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BEGIN:VEVENT
SUMMARY:Ieke Moerdijk (Universiteit Utrecht)
DTSTART:20210201T083000Z
DTEND:20210201T100000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/1
DESCRIPTION:Title: Varieties of trees.\nby Ieke Moerdijk (Universiteit Utrecht) as part
of Opening Workshop (IRP Higher Homotopy Structures 2021\, CRM-Bellaterra
)\n\n\nAbstract\nIn this lecture we will present various categories of tre
es\, together with Quillen model structures on the associated categories o
f simplicial presheaves. The central example is formed by the category Ome
ga and the complete Segal model structure on its simplicial presheaves\, w
hich provides a model for the homotopy theory of infinity-operads analogou
s to (and in fact\, in a strict sense\, containing) Rezk's complete Segal
model for infinity categories. But unlike this simplicial case\, the case
of trees allows for many variations of the underlying category of trees as
well as of the model structure. To mention a few variations\, one obtains
in this way model categories for the homotopy category of unital operads\
, that of algebras over a given infinity-operad\, and that of infinite loo
p spaces\, for example.\n\nThe lecture will survey these topics in a hopef
ully generally accessible way. Later in the programme\, we will go into so
me more technical aspects of the theory.\n\nReference: G. Heuts\, I. Moerd
ijk\, Trees in Algebra and Geometry – An introduction to dendroidal homo
topy theory (draft of book available on our websites).\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martin Markl (Akademie věd České republiky)
DTSTART:20210201T100000Z
DTEND:20210201T113000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/2
DESCRIPTION:Title: Operads\, properads\, and more.\nby Martin Markl (Akademie věd Čes
ké republiky) as part of Opening Workshop (IRP Higher Homotopy Structures
2021\, CRM-Bellaterra)\n\n\nAbstract\nWe will provide a guided tour throu
gh the menagerie of various operad- and PROP-like structures. Our approach
will be based on pasting schemes\, although other approaches will also be
mentioned.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarah Whitehouse (University of Sheffield)
DTSTART:20210202T083000Z
DTEND:20210202T100000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/3
DESCRIPTION:Title: Model category structures and spectral sequences\nby Sarah Whitehous
e (University of Sheffield) as part of Opening Workshop (IRP Higher Homoto
py Structures 2021\, CRM-Bellaterra)\n\n\nAbstract\nModel categories give
an abstract setting for homotopy theory\, allowing study of different noti
ons of equivalence. I'll give a very brief introduction and review some st
andard examples\, such as the projective model structure on chain complexe
s. Then I'll discuss various categories with associated functorial spectra
l sequences. In such settings\, one can consider a hierarchy of notions of
equivalence\, given by morphisms inducing an isomorphism at a fixed stage
of the associated spectral sequence. I'll discuss model structures with t
hese weak equivalences for filtered complexes\, for bicomplexes and for mu
lticomplexes. This involves joint work with subsets of Joana Cirici\, Dani
ela Egas Santander\, Xin Fu\, Ai Guan and Muriel Livernet.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Urtzi Buijs (Universidad de Málaga)
DTSTART:20210202T100000Z
DTEND:20210202T113000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/4
DESCRIPTION:Title: Rational homotopy theory and higher algebra.\nby Urtzi Buijs (Univer
sidad de Málaga) as part of Opening Workshop (IRP Higher Homotopy Structu
res 2021\, CRM-Bellaterra)\n\n\nAbstract\nRational homotopy theory is a br
anch of topology which studies the “non-torsion” behaviour of the homo
topy type of topological spaces. Despite having its origins in the 60’s
in the work of Daniel Quillen and Dennis Sullivan\, new approaches using h
igher algebra have been developed in the last decades. \n\nIn this talk\,
we will revise some classical constructions in terms of infinity structure
s and show some recent results in this direction. Summarizing\, we will of
fer old wine in a new bottle with some fancy tapas.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:José Manuel Moreno-Fernández (Trinity College\, Dublin)
DTSTART:20210202T113000Z
DTEND:20210202T130000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/5
DESCRIPTION:Title: A spectral sequence for tangent cohomology of algebras over algebraic op
erads.\nby José Manuel Moreno-Fernández (Trinity College\, Dublin) a
s part of Opening Workshop (IRP Higher Homotopy Structures 2021\, CRM-Bell
aterra)\n\n\nAbstract\nWe produce a spectral sequence that converges to th
e operadic cohomology of a fixed algebra over an algebraic operad. Our mai
n tool is that of filtrations arising from towers of cobrations of algebra
s. These play the role in algebra that cell attaching maps and skeletal fi
ltrations do for topological spaces.\n\nAs an application\, we consider th
e rational Adams–Hilton construction on topological spaces\, where our s
pectral sequence is multiplicative and converges to the Chas–Sullivan lo
op product. We also consider relative Sullivan models of a fibration $p$\,
where our spectral sequence converges to the rational homotopy groups of
the identity component of the space of self-fiber-homotopy equivalences of
$p$\; and the Quillen model of a space\, where our spectral sequence conv
erges to the homotopy groups of the classifying space of the identity comp
onent of the self-equivalences of the space.\n\nReferences:\n\n[1] Moreno-
