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BEGIN:VEVENT
SUMMARY:Robert Paré (Dalhousie University)
DTSTART;VALUE=DATE-TIME:20221128T173000Z
DTEND;VALUE=DATE-TIME:20221128T183000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/1
DESCRIPTION:Title: Some things about double categories\nby Robert
Paré (Dalhousie University) as part of Virtual Double Categories Worksho
p\n\n\nAbstract\nThe title says it all. I will look at some examples as an
excuse to introduce and motivate the basic concepts of double category th
eory. This way I hope to bring them to life so that\, in the end\, they mi
ght be considered as friends and not mere acquaintances (or worse). No pri
or knowledge of double categories will be assumed.\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthew Di Meglio (University of Edinburgh)
DTSTART;VALUE=DATE-TIME:20221128T183000Z
DTEND;VALUE=DATE-TIME:20221128T193000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/2
DESCRIPTION:Title: Recognising retromorphisms retrospectively\nby
Matthew Di Meglio (University of Edinburgh) as part of Virtual Double Cat
egories Workshop\n\n\nAbstract\nWhat do categories\, preordered sets\, met
ric spaces\, and topological spaces have in common? They all have an under
lying set\, and\, when viewed in the right way\, they all have some additi
onal data that relates the elements of the underlying set. For each of the
se kinds of mathematical objects\, there are two natural kinds of morphism
s. The usual kind of morphisms\, which we will refer to merely as morphism
s\, consist of a function between the underlying sets\, and a mapping of t
he additional data in the same direction as the function. For example\, fu
nctors and continuous maps are morphisms of categories and topological spa
ces\, respectively. The other kind of morphisms\, which we will refer to a
s retromorphisms\, consist of a function between the underlying sets\, and
a lifting of the additional data in the opposite direction to that of the
function. For example\, cofunctors and open maps are retromorphisms of ca
tegories and topological spaces\, respectively. In all cases\, the morphis
ms and retromorphisms assemble into a double category whose cells capture
a notion of compatibility between the two kinds of morphisms.\n\nActually\
, for any double category with chosen companions\, there is an associated
double category of monad morphisms and monad retromorphisms\, and all of t
he above examples arise in this way. My talk will give a gentle introducti
on to these double-categorical concepts\, focusing on how the above exampl
es and several more fit into the general theory. No prior knowledge about
double categories will be assumed. This extends recent work (arXiv:2209.01
144) on enriched cofunctors beyond the setting of enriched category theory
.\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paula Verdugo (Macquarie University)
DTSTART;VALUE=DATE-TIME:20221128T193000Z
DTEND;VALUE=DATE-TIME:20221128T203000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/3
DESCRIPTION:Title: An homotopical way to compare 2-categories and dou
ble categories\nby Paula Verdugo (Macquarie University) as part of Vir
tual Double Categories Workshop\n\n\nAbstract\nIt has proven useful to stu
dy some aspects of 2-categories within the framework of double categories
– for example\, there are some 2-categorical universal properties that e
lude a purely 2-categorical description but show themselves naturally expr
essed in the double categorical world. In this talk we present ways to com
pare these two 2-dimensional worlds homotopically by means of model catego
ries. If time permits\, we will explore how we may generalize this idea to
higher dimensions. This is joint work with Lyne Moser and Maru Sarazola.
