## Talks

Thursday, July 15, 11:00 ~ 11:30 UTC-3

## A cartesian closed category of algebraic theories

### André Joyal

#### Université du Québec à Montréal (UQAM), Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak008463085cf4549bbd8c41a74c6aaa4c').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy008463085cf4549bbd8c41a74c6aaa4c = 'j&#111;y&#97;l.&#97;ndr&#101;' + '&#64;'; addy008463085cf4549bbd8c41a74c6aaa4c = addy008463085cf4549bbd8c41a74c6aaa4c + '&#117;q&#97;m' + '&#46;' + 'c&#97;'; var addy_text008463085cf4549bbd8c41a74c6aaa4c = 'j&#111;y&#97;l.&#97;ndr&#101;' + '&#64;' + '&#117;q&#97;m' + '&#46;' + 'c&#97;';document.getElementById('cloak008463085cf4549bbd8c41a74c6aaa4c').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy008463085cf4549bbd8c41a74c6aaa4c + '\'>'+addy_text008463085cf4549bbd8c41a74c6aaa4c+'<\/a>';

By an "algebraic theory" we mean a small category with finite products. A "combinatorial morphism" of algebraic theories $A\to B$ is defined to be a functor $Mod(A)\to Mod(B)$ preserving sifted colimits. For example, if $u:A\to B$ is a functor preserving products, then the pullback functor $u^\star:Mod(B)\to Mod(A)$ is a combinatorial morphism $B\to A$ and the pushforward functor $u_!:Mod(A)\to Mod(B)$ is a combinatorial morphism $A\to B$. We show that the 2-category of algebraic theories and combinatorial morphisms is cartesian closed.

Joint work with Marcelo Fiore (University of Cambridge, England).

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Thursday, July 15, 11:30 ~ 12:00 UTC-3

## Towards cotangent categories

### Geoff Cruttwell

#### Mount Allison University, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakb847d2d4eda79ac175e7fe00eee86830').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyb847d2d4eda79ac175e7fe00eee86830 = 'gcr&#117;ttw&#101;ll' + '&#64;'; addyb847d2d4eda79ac175e7fe00eee86830 = addyb847d2d4eda79ac175e7fe00eee86830 + 'mt&#97;' + '&#46;' + 'c&#97;'; var addy_textb847d2d4eda79ac175e7fe00eee86830 = 'gcr&#117;ttw&#101;ll' + '&#64;' + 'mt&#97;' + '&#46;' + 'c&#97;';document.getElementById('cloakb847d2d4eda79ac175e7fe00eee86830').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyb847d2d4eda79ac175e7fe00eee86830 + '\'>'+addy_textb847d2d4eda79ac175e7fe00eee86830+'<\/a>';

Tangent categories, first defined by Rosicky (and further developed in a series of recent papers) are a minimal setting for differential geometry (as opposed to synthetic differential geometry, which aims to be a "nicest" setting for differential geometry). They involve a category with an endofunctor on it which behaves like the tangent bundle. Many ideas and results from differential geometry have been generalized to tangent categories, including vector bundles, connections, and differential forms.

In this talk I'll discuss recent progress J.-S. Lemay and I have made towards defining and working with cotangent bundles in the setting of tangent categories. We'll also consider how one could define cotangent categories - a separate axiomatic structure consisting of a category equipped with an abstract cotangent bundle.

Joint work with J.-S. Lemay (Mount Allison University).

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Thursday, July 15, 12:00 ~ 12:30 UTC-3

## A higher Grothendieck construction

### Matias del Hoyo

#### Universidade Federal Fluminense, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak9e6532b425d681b391e27e41fd112f4e').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy9e6532b425d681b391e27e41fd112f4e = 'mld&#101;lh&#111;y&#111;' + '&#64;'; addy9e6532b425d681b391e27e41fd112f4e = addy9e6532b425d681b391e27e41fd112f4e + '&#105;d' + '&#46;' + '&#117;ff' + '&#46;' + 'br'; var addy_text9e6532b425d681b391e27e41fd112f4e = 'mld&#101;lh&#111;y&#111;' + '&#64;' + '&#105;d' + '&#46;' + '&#117;ff' + '&#46;' + 'br';document.getElementById('cloak9e6532b425d681b391e27e41fd112f4e').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy9e6532b425d681b391e27e41fd112f4e + '\'>'+addy_text9e6532b425d681b391e27e41fd112f4e+'<\/a>';

In joint work with G. Trentinaglia, motivated by the study of Lie groupoids, we develop a construction relating simplicial vector bundles and representations up to homotopy. Our construction can be seen both as a higher analog of the Grothendieck correspondence between fibered categories and pseudo-functors in Cat, and also as a relative version of classic Dold-Kan correspondence between simplicial objects and chain complexes. In this short talk, I will briefly discuss the background, present our formulas, and mention some applications.

Joint work with Giorgio Trentinaglia (Instituto Superior Técnico, Portugal).

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Thursday, July 15, 12:30 ~ 13:00 UTC-3

## The Double Category of Measurable Functions and Stochastic Maps

### Evangelia Aleiferi

#### Columbia College, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak1bc0c232f78cb4b309eb66d076aae4b1').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy1bc0c232f78cb4b309eb66d076aae4b1 = '&#101;v&#97;ng&#101;l&#105;&#97;.&#97;l&#101;&#105;f&#101;r&#105;' + '&#64;'; addy1bc0c232f78cb4b309eb66d076aae4b1 = addy1bc0c232f78cb4b309eb66d076aae4b1 + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_text1bc0c232f78cb4b309eb66d076aae4b1 = '&#101;v&#97;ng&#101;l&#105;&#97;.&#97;l&#101;&#105;f&#101;r&#105;' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloak1bc0c232f78cb4b309eb66d076aae4b1').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy1bc0c232f78cb4b309eb66d076aae4b1 + '\'>'+addy_text1bc0c232f78cb4b309eb66d076aae4b1+'<\/a>';

The category of measurable spaces and stochastic maps, $\mathbf{Stoch}$, was introduced by Giry in 1982, following some highly referenced unpublished manuscripts by Lawvere back in 1962. Giry was able to show that the Kleisli category of the Giry monad on the category of measurable functions $\mathbf{Meas}$, is exactly the same as the category $\mathbf{Stoch}$. This shows that $\mathbf{Stoch}$ behaves for $\mathbf{Meas}$ as the category $\mathbf{Rel}$ behaves for the category $\mathbf{Set}$. Inspired by this, and the advantages of combining relations and functions into one structure, that of the double category of sets, relations horizontally, and functions vertically, we are defining the double category of measurable spaces, stochastic maps horizontally, and measurable functions vertically. In this talk we will explore some of the properties of this double category.