Fernández\, J.\, Tamaroff\, P.\, A spectral sequence for tangent cohomolo
gy of algebraic operads\, arXiv:2008.00876 (2020).\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lennart Meiert (Universiteit Utrecht)
DTSTART:20210203T083000Z
DTEND:20210203T100000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/6
DESCRIPTION:Title: Chromatic localizations of algebraic K-theory.\nby Lennart Meiert (U
niversiteit Utrecht) as part of Opening Workshop (IRP Higher Homotopy Stru
ctures 2021\, CRM-Bellaterra)\n\n\nAbstract\nA classic result of Waldhause
n says essentially that algebraic K-theory preserves rational equivalences
between connective ring spectra. From the viewpoint of chromatic homotopy
theory\, rationalization is just the zeroth level of chromatic localizati
ons. Based on work of Clausen–Mathew–Naumann–Noel we showed in joint
work with Land\, Mathew and Tamme that in general the $n$-th chromatic le
vel of the algebraic $K$-theory of a ring spectrum depends only on the $n$
-th and $(n – 1)$-st chromatic level of the ring spectrum. This has in p
articular implications for red shift questions in the spirit of Ausoni and
Rognes.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Nikolaus (Westfälische Universität Münster)
DTSTART:20210203T100000Z
DTEND:20210203T113000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/7
DESCRIPTION:Title: Polynomial functors and K-theory\nby Thomas Nikolaus (Westfälische
Universität Münster) as part of Opening Workshop (IRP Higher Homotopy St
ructures 2021\, CRM-Bellaterra)\n\n\nAbstract\nWe will report on (long ove
rdue) joint work with Clark Barwick\, Saul Glasman and Akhil Mathew. Algeb
raic $K$-theory of a ring or more generally an additive category is\, by i
ts definition as a group completion\, functorial in additive functors. We
prove that it is in fact functorial in more functors: the so-called polyno
mial functors (in the sense of Eilenberg–Mac Lane) and still satisfies a
universal property. This generalizes previous results by Passi\, Dold and
others. We will in fact show this for a stable $\\infty$-category and pol
ynomial ($= n$-excisive) functors in the sense of Goodwillie. If time perm
its\, we will explain applications of this result for lambda-ring structur
es on algebraic $K$-theory and give the definition of a spectral lambda ri
ng (i.e.\, a higher algebra version of a lambda ring).\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tobias Dyckerhoff (Universität Hamburg)
DTSTART:20210204T083000Z
DTEND:20210204T100000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/8
DESCRIPTION:Title: A categorified Dold–Kan correspondence\nby Tobias Dyckerhoff (Univ
ersität Hamburg) as part of Opening Workshop (IRP Higher Homotopy Structu
res 2021\, CRM-Bellaterra)\n\n\nAbstract\nThe transition from Betti number
s to homology groups was a decisive step turning the subject previously kn
own as combinatorial topology into what is nowadays called algebraic topol
ogy. Further\, the accompanying foundations of homological algebra are of
universal nature so that they can be applied in a wide range of other math
ematical subjects where they have come to play an essential role. \n\nIn t
his talk\, we discuss the idea of categorifying homological algebra one st
ep further\, replacing complexes of abelian groups by complexes of enhance
d triangulated categories\, illustrated by a concrete result: a categorifi
cation of the classical Dold–Kan correspondence\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rune Haugseng (NTNU Trondheim)
DTSTART:20210204T100000Z
DTEND:20210204T113000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/9
DESCRIPTION:Title: Homotopy-coherent distributivity and the universal property of bispans\nby Rune Haugseng (NTNU Trondheim) as part of Opening Workshop (IRP Hig
her Homotopy Structures 2021\, CRM-Bellaterra)\n\n\nAbstract\nStructures w
here we have both a contravariant (pullback) and a covariant (pushforward)
functoriality that satisfy base change can be encoded by functors out of
($\\infty$-)categories of spans (or correspondences). In some cases we hav
e two pushforwards (an 'additive' and a 'multiplicative' one)\, satisfying
a distributivity relation. Such structures can be described in terms of b
ispans (or polynomial diagrams). For example\, commutative semirings can b
e described in terms of bispans of finite sets\, while bispans in finite $
G$-sets can be used to encode Tambara functors\, which are the structure o
n $\\pi_0$ of $G$-equivariant commutative ring spectra. \n\nMotivated by a
pplications of the ∞-categorical upgrade of such descriptions to motivic
and equivariant ring spectra\, I will discuss the universal property of $
(\\infty\,2)$-categories of bispans [1]. This gives a universal way to obt
ain functors from bispans\, which amounts to upgrading 'monoid-like' struc
tures to 'ring-like' ones. In the talk I will focus on the simplest case o
f bispans in finite sets\, where this gives a new construction of the semi
ring structure on a symmetric monoidal $\\infty$-category whose tensor pro
duct commutes with coproducts. \n\nReferences:\n\n[1] Elden Elmanto and Ru
ne Haugseng\, On distributivity in higher algebra I: The universal propert
y of bispans\, arXiv:2010.15722.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Irakli Patchkoria (University of Aberdeen)
DTSTART:20210205T083000Z
DTEND:20210205T100000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/10
DESCRIPTION:Title: Classification of module spectra and Franke's algebraicity conjecture\nby Irakli Patchkoria (University of Aberdeen) as part of Opening Works
hop (IRP Higher Homotopy Structures 2021\, CRM-Bellaterra)\n\n\nAbstract\n