Basic knowledge about (2- and double) categories will be useful\, together
with the idea behind model categories.\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seerp Roald Koudenburg (Middle East Technical University\, Norther
n Cyprus Campus)
DTSTART;VALUE=DATE-TIME:20221129T150000Z
DTEND;VALUE=DATE-TIME:20221129T160000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/4
DESCRIPTION:Title: Formalising size in formal category theory\nby
Seerp Roald Koudenburg (Middle East Technical University\, Northern Cypru
s Campus) as part of Virtual Double Categories Workshop\n\n\nAbstract\nFor
malising the classical notion of Yoneda embedding necessarily includes for
malising the notion of "locally small functor"\, that is functors $f\\colo
n A\\to C$ such that all homsets $C(fa\, c)$ are small. A Yoneda structure
on a 2-category for instance includes the notion of "admissible morphism"
which formalises that of locally small functor.\n\nIn a (generalised) dou
ble categorical approach to formal Yoneda embeddings\, instead of postulat
ing a notion of "locally small morphism"\, it is natural to regard *all* h
orizontal (say) morphisms to be "locally small". This gives an intrisic no
tion of local smallness which fruitfully interacts with other formal notio
ns such as that of adjunct vertical morphism\, fully faithful vertical mor
phism and restriction of horizontal morphisms.\n\nThe aim of this talk is
to introduce a double categorical approach to formal Yoneda embeddings wit
hout presuming familiarity with formal category theory. Motivated by obtai
ning a common formalisation of the classical notions of generic subobject
(of topos theory) and Yoneda embedding\, we are led to consider "augmented
virtual double categories" as the right setting for doing so. These gener
alise double categories by allowing cells with paths of horizontal morphis
ms as horizontal source and\, as horizontal target\, either a single horiz
ontal morphism or an empty path. We conclude by looking at several formal
results that involve formal notions of smallness such as\, for instance\,
a description of the sense in which a formal presheaf object $P$\, defined
by a formal Yoneda embedding $y\\colon A\\to P$\, is cocomplete.\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christina Vasilakopoulou (National Technical University of Athens\
, School of Applied Mathematics and Physical Sciences)
DTSTART;VALUE=DATE-TIME:20221129T160000Z
DTEND;VALUE=DATE-TIME:20221129T170000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/5
DESCRIPTION:Title: Monoidal Kleisli double categories\nby Christi
na Vasilakopoulou (National Technical University of Athens\, School of App
lied Mathematics and Physical Sciences) as part of Virtual Double Categori
es Workshop\n\n\nAbstract\nIn this talk\, we will describe how to formally
extend the so-called arithmetic product of symmetric sequences to colored
symmetric sequences\, namely their many-object analogue. This requires ge
neral results on extending monoidal structures to Kleisli constructions\,
specifically in the world of double categories. Some basic knowledge of fi
brant double categories is assumed\, but involved concepts and tools shall
be in general described throughout the talk.\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simona Paoli (University of Aberdeen)
DTSTART;VALUE=DATE-TIME:20221201T160000Z
DTEND;VALUE=DATE-TIME:20221201T170000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/6
DESCRIPTION:Title: Double categories and weak units\nby Simona Pa
oli (University of Aberdeen) as part of Virtual Double Categories Workshop
\n\n\nAbstract\nSegal-type models of weak 2-categories are simplicial mode
ls of bicategories\, and they comprise the Tamsamani model and the more re
cent weakly globular double categories\, introduced by Paoli and Pronk. Fa
ir 2-categories\, introduced by J. Kock\, model weak 2-categories with str
ictly associative compositions and weak unit laws. This model has some fea
tures in common with the Segal-type models\, but with the simplicial delta
replaced by the fat delta\, which encodes weak units.\n\nI will illustrat
e a direct comparison of fair 2-categories with weakly globular double cat
egories: this result sheds new light on the notion of weak globularity as
encoding weak units and has potential for higher dimensional generalizatio
ns.\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brandon Shapiro (Topos Institute)
DTSTART;VALUE=DATE-TIME:20221129T183000Z
DTEND;VALUE=DATE-TIME:20221129T193000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/7
DESCRIPTION:Title: Double Presheaf Categories via Polynomial Functors
\nby Brandon Shapiro (Topos Institute) as part of Virtual Double Categ
ories Workshop\n\n\nAbstract\nThe category Poly of polynomial functors on
sets and natural transformations between them forms a monoidal category un
der composition. By results of Ahman–Uustalu and Garner\, the bicategory
of comonoids and bicomodules in Poly has as objects small categories and
as morphisms from C to D familial (aka parametric right adjoint) functors
from C-Set to D-Set\, the categories of Set-valued functors from C and D.