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Thursday, July 15, 13:30 ~ 14:00 UTC-3

### Darien DeWolf

#### Brandon University, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak3d74a66afee12dcc1415e2b1081f84f9').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy3d74a66afee12dcc1415e2b1081f84f9 = 'D&#101;W&#111;lfD' + '&#64;'; addy3d74a66afee12dcc1415e2b1081f84f9 = addy3d74a66afee12dcc1415e2b1081f84f9 + 'Br&#97;nd&#111;nU' + '&#46;' + 'c&#97;'; var addy_text3d74a66afee12dcc1415e2b1081f84f9 = 'D&#101;W&#111;lfD' + '&#64;' + 'Br&#97;nd&#111;nU' + '&#46;' + 'c&#97;';document.getElementById('cloak3d74a66afee12dcc1415e2b1081f84f9').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy3d74a66afee12dcc1415e2b1081f84f9 + '\'>'+addy_text3d74a66afee12dcc1415e2b1081f84f9+'<\/a>';

My thesis introduced restriction monads in bicategories containing a suitable 0-cell $E$ and a restriction operator defined by a family of functions indexed by so-called $E$-elemental 1-cells. In $\mathrm{Span}(\mathbf{Set})$, this suitably-defined $E$ works out to be a single-element set, or a terminal object in $\mathbf{Set}$. This talk extends this notion to generalized objects to include bicategories without enough elements. We will then give examples and applications.

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Thursday, July 15, 14:00 ~ 14:30 UTC-3

## Length 1 double categories via monoidal End-indexings

### Juan Orendain

#### National University of Mexico, UNAM, Mexico   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak67f31d2e22bef9b9339e16a10dee1b05').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy67f31d2e22bef9b9339e16a10dee1b05 = 'j&#111;r&#101;nd&#97;&#105;n' + '&#64;'; addy67f31d2e22bef9b9339e16a10dee1b05 = addy67f31d2e22bef9b9339e16a10dee1b05 + 'm&#97;tm&#111;r' + '&#46;' + '&#117;n&#97;m' + '&#46;' + 'mx'; var addy_text67f31d2e22bef9b9339e16a10dee1b05 = 'j&#111;r&#101;nd&#97;&#105;n' + '&#64;' + 'm&#97;tm&#111;r' + '&#46;' + '&#117;n&#97;m' + '&#46;' + 'mx';document.getElementById('cloak67f31d2e22bef9b9339e16a10dee1b05').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy67f31d2e22bef9b9339e16a10dee1b05 + '\'>'+addy_text67f31d2e22bef9b9339e16a10dee1b05+'<\/a>';

Symmetric monoidal structures on framed bicategories descend to symmetric monoidal structures on horizontal bicategories. The axioms defining symmetric monoidal double categories are significantly more tractable than those defining symmetric 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 classically been useful in expressing Kelly and Street's mates correspondence and in proving the 2-dimensional Seifert-van Kampen theorem of Brown et. al., amongst many other applications.

Globularly generated double categories are minimal solutions to lifting problems of bicategories into double categories along given categories of vertical arrows. Globularly generated double categories form a coreflective sub-2-category of general double categories. This, together with an analysis of the internal structure of globularly generated double categories yields a numerical invariant on general double categories. We call this invariant the length. The length of a double category C measures the complexity of lifting decorated bicategories into C.

It is conjectured that framed bicategories are of length 1. Motivated by this I present a general method for constructing globularly generated double categories of length 1 through extra data in the form of what I will call End-monoidal indexings of decoration categories. The methods presented are related to Moeller and Vasilakopoulou's monoidal Grothendieck construction, to Shulman's construction of framed bicategories from monoidal fibrations on cocartesian categories, and in the case of strict single object and single horizontal morphism 2-groupoids decorated by groups, specialize to semidirect products.

Bibliography

[1] J. Orendain. Internalizing decorated bicategories: The globularily generated condition. Theory and Applications of Categories, Vol.34, 2019, No. 4, pp 80-108.2.

[2]J. Orendain. Free globularily generated double categories. Theory and Applications of Categories, Vol. 34, 2019, No. 42, pp1343-1385.3.

[3] J. Orendain. Free Globularly Generated Double Categories II: The Canonical Double Projection. To appear in Cahiers de topologie et geometrie differentielle categoriques. arXiv:1905.02888

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Thursday, July 15, 14:30 ~ 15:00 UTC-3

## Hopf G-coalgebras and the cobar functor

### Eduardo Hoefel

#### UFPR - Curitiba, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakfed7b0ae9b1289cae6090eea79a41cc1').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyfed7b0ae9b1289cae6090eea79a41cc1 = 'h&#111;&#101;f&#101;l' + '&#64;'; addyfed7b0ae9b1289cae6090eea79a41cc1 = addyfed7b0ae9b1289cae6090eea79a41cc1 + '&#117;fpr' + '&#46;' + 'br'; var addy_textfed7b0ae9b1289cae6090eea79a41cc1 = 'h&#111;&#101;f&#101;l' + '&#64;' + '&#117;fpr' + '&#46;' + 'br';document.getElementById('cloakfed7b0ae9b1289cae6090eea79a41cc1').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyfed7b0ae9b1289cae6090eea79a41cc1 + '\'>'+addy_textfed7b0ae9b1289cae6090eea79a41cc1+'<\/a>';

This talk will begin by reviewing the Turaev category. Hopf G-coalgebras can be simply defined as Hopf algebras in that category. After presenting some interesting examples, I will communicate the results of my ongoing project involving the cobar functor in the context of G-coalgebras. This is work in progress, and we expect to extend the classical adjunction between the cobar functor and the loop space functor to the context of Turaev categories.

