory\, in a double-categorica
l setting. We obtain generalizations of some of the existing results: we c
an easily specialize our results to V-categories\, even without assuming V
is symmetric. But we also learn more about how cocompletions behave. The
usual universal property of a cocompletion turns out to be too weak\, so w
e find the appropriate generalization. The formal theories of presheaves a
nd cocompletions also begin to diverge\, exposing differences that are inv
isible in the enriched setting\, though we find conditions that enable us
to construct cocompletions from presheaves.\n\nThis talk is based on joint
work with Nathanael Arkor (Tallinn University of Technology).\n
LOCATION:https://researchseminars.org/talk/SecondVDCW/10/
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