Familial functors between (co)presheaf categories are defined as disjoint
unions of representables on each component\, and include the free category
monad on graphs and data migration functors between categories of databas
es.\n\nPolynomial functors on categories also have a monoidal structure by
composition\, where the comonoids are precisely small double categories s
atisfying a factorization condition that includes most double categories i
n common use. Bicomodules between such double categories \\(\\bf C\\) and
\\(\\bf D\\) induce "familial" double functors from \\({\\bf C}^t\\)-Set t
o \\({\\bf D}^t\\)-Set\, the copresheaf double categories of lax double fu
nctors from the transpose of \\(\\bf C\\) and \\(\\bf D\\) to the double c
ategory of sets\, functions\, spans\, and maps of spans.\n\nI will introdu
ce polynomial functors\, comonoids and bicomodules\, familial functors\, a
nd double copresheaf categories to build up to this result (joint with Dav
id Spivak) and its early implications for double category theory and categ
orical databases\, assuming background only in double categories and ordin
ary (co)presheaf categories.\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chad Nester (Tallinn University of Technology)
DTSTART;VALUE=DATE-TIME:20221130T150000Z
DTEND;VALUE=DATE-TIME:20221130T160000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/8
DESCRIPTION:Title: The Free Cornering of a Monoidal Category\nby
Chad Nester (Tallinn University of Technology) as part of Virtual Double C
ategories Workshop\n\n\nAbstract\nThe free cornering of a monoidal categor
y is a single-object double category obtained by adding companion and conj
oint structure to it. If the morphisms of the monoidal category admit inte
rpretation as processes\, then this interpretation extends to the cells of
the free cornering\, which then admit interpretation as interacting proce
sses. In this talk I will introduce the free cornering construction\, expl
ain its interactive interpretation\, and survey a number of results surrou
nding the free cornering from both published and ongoing work. \n\nSingle-
object double categories are much simpler than their more general counterp
arts. I expect that most of the talk will be accessible to anyone with eve
n a surface understanding of monoidal categories\, and in particular their
string diagrams.\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Bourke (Masaryk University)
DTSTART;VALUE=DATE-TIME:20221130T160000Z
DTEND;VALUE=DATE-TIME:20221130T170000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/9
DESCRIPTION:Title: A double-categorical look at different flavours of
factorisation system\nby John Bourke (Masaryk University) as part of
Virtual Double Categories Workshop\n\n\nAbstract\nOrthogonal factorisation
systems are a simple and natural categorical structure that is not hard t
o grasp. Algebraic weak factorisation systems\, which generalise the orth
ogonal ones\, are not so straightforward\, involving monads and comonads a
nd quite a lot of structure to keep track of. However these structures ca
n be approached via double categories and this double-categorical perspect
ive is very useful. I will tell a bit of the story of the double-categori
cal approach to these structures.\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolas Behr (CNRS\, IRIF\, Université Paris Cité)
DTSTART;VALUE=DATE-TIME:20221130T173000Z
DTEND;VALUE=DATE-TIME:20221130T183000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/10
DESCRIPTION:Title: Double-categorical Compositional Rewriting Theory
\nby Nicolas Behr (CNRS\, IRIF\, Université Paris Cité) as part of V
irtual Double Categories Workshop\n\n\nAbstract\nReporting on recent resul
ts of joint work with R. Harmer\, P.-A. Melliès and N. Zeilberger\, I wil
l present a novel formalization of compositional rewriting theory via doub
le categories. For a given rewriting theory\, individual rewriting steps a
re formalized as 2-cells of a double category. One of the crucial aspects
of compositionally consists then in providing a set of axioms that the dou
ble category of the rewriting system must satisfy in order to ensure the e
xistence of concurrency and associativity theorems.\, which are quintessen
tial for developing important applications of rewriting systems such as In
combinatorics and Markov chain theory. Another concept central to this en
d\, I.e.\, “counting modulo universal properties”\, may be implemented
via a certain presheaf and coend calculus. Time permitting\, I will sketc
h how the counting calculus then leads to a categorification of the concep
t of rule algebras (which capture the combinatorics of interactions of rew
riting steps).\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Susan Niefield (Union College\, Schenectady\, NY)
DTSTART;VALUE=DATE-TIME:20221130T183000Z
DTEND;VALUE=DATE-TIME:20221130T193000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/11
DESCRIPTION:Title: Cartesian Closed Double Categories\nby Susan
Niefield (Union College\, Schenectady\, NY) as part of Virtual Double Cate
gories Workshop\n\n\nAbstract\nIn this talk\, we consider approaches to ex
ponentiability in double categories. For a 1-category $\\mathcal D$\, one
can define cartesian closure via a pointwise or a 2-variable adjunction\,
and arrive at equivalent definitions. The pointwise approach requires the
existence of an exponential object $Z^Y$\, for every object $Y$ of $\\math
cal D$\, whereas for the latter\, the exponentials are given by a bifuncto
r ${\\mathcal D}^{\\rm op}\\times {\\mathcal D}\\to \\mathcal D$. We will
see that these two approaches differ for a double categories $\\mathbb D$.