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Thursday, July 15, 15:00 ~ 15:30 UTC-3

## Categorical differentiation of homotopy functors and applications

### Kristine Bauer

#### University of Calgary, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak8e7624ee0e4dc3f762b256b564162df2').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy8e7624ee0e4dc3f762b256b564162df2 = 'B&#97;&#117;&#101;rk' + '&#64;'; addy8e7624ee0e4dc3f762b256b564162df2 = addy8e7624ee0e4dc3f762b256b564162df2 + '&#117;c&#97;lg&#97;ry' + '&#46;' + 'c&#97;'; var addy_text8e7624ee0e4dc3f762b256b564162df2 = 'B&#97;&#117;&#101;rk' + '&#64;' + '&#117;c&#97;lg&#97;ry' + '&#46;' + 'c&#97;';document.getElementById('cloak8e7624ee0e4dc3f762b256b564162df2').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy8e7624ee0e4dc3f762b256b564162df2 + '\'>'+addy_text8e7624ee0e4dc3f762b256b564162df2+'<\/a>';

The Goodwillie functor calculus tower is an approximation of a homotopy functor which resembles the Taylor series approximation of a function in ordinary calculus. In 2017, Johnson, Osborne, Riehl, Tebbe and I (BJORT, collectively) showed that the directional derivative for functors of an abelian category are an example of a categorical derivative in the sense of Blute, Cockett and Seely. The BJORT result relied on the fact that the target and source of the functors in question were both abelian categories. This leads one to the question of whether or not other sorts of homotopy functors have a similar structure.

To address this question, Burke and Ching and I instead use the notion of tangent category, due to Rosicky, Cockett-Cruttwell and via an incarnation due to Leung. In recent work with Burke and Ching, we make precise the notion of a tangent infinity category, and show that the directional derivative for homotopy functors from Goodwillie's calculus of functors appears as the associated categorical derivative of a particular tangent infinity category.

In this talk I will give an overview of these structure, with an eye towards possible applications. These applications are being developed by Johnson, Yeakel and I.

Joint work with Matthew Burke (Lyryx Learning Inc.), Michael Ching (Amherst College), Brenda Johnson (Union College), Christina Osborne (Cedarville University), Emily Riehl (Johns Hopkins University), Amelia Tebbe (Indiana University Kokomo) and Sarah Yeakel (University of California, Riverside).

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Friday, July 16, 12:00 ~ 12:30 UTC-3

## Connectivity of random simplicial complexes

### Jonathan Barmak

#### IMAS/Universidad de Buenos Aires, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakd1d0f79b3569bf49483a4908e5539b9a').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyd1d0f79b3569bf49483a4908e5539b9a = 'jb&#97;rm&#97;k' + '&#64;'; addyd1d0f79b3569bf49483a4908e5539b9a = addyd1d0f79b3569bf49483a4908e5539b9a + 'dm' + '&#46;' + '&#117;b&#97;' + '&#46;' + '&#97;r'; var addy_textd1d0f79b3569bf49483a4908e5539b9a = 'jb&#97;rm&#97;k' + '&#64;' + 'dm' + '&#46;' + '&#117;b&#97;' + '&#46;' + '&#97;r';document.getElementById('cloakd1d0f79b3569bf49483a4908e5539b9a').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyd1d0f79b3569bf49483a4908e5539b9a + '\'>'+addy_textd1d0f79b3569bf49483a4908e5539b9a+'<\/a>';

A simplicial complex is $r$-conic if every subcomplex of at most $r$ vertices is contained in a cone. We prove that for any $d\ge 0$ there exists $r$ such that $r$-conicity implies $d$-connectivity of the polyhedron. On the other hand, for a fixed $r$, the probability of a random simplicial complex being $r$-conic tends to $1$ as the number of vertices tends to $\infty$. Thus, random complexes are $d$-connected with high probability.

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Friday, July 16, 12:30 ~ 13:00 UTC-3

## The Interleaving Distance for Graphical Signatures

### Elizabeth Munch

#### Michigan State University, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak514ff0938c68a53cd316ddb9e8708a22').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy514ff0938c68a53cd316ddb9e8708a22 = 'm&#117;nch&#101;l&#105;' + '&#64;'; addy514ff0938c68a53cd316ddb9e8708a22 = addy514ff0938c68a53cd316ddb9e8708a22 + 'ms&#117;' + '&#46;' + '&#101;d&#117;'; var addy_text514ff0938c68a53cd316ddb9e8708a22 = 'm&#117;nch&#101;l&#105;' + '&#64;' + 'ms&#117;' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak514ff0938c68a53cd316ddb9e8708a22').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy514ff0938c68a53cd316ddb9e8708a22 + '\'>'+addy_text514ff0938c68a53cd316ddb9e8708a22+'<\/a>';

Reeb graphs and other related graphical signatures have extensive use in applications, but only recently has there been intense interest in finding metrics for these objects. The idea is that graphical signatures such as Reeb graphs, merge trees, and contour trees encode data in both a space and a real valued function, and we want to build metrics that are sensitive to this information. In this talk, we will focus on a particular metric for comparing Reeb graphs known as the interleaving distance which is a categorical reformulation of the eponymous metric from persistence modules arising in Topological Data Analysis, and show how it can be used as input to statistical and machine learning problems.