This is all joint work with Piotr Pstrągowski. Given an $E_1$-ring $R$ su
ch that the graded homotopy ring $\\pi_* R$ is $q$-sparse and the global p
rojective dimension $d$ of $\\pi_* R$ is less than $q$\, we show that the
homotopy $(q – d)$-category of Mod($R$) is equivalent to the homotopy $(
q – d)$-category of differential graded modules over $\\pi_*R$. Thus for
such $E_1$-rings the homotopy theory of their modules is algebraic up to
the level $(q – d)$. Examples include appropriate Morava $K$-theories\,
Johnson–Wilson theories\, truncated Brown–Peterson theories and some v
ariations of topological $K$-theory spectra. We also show that the result
is optimal in the sense that $(q – d)$ is the best possible level in gen
eral where algebraicity happens. At the end of the talk we will outline ho
w the results for modules can be generalised to the settings where we do n
ot have compact projective generators. This proves Franke's algebraicity c
onjecture\, which provides a general result when certain nice homology the
ories provide algebraic models for homotopy theories.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Natàlia Castellana (Universitat Autònoma de Barcelona)
DTSTART:20210205T100000Z
DTEND:20210205T113000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/11
DESCRIPTION:Title: Local Gorenstein duality\nby Natàlia Castellana (Universitat Autò
noma de Barcelona) as part of Opening Workshop (IRP Higher Homotopy Struct
ures 2021\, CRM-Bellaterra)\n\n\nAbstract\nThere are several equivalent ch
aracterizatons of local commutative Gorenstein rings\, which\, in stable h
omotopy theory\, have inspired the notions of Gorenstein ring spectrum (Dw
yer–Greenlees–Iyengar [4]) and Gorenstein duality [6]. We investigate
when a commutative ring spectrum $R$ satisfies a homotopical version of lo
cal Gorenstein duality introduced by Barthel–Heard–Valenzuela [2]\, ex
tending the notion previously studied by Greenlees\, and which has structu
ral implications on the homotopy groups of the ring spectrum.\n\nIn order
to do this\, we prove an ascent theorem for local Gorenstein duality along
morphisms of $k$-algebras [1]. Our main examples are of the form $R = C^*
(X\; k)$\, the ring spectrum of cochains on a space $X$ for a field $k$. I
n particular\, we establish local Gorenstein duality in characteristic $p$
for $p$-compact groups [5] and $p$-local finite groups [3] as well as for
$k = \\mathbb Q$ and $X$ a simply connected space which is Gorenstein in
the sense of Dwyer\, Greenlees and Iyengar.\n\nReferences:\n\n[1] Tobias B
arthel\, Natàlia Castellana\, Drew Heard\, and Gabriel Valenzuela\, J. Pu
re Appl. Algebra 225(2) (2021).\n\n[2] Tobias Barthel\, Drew Heard\, and G
abriel Valenzuela\, Local duality for structured ring spectra\, J. Pure Ap
pl. Algebra 222(2) (2018)\, 433–463.\n\n[3] Carles Broto\, Ran Levi\, an
d Bob Oliver\, The homotopy theory of fusion systems\, J. Amer. Math. Soc.
16(4) (2003)\, 779–856.\n\n[4] William Dwyer\, John Greenlees\, and Sri
kanth Iyengar\, Duality in algebra and topology\, Adv. Math. 200(2) (2006)
\, 357–402.\n\n[5] William Dwyer and Clarence Wilkerson\, Homotopy fixed
-point methods for Lie groups and finite loop spaces\, Ann. of Math. (2) 1
39(2) (1994)\, 395–442.\n\n[6] John Greenlees\, Homotopy invariant commu
tative algebra over fields\, In: Building Bridges between Algebra and Topo
logy\, Advanced Courses in Math. CRM Barcelona\, pages 1031–69\, Birkhä
user/Springer\, Cham\, 2018.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jovana Obradović (Matematički institut SANU)
DTSTART:20210201T143000Z
DTEND:20210201T160000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/12
DESCRIPTION:Title: Minimal models of graphs-related operadic algebras\nby Jovana Obrad
ović (Matematički institut SANU) as part of Opening Workshop (IRP Higher
Homotopy Structures 2021\, CRM-Bellaterra)\n\n\nAbstract\nWe construct ex
plicit minimal models for the (hyper)operads governing modular\, cyclic an
d ordinary operads. Algebras for these models are homotopy versions of the
corresponding structures. \n\nThis is joint work with Batanin and Markl.\
n
LOCATION:https://researchseminars.org/talk/OWHHS2021/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manuel Rivera (University of Purdue)
DTSTART:20210201T160000Z
DTEND:20210201T173000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/13
DESCRIPTION:Title: The fundamental group of a simplicial cocommutative coalgebra\nby M
anuel Rivera (University of Purdue) as part of Opening Workshop (IRP Highe
r Homotopy Structures 2021\, CRM-Bellaterra)\n\n\nAbstract\nIn this talk I
will describe a functor F from the category of connected simplicial cocom
mutative coalgebras to differential graded bialgebras satisfying the follo
wing properties:\n\n1) if $C$ is the simplicial cocomutative coalgebra of
chains on a reduced simplicial set $X$\, then the dg bialgebra $F (C )$ is
naturally quasi-isomorphic to the chains on the based loop space of $|X|$
\;\n\n2) $F$ is homotopical in the sense that it sends 'Koszul weak equiva
lences' (also called '$\\Omega$-quasi-isomorphisms') of simplicial cocommu
tative coalgebras to quasi-isomorphisms of dg bialgebras.\n\nThe compositi
on $\\Pi_1=G\\circ H_0\\circ F$\, where $H_0$ denotes zero-th homology and
$G$ denotes group-like elements\, gives rise to a functor from connected
simplicial cocomutative coalgebras to the category of groups\, which recov
ers the fundamental group when applied to chains on a simplicial set. We u
se this construction to extend theorems of Quillen\, Sullivan\, Mandell\,
and Goerss to the setting of non-simply connected spaces. The end goal of