In particular\, a bifunctor ${\\mathbb D}^{\\rm op}\\times {\\mathbb D}\\
to {\\mathbb D}$ would involve not only objects of $\\mathbb D_0$\, but al
so those of $\\mathbb D_1$\, i.e.\, vertical morphisms of $\\mathbb D$.\n\
nWe will assume some familiarity with double categories\, but recall the d
efinitions and properties of adjoints on double categories\, as well as th
e relevant examples.\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lyne Moser (University of Regensburg)
DTSTART;VALUE=DATE-TIME:20221201T150000Z
DTEND;VALUE=DATE-TIME:20221201T160000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/12
DESCRIPTION:Title: Representation theorem for enriched categories\nby Lyne Moser (University of Regensburg) as part of Virtual Double Cate
gories Workshop\n\n\nAbstract\nUniversal properties play an important role
in mathematics\, as they allow us to make many constructions such as (co)
limits\, Kan extensions\, adjunctions\, etc. In particular\, a universal p
roperty is often formulated by requiring that a certain presheaf is repres
entable. The representation theorem gives a useful characterization of the
se representable presheaves in terms of initial objects in their category
of elements. Going one dimension up and considering 2-categories\, with ts
lil clingman we showed that the same results is true for Cat-valued preshe
aves if one considers their "double category of elements". In this talk\,
I will explain how to generalize the representation theorem to the more ge
neral framework of V-enriched categories\, where V is a cartesian closed c
ategory. This is joint work with Maru Sarazola\, and Paula Verdugo. Only b
asic knowledge of ordinary category theory will be assumed.\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bojana Femić (Mathematical Institute of Serbian Academy of Scienc
es and Arts)
DTSTART;VALUE=DATE-TIME:20221129T173000Z
DTEND;VALUE=DATE-TIME:20221129T183000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/13
DESCRIPTION:Title: Gray-type monoidal product and Bifunctor Theorem
for double categories\nby Bojana Femić (Mathematical Institute of Ser
bian Academy of Sciences and Arts) as part of Virtual Double Categories Wo
rkshop\n\n\nAbstract\nIn the introductory part we will recall what double
categories are and we will\nshow some examples that illustrate why double
categories present a more\nsuitable framework to work in than bicategories
. We proceed by describing the\nconstruction of a Gray type monoidal produ
ct for double categories for two\nversions of morphisms: strict and lax do
uble functors. Along the way we\ncharacterize (lax double) quasi-functors(
in analogy to "quasi-functors of two\nvariables" for 2-categories of Gray
). We introduce their 2-category\n$q\\text{-}\\operatorname{Lax}_{hop}(\\m
athbb{A}\\times\\mathbb{B}\,\\mathbb{C})$\nand construct a 2-functor\n$\\m
athcal{F}: q\\text{-}\\operatorname{Lax}_{hop}(\\mathbb{A}\\times\\mathbb{
B}\,\\mathbb{C}) \\to \\operatorname{Lax}_{hop}(\\mathbb{A}\\times\\mathbb
{B}\,\\mathbb{C})$\nto the 2-category of lax\ndouble bifunctors. This is a
double category version of the Bifunctor Theorem\,\nrecently proved for 2
-categories. We will show when this 2-functor $\\mathcal{F}$\nrestricts to
a 2-equivalence. For a consequence we derive 2-functors known as\ncurryin
g and uncurrying functors in Computer Science\, in the context of double\n
categories. We finish by showing the application of the 2-functor $\\mathc
al{F}$ on\n2-monads. Concretely: we obtain that $\\mathcal{F}$ is a genera
lization to non-trivial\ndouble categories of the classical Beck's result\
, that a distributive law\nbetween two monads makes it possible for them t
o compose. It turns out that a\ndistributive law is a specification of a (
lax double) quasi-functor to trivial\ndouble categories. No prior knowledg
e of double categories is necessary\, but\nknowing 2-categories\, their fu