Joint work with Anastasios Stefanou (Ohio State University), Erin Chambers (St Louis University) and Tim Ophelders (TU Eindhoven).

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Friday, July 16, 13:00 ~ 13:30 UTC-3

## Group actions on contractible 2-complexes

#### IMAS - Universidad de Buenos Aires, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakf2f71f8b60899fc661a12ed379146e61').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyf2f71f8b60899fc661a12ed379146e61 = '&#105;v&#97;ns&#97;d&#111;fsch&#105;' + '&#64;'; addyf2f71f8b60899fc661a12ed379146e61 = addyf2f71f8b60899fc661a12ed379146e61 + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_textf2f71f8b60899fc661a12ed379146e61 = '&#105;v&#97;ns&#97;d&#111;fsch&#105;' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloakf2f71f8b60899fc661a12ed379146e61').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyf2f71f8b60899fc661a12ed379146e61 + '\'>'+addy_textf2f71f8b60899fc661a12ed379146e61+'<\/a>';

In this talk I will discuss the proof of a recent result: every action of a finite group G on a finite and contractible 2-complex X has a fixed point.

This was conjectured by Carles Casacuberta and Warren Dicks and was also posed as a question by Michael Aschbacher and Yoav Segev. We build on the classification, due to Bob Oliver and Yoav Segev, of the groups G which act without fixed points on an acyclic 2-complex. Part of this work is joint with Kevin Piterman.

Joint work with Kevin Piterman (IMAS - Universidad de Buenos Aires).

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Friday, July 16, 13:30 ~ 14:00 UTC-3

## Morse theory for group presentations

### Ximena Fernandez

#### Swansea University, United Kingdom   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakdd049cc9c547bb6c8493a1870b4da7a0').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addydd049cc9c547bb6c8493a1870b4da7a0 = 'x.l.f&#101;rn&#97;nd&#101;z' + '&#64;'; addydd049cc9c547bb6c8493a1870b4da7a0 = addydd049cc9c547bb6c8493a1870b4da7a0 + 'sw&#97;ns&#101;&#97;' + '&#46;' + '&#97;c' + '&#46;' + '&#117;k'; var addy_textdd049cc9c547bb6c8493a1870b4da7a0 = 'x.l.f&#101;rn&#97;nd&#101;z' + '&#64;' + 'sw&#97;ns&#101;&#97;' + '&#46;' + '&#97;c' + '&#46;' + '&#117;k';document.getElementById('cloakdd049cc9c547bb6c8493a1870b4da7a0').innerHTML += '<a ' + path + '\'' + prefix + ':' + addydd049cc9c547bb6c8493a1870b4da7a0 + '\'>'+addy_textdd049cc9c547bb6c8493a1870b4da7a0+'<\/a>';

The Andrews--Curtis conjecture (1965) is one the most relevant open problems in low-dimensional topology, with roots in Whitehead’s simple homotopy theory and combinatorial group theory. It is closely related to other important problems in algebraic topology such as the Whitehead asphericity conjecture, the Zeeman conjecture and the Poincaré conjecture (now a theorem).

The Andrews--Curtis conjecture states that any balanced presentation of the trivial group can be transformed into the empty presentation through a sequence of a class of movements (called $Q^{**}$-transformations) that do not change its deficiency. The geometric equivalent formulation states that if $K$ is a contractible complex of dimension 2, then it 3-deforms to a point. Although the conjecture is known to be true for some classes of complexes, it still remains open.

In this talk, I will introduce a novel combinatorial method to study $Q^{**}$-transformations of group presentations. The procedure is based on a new version of discrete Morse theory that provides a simple homotopy equivalence between a given regular CW-complex and its Morse complex, with an explicit description of the attaching maps and bounds on the deformation. I will present applications of this technique to the study of potential counterexamples to the Andrews--Curtis conjecture, showing that many of them do satisfy the conjecture.

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Friday, July 16, 14:30 ~ 15:00 UTC-3

## Locally bounded enriched categories

### Rory Lucyshyn-Wright

#### Brandon University, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak826981b09d133f70d193ca8854b73ee8').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy826981b09d133f70d193ca8854b73ee8 = 'l&#117;cyshyn-wr&#105;ghtr' + '&#64;'; addy826981b09d133f70d193ca8854b73ee8 = addy826981b09d133f70d193ca8854b73ee8 + 'br&#97;nd&#111;n&#117;' + '&#46;' + 'c&#97;'; var addy_text826981b09d133f70d193ca8854b73ee8 = 'l&#117;cyshyn-wr&#105;ghtr' + '&#64;' + 'br&#97;nd&#111;n&#117;' + '&#46;' + 'c&#97;';document.getElementById('cloak826981b09d133f70d193ca8854b73ee8').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy826981b09d133f70d193ca8854b73ee8 + '\'>'+addy_text826981b09d133f70d193ca8854b73ee8+'<\/a>';

Locally bounded categories [3] have some of the convenient features of locally presentable categories, such as reflectivity results for orthogonal subcategories [1,2] as well as results on the existence of free monads, colimits of monads, and colimits in categories of algebras [2]. The axioms for locally bounded categories are formulated in terms of factorization systems and are general enough to admit a vast array of categories that are not locally presentable but provide important backgrounds for topology and analysis. In particular, there are various categories of topological structures and many quasitoposes that are locally bounded but not locally presentable. Based on ideas of Freyd and Kelly [1], the notion of locally bounded category was introduced by Kelly [3], who employed a given locally bounded closed category $\mathcal{V}$ as the basis for a general treatment of enriched limit theories. The latter theories generalize Kelly's enriched finite limit theories, which Kelly introduced in [4] along with notions of locally presentable closed category and locally presentable $\mathcal{V}$-category. However, notably absent from the literature is a notion of locally bounded $\mathcal{V}$-category that would complete the parallel between the locally presentable and locally bounded settings.