the program is to provide a complete algebraic (homological) characterizat
ion of homotopy types. Some of the results discussed are a joint work with
Mahmoud Zeinalian and Felix Wierstra [1]. \n\nReferences:\n\n[1] Manuel R
ivera\, Felix Wierstra\, Mahmoud Zeinalian\, The simplicial coalgebra of c
hains determines homotopy types rationally and one prime at a time\, arXiv
:2006.05362.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tashi Walde (Technische Universität München)
DTSTART:20210202T143000Z
DTEND:20210202T160000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/14
DESCRIPTION:Title: Higher Segal spaces via higher excision\nby Tashi Walde (Technische
Universität München) as part of Opening Workshop (IRP Higher Homotopy S
tructures 2021\, CRM-Bellaterra)\n\n\nAbstract\nStarting from the classica
l Segal spaces\, Dyckerhoff and Kapranov introduced a hierarchy of what th
ey call higher Segal structures. While the first new level (2-Segal spaces
) has been well studied in recent years\, not much is known about the high
er levels and the hierarchy as a whole.\n\nIn this talk I explain how this
hierarchy can be understood conceptually in close analogy to the manifold
calculus of Goodwillie and Weiss. I describe a natural “discrete manifo
ld calculus" on the simplex category and on the cyclic category\, for whic
h the polynomial functors are precisely the higher Segal objects. Furtherm
ore\, this perspective yields intrinsic categorical characterizations of h
igher Segal objects in the spirit of higher excision.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gustavo Jasso (Universität Bonn)
DTSTART:20210202T160000Z
DTEND:20210202T173000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/15
DESCRIPTION:Title: The Waldhausen S-construction and the symplectic geometry of surfaces a
nd their symmetric products\nby Gustavo Jasso (Universität Bonn) as p
art of Opening Workshop (IRP Higher Homotopy Structures 2021\, CRM-Bellate
rra)\n\n\nAbstract\nIn this talk I will describe how the Waldhausen S-cons
truction and its higher-dimensional variants arise in relation to the symp
lectic geometry of surfaces and their symmetric products. More concretely\
, I will discuss the role of Auroux's partially wrapped Fukaya categories
in this context\, with emphasis in the special case of disks and the $A_\\
infty$-structures that arise in this case.\n\nThe talk is based on joint w
ork with T. Dyckerhoff and Y. Lekili and with T. Dyckerhoff and T. Walde.\
n
LOCATION:https://researchseminars.org/talk/OWHHS2021/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Berwick-Evans (University of Illinois at Urbana-Champaign)
DTSTART:20210201T173000Z
DTEND:20210201T190000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/16
DESCRIPTION:Title: How do field theories detect torsion in topological modular forms?\
nby Daniel Berwick-Evans (University of Illinois at Urbana-Champaign) as p
art of Opening Workshop (IRP Higher Homotopy Structures 2021\, CRM-Bellate
rra)\n\n\nAbstract\nSince the mid 1980s\, there have been hints of a conne
ction between 2-dimensional field theories and elliptic cohomology. This l
ed to Stolz and Teichner's conjectured geometric model for the universal e
lliptic cohomology theory of topological modular forms (TMF)\, for which c
ocycles are 2-dimensional (supersymmetric) field theories. Properties of t
hese field theories lead naturally to the expected integrality and modular
ity properties of classes in TMF. However\, the abundant torsion in TMF ha
s always been mysterious from the field theory point of view. In this talk
\, we will describe a map from 2-dimensional field theories to a cohomolog
y theory that approximates TMF. This map affords a cocycle description of
certain torsion classes. In particular\, we will explain how a choice of a
nomaly cancelation for the supersymmetric sigma model with target $S^3$ de
termines a cocycle representative of the generator of $\\pi_3($TMF$)=\\mat
hbb Z/24$.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Drew Heard (NTNU Trondheim)
DTSTART:20210203T113000Z
DTEND:20210203T130000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/17
DESCRIPTION:Title: Support theory for triangulated categories in algebra and topology\
nby Drew Heard (NTNU Trondheim) as part of Opening Workshop (IRP Higher Ho
motopy Structures 2021\, CRM-Bellaterra)\n\n\nAbstract\nWe will survey the
support theory of triangulated categories through the machinery of tensor
-triangulated geometry. We will discuss the stratification theory of Benso
n–Iyengar–Krause for triangulated categories\, the construction by Bal
mer of the spectrum of a tensor-triangulated category\, and the relation b
etween the two. Time permitting\, we will discuss a recent application to
the category of derived Mackey functors.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Walker Stern (University of Virginia)
DTSTART:20210204T150000Z
DTEND:20210204T163000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/21
DESCRIPTION:Title: Generalizing Quillen's theorem A\nby Walker Stern (University of Vi
rginia) as part of Opening Workshop (IRP Higher Homotopy Structures 2021\,
CRM-Bellaterra)\n\n\nAbstract\nQuillen's Theorem A\, introduced and prove
d in [2]\, provides conditions under which a functor $F : C \\to D$ of 1-c
ategories induces a weak equivalence $ |N(C)| \\to |N(D)|$ of classifying
spaces. In this talk\, we will discuss two possible ways of generalizing t
his criterion: working with functors of 2-categories\, or finding conditio
ns under which $F$ induces an equivalence between some $\\infty$-categoric