nctors\, transformations and modifications is\nhelpful.\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juan Orendain (Case Western Reserve University)
DTSTART;VALUE=DATE-TIME:20221201T173000Z
DTEND;VALUE=DATE-TIME:20221201T183000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/14
DESCRIPTION:Title: How long does it take to frame a bicategory?\
nby Juan Orendain (Case Western Reserve University) as part of Virtual Dou
ble Categories Workshop\n\n\nAbstract\nFramed bicategories are double cate
gories having all companions and conjoints. Many structures naturally orga
nize into framed bicategories\, e.g. open Petri nets\, polynomials functor
s\, polynomial comonoids\, structured cospans\, etc. Symmetric monoidal st
ructures on framed bicategories descend to symmetric monoidal structures o
n bicategories. The axioms defining symmetric monoidal double categories a
re more tractible\, and arguably more natural\, than those defining symmet
ric monoidal bicategories. It is thus convenient to study ways of lifting
a given bicategory into a framed bicategory along an appropriate category
of vertical morphisms. Solutions to the problem of lifting bicategories to
double categories have been useful in expressing Kelly and Street's mates
correspondence and in proving the higher dimensional Seifert-van Kampen t
heorem of Brown et. al.\, amongst many other applications. We consider lif
ting problems in their full generality.\n\nGlobularly generated double cat
egories are minimal solutions to lifting problems of bicategories into dou
ble categories along given categories of vertical arrows. Globularly gener
ated double categories form a coreflective sub-2-category of general doubl
e categories. This\, together with an analysis of the internal structure o
f globularly generated double categories yields a numerical invariant on g
eneral double categories. We call this number the vertical length. The ver
tical length of a double category C measures the complexity of composition
s of globular and horizontal identity squares of C and thus provides a mea
sure of complexity for lifting problems on the horizontal bicategory of C.
I will explain recent results on the theory of globularly generated doubl
e categories and the vertical length invariant. Minimal knowledge of doubl
e categories and bicategories will be assumed.\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dorette Pronk (Dalhousie University\, Department of Mathematics an
d Statistics)
DTSTART;VALUE=DATE-TIME:20221201T183000Z
DTEND;VALUE=DATE-TIME:20221201T193000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/15
DESCRIPTION:Title: Fibrations and doubly lax colimits of double cate
gories\nby Dorette Pronk (Dalhousie University\, Department of Mathema
tics and Statistics) as part of Virtual Double Categories Workshop\n\n\nAb
stract\nClassically\, the Grothendieck construction\, or the category of e
lements\, of a pseudo functor \\(F\\colon {\\mathbf D}\\to{\\bf Cat}\\) (a
lso called the indexing functor) has two properties that define it (up to
equivalence):\n\n(A) it is the object part of the adjoint equivalence bet
ween the category of such pseudo functors\, also called indexing functors\
, and the category of fibrations over \\({\\mathbf D}\\)\;\n\n(B) it is th
e lax colimit for the diagram defined by \\(F\\).\n\nThese two results hav
e been shown to hold for suitably defined Grothendieck constructions for b
icategories and \\(\\infty\\)-categories and has been conjectured for weak
\\(n\\)-categories. So when one is interested in colimits of double categ
ories or fibrations between double categories we want to look for a suitab
le Grothendieck construction.\n\nWhen we replace the category \\({\\mathbf
D}\\) above by a double category \\({\\mathbb D}\\)\, the first question
is what the codomain of the indexing functor should be. The issue is that