In this talk on joint work with Jason Parker, we introduce a notion of locally bounded $\mathcal{V}$-category enriched over a locally bounded closed category $\mathcal{V}$. Locally bounded $\mathcal{V}$-categories are examples of the more basic notion of $\mathcal{V}$-factegory, by which we mean a $\mathcal{V}$-category equipped with an enriched proper factorization system $(\mathcal{E},\mathcal{M})$ that is suitably compatible with the given factorization system carried by $\mathcal{V}$, and we say that a $\mathcal{V}$-factegory $\mathcal{C}$ is cocomplete if it is cocomplete as a $\mathcal{V}$-category and has arbitrary cointersections of $\mathcal{E}$-morphisms. We then say that a cocomplete $\mathcal{V}$-factegory $\mathcal{C}$ is locally bounded if it has a small $(\mathcal{E},\mathcal{M})$-generator $\mathcal{G}$ consisting of objects $G$ that are $\alpha$-bounded for some regular cardinal $\alpha$, meaning that $\mathcal{C}(G,-):\mathcal{C} \rightarrow \mathcal{V}$ preserves $\alpha$-filtered unions of $\mathcal{M}$-subobjects.

We show that locally bounded $\mathcal{V}$-categories support several adjoint functor theorems and a reflectivity result for enriched orthogonal subcategories, as well as results on the existence of algebraically free $\mathcal{V}$-monads, colimits of $\mathcal{V}$-monads, and enriched colimits in $\mathcal{V}$-categories of algebras. As a technique for constructing examples of locally bounded $\mathcal{V}$-categories, we show that a cocomplete $\mathcal{V}$-factegory $\mathcal{C}$ is locally bounded as soon as there exists a right adjoint $\mathcal{V}$-functor $G:\mathcal{C} \rightarrow \mathcal{B}$ that is valued in some locally bounded $\mathcal{V}$-category $\mathcal{B}$ and satisfies certain axioms; we call such a $\mathcal{V}$-functor $G$ a bounding right adjoint for $\mathcal{C}$. We characterize locally bounded $\mathcal{V}$-categories as precisely those cocomplete $\mathcal{V}$-factegories $\mathcal{C}$ that admit a bounding right adjoint valued in a presheaf $\mathcal{V}$-category.

We show that bounding right adjoints give rise to various classes of examples of locally bounded $\mathcal{V}$-categories, including certain orthogonal subcategories of locally bounded $\mathcal{V}$-categories as well as certain $\mathcal{V}$-categories of algebras/models for monads/theories on locally bounded $\mathcal{V}$-categories.

By analogy with Kelly's notion of finite weighted limit enriched in a locally finitely presentable closed category $\mathcal{V}$, we define a notion of $\alpha$-small weighted limit enriched in a locally $\alpha$-bounded closed category $\mathcal{V}$. We show that the $\mathcal{V}$-category of models for an $\alpha$-small-limit sketch or theory is locally $\alpha$-bounded as soon as $\mathcal{V}$ is locally $\alpha$-bounded and $\mathcal{E}$-cowellpowered.

We also discuss applications of locally bounded $\mathcal{V}$-categories to the study of enriched monads for subcategories of arities.

[1] P. J. Freyd and G. M. Kelly, Categories of continuous functors, Journal of Pure and Applied Algebra 2 (1972) 169-191.

[2] G. M. Kelly, A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on. Bulletin of the Australian Mathematical Society 22 (1980) 1-83.

[3] G. M. Kelly, Basic concepts of enriched category theory. Repr. Theory and Applications of Categories, No. 10, 2005, Reprint of the 1982 original [Cambridge University Press].

[4] G. M. Kelly, Structures defined by finite limits in the enriched context, I, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 23 (1982), 3-42.

Joint work with Jason Parker (Brandon University, Canada).

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Friday, July 16, 15:00 ~ 15:30 UTC-3

## Presentations and Algebraic Colimits of Enriched Monads

### Jason Parker

#### Brandon University, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak325aae83291694f4ab8c0010771f988e').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy325aae83291694f4ab8c0010771f988e = 'p&#97;rk&#101;rj' + '&#64;'; addy325aae83291694f4ab8c0010771f988e = addy325aae83291694f4ab8c0010771f988e + 'br&#97;nd&#111;n&#117;' + '&#46;' + 'c&#97;'; var addy_text325aae83291694f4ab8c0010771f988e = 'p&#97;rk&#101;rj' + '&#64;' + 'br&#97;nd&#111;n&#117;' + '&#46;' + 'c&#97;';document.getElementById('cloak325aae83291694f4ab8c0010771f988e').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy325aae83291694f4ab8c0010771f988e + '\'>'+addy_text325aae83291694f4ab8c0010771f988e+'<\/a>';

Signatures and presentations for monads and theories in (enriched) universal algebra have been previously studied mainly in the context of locally presentable enriched categories over a locally presentable symmetric monoidal closed category $\mathscr{V}$; see e.g. [6] and its successor [7], as well as the recent [2]. In this talk, summarizing joint work with Rory Lucyshyn-Wright, we develop a framework for studying such phenomena that subsumes a much wider class of enriched categories, including the locally bounded categories of [5, Chapter 6] and the symmetric monoidal closed $\pi$-categories of [1]. Locally bounded categories in particular provide a vast generalization of locally presentable categories and include many quasitoposes and categories of topological spaces as examples. Moreover, our framework is sufficiently general to encompass the Lawvere $\Phi$-theories of [8] even when the base $\mathscr{V}$ is not locally presentable.