al localizations of $C$ and $D$. Combining these two approaches will lead
us to a single generalization of the classical Theorem A to 2-categories e
quipped with a set of marked morphisms. We will sketch the proof of this g
eneralization provided in [1]\, and discuss implications.\n\nThis work is
part of a broader project\, aimed at providing computational techniques fo
r $(\\infty\,2)$-categories in their avatar as scaled simplicial sets. In
the next talk\, Fernando Abellán García will discuss a related facet of
this project\, related to (co)limits in $(\\infty\,2)$-categories.\n\nRefe
rences:\n\n[1] Fernando Abellán García and Walker H. Stern\, Theorem A f
or marked 2-categories\, arXiv:2002.12817.\n\n[2] Daniel Quillen\, Higher
Algebraic K-theory I: Higher K-Theories\, Lecture Notes in Mathematics\, v
ol. 341. Springer\, Berlin\, Heidelberg.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fernando Abellán (Universität Hamburg)
DTSTART:20210204T163000Z
DTEND:20210204T180000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/22
DESCRIPTION:Title: Marked colimits and higher cofinality\nby Fernando Abellán (Univer
sität Hamburg) as part of Opening Workshop (IRP Higher Homotopy Structure
s 2021\, CRM-Bellaterra)\n\n\nAbstract\nIn this talk I will introduce the
notion of marked colimits in $\\infty$-bicategories\, providing a natural
framework to interpret Theorem A$^\\dagger$ from the previous talk as a hi
gher cofinality statement. Relevant examples will be discussed\, as a well
the relation of marked colimits with weighted colimits. The main result t
hat I will discuss is the following:\n\nLet $C^\\dagger$\, $D^\\dagger$ be
$\\infty$-categories equipped with a collection of marked morphisms and l
et $f$ be a functor from $C^\\dagger$ to $D^\\dagger$ that preserves the m
arking. Then $f$ is a marked cofinal functor (i.e.\, restriction along $f$
preserves marked colimits) if and only if the conditions of Theorem A$^\\
dagger$ are satisfied.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emanuele Dotto (University of Warwick)
DTSTART:20210205T141500Z
DTEND:20210205T154500Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/24
DESCRIPTION:Title: Hermitian K-theory of stable infinity-categories\nby Emanuele Dotto
(University of Warwick) as part of Opening Workshop (IRP Higher Homotopy
Structures 2021\, CRM-Bellaterra)\n\n\nAbstract\nThe talk will give an ove
rview of Grothendieck–Witt theory in the higher categorical formalism of
stable infinity-categories equipped with a Poincaré structure. As an exa
mple of the flexibility of this framework\, we will see how to relate the
Grothendieck–Witt groups to Ranicki's $L$-groups and how to prove a stro
ng version of Karoubi's periodicity theorem without assuming that 2 is inv
ertible in the base ring. \n\nThis is joint work with Calmès\, Harpaz\, H
ebestreit\, Land\, Moi\, Nardin\, Nikolaus and Steimle.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guillermo Carrión (Universitat Autònoma de Barcelona)
DTSTART:20210205T154500Z
DTEND:20210205T171500Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/25
DESCRIPTION:Title: Approaching higher limits from homotopy theory\nby Guillermo Carri
ón (Universitat Autònoma de Barcelona) as part of Opening Workshop (IRP
Higher Homotopy Structures 2021\, CRM-Bellaterra)\n\n\nAbstract\nHigher li
mits are the higher derived functors of the inverse limit construction for
functors taking values in abelian groups. Classically\, they are computed
using tools from homological algebra.\n\nIn algebraic topology\, the coho
mology of a homotopy colimit of a diagram spaces can be approached via a s
pectral sequence whose $E_2$ page consists precisely of the higher limits
of the functor obtained from applying cohomology to the diagram of spaces.
In particular\, if the higher limits vanish\, then the cohomology of the
homotopy colimit is just the inverse limit of the cohomologies. There are
many vanishing results\, for example the Mittag-Leffler property [2\, Sect
ion 3.5] or the pseudo-projectivity property [1].\n\nWe study the case whe
re the category $P$ is a poset with an order preserving map $d: P\\to\\mat
hbb N$. If we consider the injective model category on the functor categor
y Fun($P$\, Ch(Ab))\, a functor is pseudo-projective if it is cofibrant.\n
\nIn this talk we will show how we can use the techniques from model categ
ories\, inspired by homotopy theory\, to describe higher limits in this si
tuation when the indexing category is a poset. We will give explicit bound
s for the vanishing of higher limits in terms of properties of the functor
improving previous results.\n\nReferences:\n\n[1] Diaz Ramos\, A.\, A fam
ily of acyclic functors\, Journal of Pure and Applied Algebra\, 213.5 (200
9)\, 783–794.\n\n[2] Weibel\, C.\, An Introduction to Homological Algebr
a\, Cambridge University Press\, 1994.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ralph Kaufmann (University of Purdue)
DTSTART:20210208T151500Z
DTEND:20210208T164500Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/38
DESCRIPTION:Title: Progress in operad-like theories with a focus on Feynman categories
\nby Ralph Kaufmann (University of Purdue) as part of Opening Workshop (IR
P Higher Homotopy Structures 2021\, CRM-Bellaterra)\n\n\nAbstract\nIn the
past years\, several genereralization of operads have been considered. We
will briefly give a quick overview and then turn to a particularly effecti
ve formulation — Feynman categories.\n\nThe basic setup is categorical\,
as the name suggests\, and this allows to consider many natural construct
ions. One important aspect are representations\, which are functors in thi