\\({\\bf DblCat}\\)\, the category of double categories\, is not a double
category. However\, it is enriched over double categories: it has two clas
ses of transformations (which I will call *inner* and *outer*) a
s local arrows and modifications as local double cells. In this talk I wi
ll present two replacements for \\({\\bf DblCat}\\) in the definition of i
ndexing functor. I will replace it with the quintet double category of the
2-category of double categories with double functors and outer transforma
tions and I will replace it by the double 2-category \\({\\mathbb S}\\math
rm{pan(Cat)}\\) (and we will work with all lax functors). The first case h
as the advantage that the objects are still double categories. The second
case is inspired by the fact that a lax double functor from the terminal d
ouble category into \\({\\mathbb S}\\mathrm{pan(Cat)}\\) corresponds to a
double category.\n\nThe last type of indexing functor has an associated Gr
othendieck construction that can be viewed as a Grothendieck construction
that is done in two layers and leads to a category object in a suitable ca
tegory of fibrations: a double fibration.\nThe first type of indexing func
tor leads us to a new notion of what we call a *doubly lax transformatio
n* between double functors of the form \\({\\mathbb A}\\to{\\mathbb Q}{
\\mathcal D}_{\\mathrm{outer}}\\) where \\({\\mathcal D}\\) is a \\({\\bf
DblCat}\\)-enriched category. These give then rise to a notion of *doubl
y lax colimit* for diagrams of double categories. These doubly lax coli
mits can be constructed using the *double Grothendieck construction*
I will introduce\, and have a universal property that is expressed in term
s of both classes of 2-cells of \\({\\bf DblCat}\\). Furthermore\, it also
has a canonical projection double functor to \\({\\mathbb D}\\)\, the ind
exing double category\, and this has the properties of a double fibration
in one direction and a weaker fibration property in the other direction.\n
\nWe will discuss how both constructions generalize various colimit and fi
bration constructions for 2-categories and bicategories and finally\, we w
ill consider how the two are related. A partial answer to this last questi
on is given by the construction of a lax double functor \\({\\mathbb Q}{\\
bf DblCat}_{\\mathrm{outer}}\\to {\\mathbb S}\\mathrm{pan(Cat)}\\).\n\nThi
s talk is based on work done with Geoff Cruttwell\, Michael Lambert and Ma
rtin Szyld in [1] and work done with Marzieh Bayeh and Martin Szyld in [2]
.\n\nBackground needed for this talk: mainly\, the basics of double catego
ries - double functors\, and horizontal and vertical transformations (here
referred to as outer and inner transformations)\; familiarity with the Gr
othendieck construction and its properties is also useful.\n\n[1] G.S.H. C
ruttwell\, M.J. Lambert\, D.A. Pronk\, M. Szyld\, Double Fibrations\, arXi
v:2205.15240\, and to appear in *Theory and Applications of Categories (2022)\n\n[2] Marzieh Bayeh\, Dorette Pronk\, Martin Szyld\, The Grothe
ndieck Construction for Double Categories\, in progress.\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matteo Capucci (University of Strathclyde)
DTSTART;VALUE=DATE-TIME:20221202T160000Z
DTEND;VALUE=DATE-TIME:20221202T170000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/16
DESCRIPTION:Title: From categorical systems theory to categorical cy
bernetics\nby Matteo Capucci (University of Strathclyde) as part of Vi
rtual Double Categories Workshop\n\n\nAbstract\nMyers' categorical system
theory is a double categorical yoga for describing the compositional struc
ture of open dynamical systems. It unifies and builds on previous work on
operadic notions of system theory\, and provides a strong conceptual scaff
olding for behavioral system theory. However\, some of the most interestin
g systems out have a richer compositional structure than that of dynamical
systems. These are cybernetic systems\, or in other words\, interactive c