Given a symmetric monoidal closed category $\mathscr{V}$ and a $\mathscr{V}$-category $\mathscr{C}$, a subcategory of arities in $\mathscr{C}$ is a dense sub-$\mathscr{V}$-category $j : \mathscr{J} \hookrightarrow \mathscr{C}$, which is said to be eleutheric if it satisfies a certain exactness condition (which in particular guarantees that arbitrary $\mathscr{V}$-functors $\mathscr{J} \to \mathscr{C}$ are equivalent to $\mathscr{J}$-ary $\mathscr{V}$-endofunctors on $\mathscr{C}$ as defined below; cf. [9, 7.1]). Examples of eleutheric subcategories of arities abound:

-The subcategory of $\alpha$-presentable objects $\mathscr{C}_{\alpha} \hookrightarrow \mathscr{C}$ in any locally $\alpha$-presentable $\mathscr{V}$-category $\mathscr{C}$;

-More generally, the subcategory $\mathscr{C}_\Phi \hookrightarrow \mathscr{C}$ of $\Phi$-presentable objects in any locally $\Phi$-presentable $\mathscr{V}$-category $\mathscr{C}$ for a class of locally small weights $\Phi$ satisfying Axiom A of [8];

-Even more generally, any free $\Psi$-cocompletion $j : \mathscr{J} \hookrightarrow \mathscr{C}$ for a class of weights $\Psi$ and a $\mathscr{V}$-category $\mathscr{J}$;

-The unrestricted subcategory of arities $1_{\mathscr{C}}: \mathscr{C} \to \mathscr{C}$ in an arbitrary $\mathscr{V}$-category $\mathscr{C}$;

-The subcategory of arities $\left\{I\right\} \hookrightarrow \mathscr{V}$ in $\mathscr{V}$ consisting of just the unit object;

-The subcategory $\left\{ n \cdot I \colon n \in \mathbb{N}\right\} \hookrightarrow \mathscr{V}$ consisting of finite copowers of the unit object in any symmetric monoidal closed $\pi$-category $\mathscr{V}$ (cf. [1]), which need not be locally presentable.

Given an eleutheric subcategory of arities $j : \mathscr{J} \hookrightarrow \mathscr{C}$ in a $\mathscr{V}$-category $\mathscr{C}$, a $\mathscr{V}$-endofunctor $T : \mathscr{C} \to \mathscr{C}$ is then said to be $\mathscr{J}$-ary if it preserves left Kan extensions along $j$, and a $\mathscr{V}$-monad $\mathbb{T} = (T, \eta, \mu)$ on $\mathscr{C}$ is said to be $\mathscr{J}$-ary if its underlying $\mathscr{V}$-endofunctor $T$ is so. For example:

-The $\mathscr{J}$-ary monads on a locally $\alpha$-presentable $\mathscr{V}$-category $\mathscr{C}$ are precisely the $\alpha$-ary monads, i.e. the monads that preserve $\alpha$-filtered colimits, and correspond to the enriched Lawvere theories of [10] when $\alpha = \aleph_0$;

-More generally, the $\mathscr{J}$-ary monads on a locally $\Phi$-presentable $\mathscr{V}$-category $\mathscr{C}$ are precisely the $\Phi$-accessible monads, i.e. the monads that preserve $\Phi$-flat colimits, and correspond to the Lawvere $\Phi$-theories of [8];

-Even more generally, the $\mathscr{J}$-ary monads on a $\mathscr{V}$-category $\mathscr{C}$ which is the free $\Psi$-cocompletion of a subcategory $\mathscr{J}$ for a class of weights $\Psi$ are precisely the $\Psi$-cocontinuous monads, and correspond to the $\mathscr{J}$-theories of [9] when $\mathscr{C} = \mathscr{V}$;

-The $\mathscr{J}$-ary monads for the unrestricted subcategory of arities are just arbitrary $\mathscr{V}$-monads, and correspond to the $\mathscr{V}$-theories of [3] when $\mathscr{C} = \mathscr{V}$;

-The $\mathscr{J}$-ary monads for the subcategory of arities on the unit object of $\mathscr{V}$ are the tensor-preserving monads on $\mathscr{V}$, and correspond to monoids in $\mathscr{V}$ (cf. [9, 4.2.5]);

-The $\mathscr{J}$-ary monads for the subcategory of arities on the finite copowers of the unit object in any $\pi$-category $\mathscr{V}$ correspond to the Borceux-Day enriched algebraic theories of [1].

Finally, given a subcategory of arities $j : \mathscr{J} \hookrightarrow \mathscr{C}$ in a $\mathscr{V}$-category $\mathscr{C}$, one can define the notion of a $\mathscr{J}$-signature in $\mathscr{C}$ as an $\mathsf{ob}\mathscr{J}$-indexed family of objects in $\mathscr{C}$, analogously to the definition of a traditional signature in universal algebra as a family of sets indexed by the finite cardinals, as well as the notion of a $\mathscr{J}$-presentation for $\mathscr{J}$-ary monads.

In this talk, we will explore the relationship between $\mathscr{J}$-ary monads on the one hand and $\mathscr{J}$-signatures and presentations on the other for a small eleutheric subcategory of arities $j : \mathscr{J} \hookrightarrow \mathscr{C}$ in a (cocomplete and cotensored) $\mathscr{V}$-category $\mathscr{C}$ that satisfies a mild boundedness condition defined in terms of factorization systems and notions from [4]. It turns out that any small and eleutheric subcategory of arities whatsoever in a locally bounded enriched category satisfies this boundedness condition, which thereby enables us to develop the following central results in a much broader context than has been traditionally studied:

-The forgetful functor from $\mathscr{J}$-ary monads on $\mathscr{C}$ to $\mathscr{J}$-signatures in $\mathscr{C}$ is monadic;

-Any small diagram of $\mathscr{J}$-ary monads on $\mathscr{C}$ has a colimit which is both $\mathscr{J}$-ary and algebraic, in the sense that its $\mathscr{V}$-category of algebras is isomorphic to the limit of the $\mathscr{V}$-categories of algebras of the monads in the diagram;

-As a consequence, any $\mathscr{J}$-presentation $P$ generates a $\mathscr{J}$-ary monad, whose $\mathscr{V}$-category of algebras is isomorphic to the (suitably defined) $\mathscr{V}$-category of $P$-algebras for the presentation. Moreover, any $\mathscr{J}$-ary monad has such a presentation.