s setting. These include algebras\, operads and other more intricate or le
ss sophisticated gadgets. In this respect the theory is analogous to repre
sentations of groups with restriction\, induction and Frobenius reciprocit
y.\n\nWe will give a gentle introduction and as time allows\, we may high
light various other constructions and applications achieved ranging from c
ategorical considerations like comprehension schemes and Hopf-algebraic as
pects to moduli spaces of curves.\n\nThese results are partly in collabora
tion with B. Ward\, J. Lucas\, I. Gálvez-Carrillo\, A. Tonks\, C. Berger\
, M. Monaco\, M. Markl and M. Batanin (in chronological order).\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pedro Tamaroff (Trinity College\, Dublin)
DTSTART:20210208T164500Z
DTEND:20210208T181500Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/40
DESCRIPTION:Title: Tangent complexes and the Diamond Lemma: homotopical methods for terms
rewriting\nby Pedro Tamaroff (Trinity College\, Dublin) as part of Ope
ning Workshop (IRP Higher Homotopy Structures 2021\, CRM-Bellaterra)\n\n\n
Abstract\nTerm rewriting has been an indispensable tool to approach variou
s computational problems involving associative algebras and algebraic oper
ads\, their homology theories and their deformation theory [1\, 2\, 3\, 5\
, 6\, 8\, 9]. One of the cornerstones of the theory\, the celebrated Diamo
nd Lemma [4]\, gives an effectively verifiable criterion of uniqueness of
normal forms for term rewriting in associative algebras.\n\nIn joint work
with V. Dotsenko [7]\, we presented a new way to interpret and prove this
result from the viewpoint of homotopical algebra. Our main result states t
hat every multiplicative free resolution of an algebra with monomial relat
ions gives rise to its own Diamond Lemma\, so that Bergman's condition of
"resolvable ambiguities" becomes the first non-trivial component of the Ma
urer–Cartan equation in the corresponding tangent complex. Our approach
works for many other algebraic structures\, such as algebraic operads\, hi
ghlighting the importance of computing multiplicative resolutions of algeb
ras presented by monomial relations\, as it was done in [10].\n\nFor those
whose intuition comes from homotopical algebra\, our work presents a conc
eptual explanation of useful (but seemingly technical) criteria of "resolv
able ambiguities" for uniqueness of normal forms. For those with a backgro
und in Gröbner bases or term rewriting\, our work offers intuition behind
both the Diamond Lemma and its optimisations\, as well as precise guidanc
e on how to generalise those for other algebraic structures. Specifically\
, our work means that computing models of algebras with monomial relations
explicitly helps both to state the relevant Diamond Lemmas and to optimis
e them.\n\nWith the aim of bringing the effective methods of term rewritin
g closer to the powerful methods of homotopical algebra and higher structu
res\, prior knowledge of the techniques involved in our work is not assume
d: they will be explained along the way\, with an emphasis in providing a
working dictionary to go from term rewriting to deformation theory and hom
otopical algebra\, and back.\n\nReferences:\n\n[1] D. J. Anick. On the hom
ology of associative algebras. Trans. Amer. Math. Soc.\, 296(2):641–659\
, 1986.\n\n[2] J. Backelin. La série de Poincaré–Betti d'une algèbre
graduée de type fini à une relation est rationnelle. C. R. Acad. Sci. Pa
ris Ser. A-B\, 287(13): A843–A846\, 1978.\n\n[3] M. J. Bardzell. Noncomm
utative Gröbner bases and Hochschild cohomology. In: Symbolic computation
: solving equations in algebra\, geometry\, and engineering (South Hadley\
, MA\, 2000)\, vol. 286 of Contemp. Math.\, pp. 227–240. Amer. Math. Soc
.\, Providence\, RI\, 2001.\n\n[4] G. M. Bergman. The diamond lemma for ri
ng theory. Adv. Math.\, 29(2):178–218\, 1978.\n\n[5] V. Dotsenko\, V. Ge
linas\, P. Tamaroff. Finite generation for Hochschild cohomology of Gorens
tein monomial algebras. arXiv:1909.00487.\n\n[6] V. Dotsenko and A. Khoros
hkin. Quillen homology for operads via Gröbner bases. Doc. Math.\, 18:707
–747\, 2013.\n\n[7] V. Dotsenko and P. Tamaroff. Tangent complexes and t
he Diamond Lemma. arXiv:2010.14792.\n\n[8] E. S. Golod. Standard bases and
homology. In Algebra – some current trends (Varna\, 1986)\, vol. 1352 o
f Lecture Notes in Math.\, pp. 88–95. Springer\, Berlin\, 1988.\n\n[9] E
. L. Green\, D. Happel\, and D. Zacharia. Projective resolutions over Arti
n algebras with zero relations. Illinois J. Math.\, 29(1):180–190\, 1985
.\n\n[10] P. Tamaroff. Minimal models for monomial algebras. Homology Homo
topy Appl.\, 23(1): 341–366\, 2021.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Fuentes-Keuthan (Johns Hopkins University)
DTSTART:20210209T151500Z
DTEND:20210209T164500Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/43
DESCRIPTION:Title: Goodwillie towers of ∞-categories and desuspension\nby Daniel Fue
ntes-Keuthan (Johns Hopkins University) as part of Opening Workshop (IRP H
igher Homotopy Structures 2021\, CRM-Bellaterra)\n\n\nAbstract\nWe reconce
ptualize the process of forming $n$-excisive approximations to $\\infty$-c
ategories\, in the sense of Heuts [1]\, as inverting the suspension functo
r lifted to $A_n$-cogroup objects. We characterize $n$-excisive $\\infty$
-categories as those $\\infty$-categories in which $A_n$-cogroup objects a
dmit desuspensions. Applying this result to pointed spaces we reprove a th
eorem of Klein–Schwänzl–Vogt [2]: every 2-connected cogroup-like $A_\
\infty$-space admits a desuspension. \n\nThis is joint work with Klaus O.