ontrol systems. Notable and motivating examples are strategic games and ma
chine learning models. In this talk I’m going to introduce the tools and
language of categorical system theory and outline how categorical cyberne
tics theory might look like. At the end\, we will briefly venture into the
triple dimension.\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claudio Pisani (Independent researcher)
DTSTART;VALUE=DATE-TIME:20221202T150000Z
DTEND;VALUE=DATE-TIME:20221202T160000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/17
DESCRIPTION:Title: Operads as double functors\nby Claudio Pisani
(Independent researcher) as part of Virtual Double Categories Workshop\n\
n\nAbstract\nThe language of double categories provides a direct abstract
approach to colored\noperads (that is\, symmetric multicategories)\, enlig
htening and simplifying\nseveral classical facts and notions. Namely\, an
operad \\(\\mathcal{O}\\)\, in its\nnon-skeletal form\, is a product-prese
rving lax double functor \n\\[\n(\\mathbb{P}\\mathrm{b}{\\mathbf{Set_f}})^
{\\mathrm{op}} \\to \\mathbb{S}\\mathrm{et}\n\\tag{1}\n\\]\nfrom the dual
of the double category of pullback squares in\nfinite sets to the double c
ategory of mappings and spans. This\ncharacterization can be rephrased in
two ways. First\, by a universal property\nof the monoid construction one
can consider instead normal functors \\((\\mathbb{P}\\mathrm{b}{\\mathbf{S
et_f}})^{\\mathrm{op}} \\to \\mathbb{C}\\mathrm{at}\\). Second\, by a doub
le category of elements\nconstruction [1\,2]\, operads are certain discret
e double\nfibrations \\(d:\\mathbb{D} \\to \\mathbb{P}\\mathrm{b}{\\mathbf
{Set_f}}\\). The idea is that the proarrow part\nof \\(\\mathbb{D}\\) is t
he category of families of objects and of arrows in \\(\\mathcal{O}\\)\n(i
ndexed by \\({\\mathbf{Set_f}}\\)) and that families of arrows in \\(\\mat
hcal{O}\\) can be\nreindexed along pullback squares in \\({\\mathbf{Set_f}
}\\).\n\nSequential operads [3]\, symmetric monoidal categories and commut
ative\nmonoids are those operads \\(\\mathcal{O}\\) for which the vertical
part of \\(d\\) is a\nfibration\, an opfibration or a discrete opfibratio
n\, respectively. In\nparticular\, we get the accompanying characterizatio
n of commutative monoids as\nproduct-preserving functors \\((\\mathbb{P}\\
mathrm{b} {\\mathbf{Set_f}})^{\\mathrm{op}} \\to \\mathbb{S}\\mathrm{q}{\\
mathbf{Set}}\\).\n\nThis approach suggests a notion of generalized operad
obtained by replacing in\n(1) the category of finite sets with any catego
ry \\(\\mathcal{C}\\). For instance\,\na category with small products (or
sums) gives a generalized monoidal category\nwith \\(\\mathcal{C} = {\\mat
hbf{Set}}\\)\, along with the corresponding generalized monoid of\nisomorp
hism classes. Furthermore\, one can define cartesian operads in this more\
ngeneral setting and prove therein the equivalence between tensor products
\,\nuniversal products and algebraic products (as in the well-known case o
f\ncategories enriched in commutative monoids).\n\nWe assume a basic knowl
edge of the language of double categories and of\nfibrations\; some acquai
ntance with operads or multicategories would be helpful\,\neven if the bas
ic ideas will be recalled.\n\n[1] M. Lambert\, Discrete double fibrations\
, TAC 37 (2021).\n\n[2] R. Paré\, Yoneda theory for double categories\, T
AC 17 (2011).\n\n[3] C. Pisani\, Sequential multicategories\, TAC 29 (2014
).\n\n[4] C. Pisani\, Fibered multicategory theory\, arXiv (2022).\n\n[5]
C. Pisani\, Operads as double functors\, arXiv (2022).\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Jaz Myers (Center for Topological and Quantum Systems\, NYU
Abu Dhabi)
DTSTART;VALUE=DATE-TIME:20221202T173000Z
DTEND;VALUE=DATE-TIME:20221202T183000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/18
DESCRIPTION:Title: The Para Construction as a Distributive Law\n