[1] F. Borceux and B. Day. Universal algebra in a closed category. Journal of Pure and Applied Algebra Vol. 16 No. 2 (1980) 133-147.

[2] J. Bourke and R. Garner. Monads and theories. Advances in Mathematics Vol. 351 (2019) 1024-1071.

[3] E.J. Dubuc. Enriched semantics-structure (meta) adjointness. Revista de la Union Matematica Argentina Vol. 25 (1970) 5-26.

[4] G.M. Kelly. A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on. Bulletin of the Australian Mathematical Society Vol. 22 (1980) 1-83.

[5] G.M. Kelly. Basic concepts of enriched category theory. Repr. Theory and Applications of Categories, No. 10, 2005, Reprint of the 1982 original [Cambridge University Press].

[6] G.M. Kelly and A.J. Power. Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads. Journal of Pure and Applied Algebra Vol. 89 (1993) 163-179.

[7] S. Lack. On the monadicity of finitary monads. Journal of Pure and Applied Algebra Vol. 140 (1999) 65-73.

[8] S. Lack and J. Rosicky. Notions of Lawvere theory. Applied Categorical Structures 19 (2011) 363-391.

[9] R.B.B. Lucyshyn-Wright. Enriched algebraic theories and monads for a system of arities. Theory and Applications of Categories Vol. 31 No. 5 (2016) 101-137.

[10] K. Nishizawa and J. Power. Lawvere theories enriched over a general base. Journal of Pure and Applied Algebra Vol. 213 Issue 3 (2009) 377-386.

Joint work with Rory Lucyshyn-Wright (Brandon University, Manitoba, Canada).

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Friday, July 16, 15:30 ~ 16:00 UTC-3

## Spaces with decidable reflection

### Matías Menni

In an extensive category with finite products we say that an object is $weakly\ locally\ connected\ (wlc)$ if it has a universal map towards a decidable object. We characterize wlc objects in ${\mathbf{Top}}$. We paraphrase a classical result by recalling that all affine schemes (over a base field) are wlc, and that the left adjoint $\pi_0$" to the subcategory of decidable objects preserves finite products. Also, we characterize the wlc objects in the (extensive) opposite of the category of MV-algebras. Moreover, if we let $\mathbf{MV}_{fp}$ be the category of finitely presentable MV-algebras then, as in the case of affine schemes, every object of $(\mathbf{MV}_{fp})^{op}$ is wlc and the associated left adjoint preserves finite products. By a well-known duality this may be seen as a statement about rational polyhedra and certain PL-maps between them.

Joint work with V. Marra.

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Wednesday, July 21, 16:00 ~ 16:30 UTC-3

## Towards 2-dimensional Tannaka duality

### Nick Gurski

Ordinary Tannaka duality recovers a coalgebra by a coend construction using the forgetful functor from comodules or finite dimensional comodules to vector spaces. In addition, structure on the forgetful functor - such as being monoidal - corresponds to additional structure on the coalgebra. In this talk, I will describe a joint project with David Yetter to generalize the abstract methods of Tannaka duality to a 2-categorical setting in which coends are replaced with codescent objects, coalgebras are replaced with pseudocomonoids, and interesting variants of the slice 2-category appear in order to produce different kinds of morphisms between 2-categories of pseudocomodules.

Joint work with David Yetter (Kansas State University).

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Wednesday, July 21, 16:30 ~ 17:00 UTC-3

## Interleavings and Gromov-Hausdorff Distance

### Jonathan Scott

One of the central notions to emerge from the study of persistent homology is that of interleaving distance. It has found recent applications in computational geometry, symplectic and contact geometry, sheaf theory, and phylogenetics. Here we present a general study of this topic, considering interleavings of functors to be solutions to a certain extension problem. By placing the problem in the context of (weighted) bicategories, we identify interleaving distance as a type of categorical generalization of Gromov--Hausdorff distance. As an application we recover a definition of shift equivalences of discrete dynamical systems.

Joint work with Vin de Silva (Pomona College) and Peter Bubenik (U Florida).

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Wednesday, July 21, 17:00 ~ 17:30 UTC-3

## Exponentiable Inclusions: Quantaloids and Ringoids

### Susan Niefield

#### Union College, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak44ed6d7dda208195d886d0486acf26be').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy44ed6d7dda208195d886d0486acf26be = 'n&#105;&#101;f&#105;&#101;ls' + '&#64;'; addy44ed6d7dda208195d886d0486acf26be = addy44ed6d7dda208195d886d0486acf26be + '&#117;n&#105;&#111;n' + '&#46;' + '&#101;d&#117;'; var addy_text44ed6d7dda208195d886d0486acf26be = 'n&#105;&#101;f&#105;&#101;ls' + '&#64;' + '&#117;n&#105;&#111;n' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak44ed6d7dda208195d886d0486acf26be').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy44ed6d7dda208195d886d0486acf26be + '\'>'+addy_text44ed6d7dda208195d886d0486acf26be+'<\/a>';

In earlier work, we showed that the inclusion of a subobject is exponentiable if and only if it is locally closed for five categories; namely, spaces (1978), locales (1980), toposes (1980), small categories (2000), and posets (2001). In 2012, we introduced the notions of locally closed inclusions and Artin-Wraith glueing in double categories. With appropriate assumptions, we showed that locally closed inclusions are exponentiable, and constructed the exponentials via the glueing condition. This provided a single theorem establishing the exponentiability of locally closed inclusions that applied to the five categories mentioned above.