Johnson and Michael River.\n\nReferences:\n\n[1] G. Heuts\, Goodwillie app
roximations to higher categories\, arXiv:1510.03304\, 2018.\n\n[2] J. Klei
n\, R. Schwänzl\, and R. Vogt\, Comultiplication and suspension\, Topolog
y and its Applications\, 1997.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kate Ponto (University of Kentucky)
DTSTART:20210209T164500Z
DTEND:20210209T181500Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/44
DESCRIPTION:Title: Traces from K-theory and zeta functions\nby Kate Ponto (University
of Kentucky) as part of Opening Workshop (IRP Higher Homotopy Structures 2
021\, CRM-Bellaterra)\n\n\nAbstract\nWhen defining mathematical invariants
there is usually give and take between computability and power. Algebraic
$K$-theory imposes a very useful additivity property but still leaves us
with significant computational difficulty. Considering homomorphisms from
$K$-theory to other groups via the Dennis trace and its spectral generaliz
ations is one way to approach this problem. In this talk I’ll describe s
ettings where this often opaque map can be connected to characteristic pol
ynomials and zeta functions.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthew B. Young (Utah State University)
DTSTART:20210209T181500Z
DTEND:20210209T194500Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/45
DESCRIPTION:Title: Real categorical representation theory in topology and physics\nby
Matthew B. Young (Utah State University) as part of Opening Workshop (IRP
Higher Homotopy Structures 2021\, CRM-Bellaterra)\n\n\nAbstract\nI will gi
ve an overview of the theory of Real categorical representations of a fini
te group\, as developed in [1\, 2\, 3]. In this theory\, a $\\mathbb Z_2$-
graded finite group acts on a category by autoequivalences or anti-autoequ
ivalences\, according to the $\\mathbb Z_2$-grading. There is a natural ge
ometric character theory of such representations which is most naturally f
ormulated in terms of unoriented mapping spaces. In this way\, one obtains
an unoriented generalization of the 2-character theory of Ganter–Kapran
ov and a candidate for a Hopkins–Kuhn–Ravenel-type character theory fo
r a conjectural Real equivariant elliptic cohomology theory. Time permitti
ng\, I will explain how Real categorical representation theory is related
to unoriented Dijkgraaf–Witten theory\, a three dimensional topological
quantum field theory [4].\n\nReferences:\n\n[1] D. Rumynin and M. Young. B
urnside rings for Real 2-representations: The linear theory. Commun. Conte
mp. Math.\, to appear. arXiv:1906.11006.\n\n[2] B. Noohi and M. Young. Twi
sted loop transgression and higher Jandl gerbes over finite groupoids. arX
iv:1910.01422\, 2019.\n\n[3] M. Young. Real representation theory of finit
e categorical groups. Higher Struct.\, to appear. arXiv:1804.09053.\n\n[4]
M. Young. Orientation twisted homotopy field theories and twisted unorien
ted Dijkgraaf–Witten theory. Commun. Math. Phys.\, 374(3):1645–1691\,
2020.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Inbar Klang (University of Columbia)
DTSTART:20210211T151500Z
DTEND:20210211T164500Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/53
DESCRIPTION:Title: Equivariant factorization homology and tools for studying it\nby In
bar Klang (University of Columbia) as part of Opening Workshop (IRP Higher
Homotopy Structures 2021\, CRM-Bellaterra)\n\n\nAbstract\nFactorization h
omology arose from Beilinson–Drinfeld's algebro-geometric approach to co
nformal field theory\, and from study of labeled configuration spaces due
to McDuff\, Segal\, Salvatore\, Andrade\, and others. In this talk\, I wil
l give an introduction to factorization homology and equivariant factoriza
tion homology. I will then discuss joint work with Asaf Horev and Foling Z
ou\, in which we prove a "non-abelian Poincaré duality" theorem for equi
variant factorization homology\, and study the equivariant factorization h
omology of equivariant Thom spectra. In particular\, this provides an aven
ue for computing certain equivariant analogues of topological Hochschild h
omology.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hiro Lee Tanaka (Texas State University)
DTSTART:20210211T164500Z
DTEND:20210211T181500Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/54
DESCRIPTION:Title: Broken lines and Floer Theory over spectra\nby Hiro Lee Tanaka (Tex
as State University) as part of Opening Workshop (IRP Higher Homotopy Stru
ctures 2021\, CRM-Bellaterra)\n\n\nAbstract\nI will discuss a program\, jo
int with Jacob Lurie\, to enrich Lagrangian Floer theory over stable homot
opy theory. Success would open new\, symplecto-geometric techniques for st
udying stable homotopy theory. In this talk I will discuss a stack of brok
en lines and explain how factorizable structures on sheaves on this stack
encode the higher homotopical data of $A_\\infty$-algebras. If time allows
\, I will discuss the deformation-theoretic data encoded in natural exampl
es\, and explain how this allows one to enrich Floer theory over spectra.\
n\nReferences:\n\n[1] Lurie\, Jacob and Tanaka\, Hiro Lee\, Associative al
gebras and broken lines\, arXiv:1805.09587.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Angélica Osorno (Reed College)