by David Jaz Myers (Center for Topological and Quantum Systems\, NYU Abu D
habi) as part of Virtual Double Categories Workshop\n\n\nAbstract\n(Joint
with Matteo Capucci)\n\nThe Para construction takes a monoidal category $M
$ and gives a category $\\mathrm{Para}(M)$ where a morphism $a \\to b$ is
a pair $(c\, f : c \\otimes a \\to b)$ of a "parameter space" $c$ and a pa
rameterized map f in M. This construction formalizes the idea of separatin
g inputs into special "control variables" or "parameters" which will be se
t separately from the other inputs to a process. The Para construction has
played an important role in categorical accounts of deep learning — whe
re it was first described by Fong\, Spivak\, and Tuyeras — open games\,
and cybernetics.\n\nThe Para construction has been generalized in a number
of ways. First\, it can take an action of a monoidal category $M$ on a ca
tegory $C$ (an "actegory"). And second\, the resulting category can be see
n as the shadow of a bicategory where 2-cells are reparameterizations. In
this talk\, we will see a further generalization of the scope of the Para
construction — we will take an actegory $\\otimes : M \\times C \\to C$
and produce a double category $\\mathbb{P}\\mathrm{ara}(\\otimes)$ whose v
ertical morphisms are parameterized by objects of $M$ and whose horizontal
morphisms are those of $C$.\n\nWe will show that in this guise\, the Para
construction arises as a (pseudo)distributive law between the action doub
le category of $\\otimes : M \\times C \\to C$ and the double category of
arrows of $C$\, each seen as (pseudo)monads in a tricategory of spans in $
\\mathsf{Cat}$. Our construction is abstract and applies in any suitably c
omplete 2-category $\\mathbb{K}$\, in particular in the 2-category of doub
le categories with vertical transformations. This lets us construct a trip
le category $\\mathrm{Para}(\\mathrm{Arena})$ whose morphisms are lenses\,
charts\, and parameterized lenses respectively. The cubes in this triple
category give representable behaviors of Capucci-Gavranovic-Hedges-Rischel
cybernetic systems\, and one of the resulting face double categories is a
variant of Shapiro and Spivak's double category $\\mathbb{O}\\mathrm{rg}$
. The method of proof also suggests generalizing the domain of the Para co
nstruction to take any "dependent actegory"\, and at this level of general
ity the double category of spans in a cartesian category is revealed to be
a sort of Para construction.\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Lambert (University of Massachusetts Boston)
DTSTART;VALUE=DATE-TIME:20221202T183000Z
DTEND;VALUE=DATE-TIME:20221202T193000Z
DTSTAMP;VALUE=DATE-TIME:20231209T105702Z
UID:VirtualDoubleCategoriesWorkshop/19
DESCRIPTION:Title: Double Categories of Relations\nby Michael La
mbert (University of Massachusetts Boston) as part of Virtual Double Categ
ories Workshop\n\n\nAbstract\nHow can we tell whether a double category is
a double category of relations on a regular category? Any such double cat
egory is at least an equipment and is cartesian\, but what else is needed?
This talk aims to present a characterization theorem that describes these
further conditions. This result should be seen as a double-categorical ve
rsion of the development due to Carboni and Walters that showed which bica
tegories occur as bicategories of relations on a regular category. This ta
lk should be accessible to anyone who knows the basics of bicategories and
double categories. Along the way\, we will review the work of Carboni and
Walters\, and define what it means for a double category to be cartesian
and to be an equipment. We will also talk about tabulators and in what way
a double category can be seen as "functionally complete".\n
LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop
/19/
END:VEVENT
END:VCALENDAR
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