DTSTART:20210211T181500Z
DTEND:20210211T194500Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/55
DESCRIPTION:Title: 2-categorical Opfibrations\, Quillen's Theorem B\, and $S^{-1}S$\nb
y Angélica Osorno (Reed College) as part of Opening Workshop (IRP Higher
Homotopy Structures 2021\, CRM-Bellaterra)\n\n\nAbstract\nQuillen recogniz
ed the higher algebraic $K$-groups of a commutative ring $R$ as the homoto
py groups of the topological group completion of the classifying space of
the category of finitely generated projective $R$-modules. He moreover pro
ved that the topological group completion could be obtained categorically
via his $S^{–1}S$ construction. In this talk we will present a 2-categor
ical version of this result. As part of the proof\, we will give a compari
son between strict and lax pullbacks for 2-categorical opfibrations\, whic
h gives a version of Quillen's Theorem B amenable to applications. This is
joint work with Nick Gurski and Niles Johnson.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Batanin (Centre de Recerca Matemàtica)
DTSTART:20210212T083000Z
DTEND:20210212T100000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/56
DESCRIPTION:Title: Grothendieck homotopy theory and polynomial monads\nby Michael Bata
nin (Centre de Recerca Matemàtica) as part of Opening Workshop (IRP Highe
r Homotopy Structures 2021\, CRM-Bellaterra)\n\n\nAbstract\nGrothendieck d
eveloped in Pursuing Stacks [5] a beautiful axiomatic approach to homotopy
theory as a theory of localisations of the category of small categories C
at. This theory was further deepened by Cisinski in [3\, 4].\n\nThe catego
ry of polynomial monads PolyMon contains Cat as a full subcategory of "lin
ear" monads. It is natural to ask how the homotopy theory of Grothendieck\
, or at least some parts of it\, can be extended to PolyMon.\n\nIn this ta
lk we show that many fundamental constructions of Grothendieck theory have
their analogues in the word of polynomial monads. This includes: Quillen
Theorem A\, Grothendieck constructions\, Thomason theorem\, homotopy left
Kan extensions\, the theory of final functors\, Cisinski localisations etc
.\n\nAs illustration we consider two applications:\n\nA theory of deloopin
g of mapping spaces between algebras of polynomial monads developed in [1]
. A seminal theorem of Dwyer–Hess–Turchin of double delooping of space
of long knots will be a consequence.\n\nA theory of locally constant alge
bras of polynomial monads from [2] which generalises Cisinski theory of lo
cally constant functors [3\, 4]. If time permits a sketch of a proof of st
abilisation theorem for higher braided operads which\, in its turn\, impli
es Baez–Dolan stabilisation for higher categories\, will be provided.\n\
nReferences:\n\n[1] Batanin M. and De Leger F.\, Polynomial monads and del
ooping of mapping spaces\, J. Noncommut. Geom. 13 (2019)\, 1521–1576.\n\
n[2] Batanin M. and White D.\, Homotopy theory of algebras of substitudes
and their localisation\, arXiv:2001.05432.\n\n[3] Cisinski D.-C.\, Les pr
éfaisceaux comme modèles des types d'homotopie\, Astérisque 308\, 2006.
\n\n[4] Cisinski D.-C.\, Locally constant functors\, Math. Proc. Cambridge
Philos. Soc. 147 (2009)\, 593–614.\n\n[5] Grothendieck A.\, Pursuing St
acks\, Manuscript\, 1983.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martina Rovelli (Australian National University)
DTSTART:20210212T100000Z
DTEND:20210212T113000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/57
DESCRIPTION:Title: Exploring (∞\, n)-categories through n-complicial sets\, part I\n
by Martina Rovelli (Australian National University) as part of Opening Wor
kshop (IRP Higher Homotopy Structures 2021\, CRM-Bellaterra)\n\n\nAbstract
\nWith the rising significance of $(\\infty\, n)$-categories\, it is impor
tant to have easy-to-handle models for those and understand them as much a
s possible. In these talks we will discuss the model of $n$-complicial set
s\, and study how one can realize convenient representatives of strict $n$
-categories\, which encode universal indexing shapes for diagrams valued i
n $(\\infty\, n)$-categories. We will focus on $n = 2$\, for which more re
sults are available\, but keep an eye towards the general case.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Viktoriya Ozornova (Ruhr Universität Bochum)
DTSTART:20210212T113000Z
DTEND:20210212T130000Z
DTSTAMP:20260415T055104Z
UID:OWHHS2021/58
DESCRIPTION:Title: Exploring (∞\, n)-categories through n-complicial sets\, part II\
nby Viktoriya Ozornova (Ruhr Universität Bochum) as part of Opening Works
hop (IRP Higher Homotopy Structures 2021\, CRM-Bellaterra)\n\n\nAbstract\n
With the rising significance of $(\\infty\, n)$-categories\, it is importa
nt to have easy-to-handle models for those and understand them as much as
possible. In these talks we will discuss the model of $n$-complicial sets\
, and study how one can realize convenient representatives of strict $n$-c
ategories\, which encode universal indexing shapes for diagrams valued in
$(\\infty\, n)$-categories. We will focus on $n = 2$\, for which more resu
lts are available\, but keep an eye towards the general case.\n
LOCATION:https://researchseminars.org/talk/OWHHS2021/58/
END:VEVENT
END:VCALENDAR