In this talk, we will show that if $\cal V$ is a cocomplete symmetric monoidal category, then categories enriched in $\cal V$ are the objects of a double category with the appropriate glueing properties, and hence, we obtain the exponentiability of locally closed inclusions of $\cal V$-categories. Furthermore, we will see that the locally closed condition is also necessary when $\cal V$ is monadic over the category of sets. Thus, we obtain a characterization of the exponentiable inclusions of quantaloids (respectively, ringoids) when $\cal V$ is the category of suplattices (respectively, abelian groups) analogous to that of the five earlier cases.

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Wednesday, July 21, 17:30 ~ 18:00 UTC-3

## Mapping Objects for Orbispaces

### Laura Scull

#### Fort Lewis College, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak94f6f985c19e7a97db1d3339501ff7c2').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy94f6f985c19e7a97db1d3339501ff7c2 = 'sc&#117;ll_l' + '&#64;'; addy94f6f985c19e7a97db1d3339501ff7c2 = addy94f6f985c19e7a97db1d3339501ff7c2 + 'f&#111;rtl&#101;w&#105;s' + '&#46;' + '&#101;d&#117;'; var addy_text94f6f985c19e7a97db1d3339501ff7c2 = 'sc&#117;ll_l' + '&#64;' + 'f&#111;rtl&#101;w&#105;s' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak94f6f985c19e7a97db1d3339501ff7c2').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy94f6f985c19e7a97db1d3339501ff7c2 + '\'>'+addy_text94f6f985c19e7a97db1d3339501ff7c2+'<\/a>';

Orbifolds, and more generally orbispaces, are spaces which have well-behaved singularities. They can be defined using atlases and charts analogously to manifolds, with charts consisting of open subsets of Euclidean space with an action of a finite group defining the local singularity structure. This approach is cumbersome and hard to work with, and instead, orbispaces are often modeled using topological groupoids. It is shown in [Moerdijk-Pronk] that orbispaces can be represented by topological groupoids with etale structure maps and proper diagonal. This representation is not unique, however, as two Morita equivalent groupoids represent the same orbispace.

Thus, to represent orbifolds and orbispaces, we turn to a bicategory of fractions where the Morita equivalences have been inverted. This gives a definition of a map between orbispaces $G \to H$ defined as a span of maps between groupoids $G \leftarrow K \to H$ where $K$ is Morita equivalent to $G$, giving an alternate representation of the domain orbispace.

The question I will address is how to create a topological mapping groupoid for orbispaces, $\mbox{OMap}(G, H)$, which encodes these spans and satisfies the properties of a mapping object. This question has been addressed in [Chen], but not in terms of orbigroupoids, and with only partial answers. Here, I will show how to define an etale proper groupoid $\mbox{OMap}(G, H)$ which is the exponential object for orbigroupoids and gives orbispaces the structure of an enriched bicategory, so that composition induces a map of orbispaces $\mbox{OMap}(G, H) \times \mbox{OMap}(H, K) \to \mbox{OMap}(G, K)$.

[Chen] Weimin Chen, On a notion of maps between orbifolds I: function spaces, Communications in Contemporary Mathematics 8 (2006), pp. 569-620.

[Moerdijk-Pronk] I. Moerdijk, D.A. Pronk, Orbifolds, sheaves and groupoids, K-Theory 12 (1997), pp. 3-21.

Joint work with Dorette Pronk (Dalhousie University).

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Wednesday, July 21, 18:00 ~ 18:30 UTC-3

## The classifying space of the 1+1 dimensional free $G$-cobordism category

### Carlos Segovia González

#### Instituto de Matemáticas UNAM-Oaxaca, México   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakc7189a90b685436e2a59a58c59cba6ce').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyc7189a90b685436e2a59a58c59cba6ce = 'cs&#101;g&#111;v&#105;&#97;' + '&#64;'; addyc7189a90b685436e2a59a58c59cba6ce = addyc7189a90b685436e2a59a58c59cba6ce + 'm&#97;t&#101;m' + '&#46;' + '&#117;n&#97;m' + '&#46;' + 'mx'; var addy_textc7189a90b685436e2a59a58c59cba6ce = 'cs&#101;g&#111;v&#105;&#97;' + '&#64;' + 'm&#97;t&#101;m' + '&#46;' + '&#117;n&#97;m' + '&#46;' + 'mx';document.getElementById('cloakc7189a90b685436e2a59a58c59cba6ce').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyc7189a90b685436e2a59a58c59cba6ce + '\'>'+addy_textc7189a90b685436e2a59a58c59cba6ce+'<\/a>';

For $G$ a finite group, we define the free $G$-cobordism category in dimension two. We show the classifying space of this category has connected components in bijection with the abelianization of $G$ and with fundamental group isomorphic to the direct sum $\mathbb{Z}+H_2(G)$, where $H_2(G)$ is the integral 2-homology group. For $G$ a finite abelian group, we study its classifying space showing an splitting as $G\times X^G\times T^{r(G)}$, where $X^G$ is a simply connected infinite loop space and $T^{r(G)}$ is the product of $r(G)$ circles. An explicit expression for the number $r(G)$ is presented. Also, a description of the classifying space of some important subcategories is provided. Finally, we present some results relative to the classification of $G$-topological quantum field theories in dimension two.

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Wednesday, July 21, 18:30 ~ 19:00 UTC-3

## Adjoint functors and symmetric monoidal categories for topological data analysis

### Peter Bubenik

#### University of Florida, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakc70d1904774dea7d3d1a344f15187f5d').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyc70d1904774dea7d3d1a344f15187f5d = 'p&#101;t&#101;r.b&#117;b&#101;n&#105;k' + '&#64;'; addyc70d1904774dea7d3d1a344f15187f5d = addyc70d1904774dea7d3d1a344f15187f5d + '&#117;fl' + '&#46;' + '&#101;d&#117;'; var addy_textc70d1904774dea7d3d1a344f15187f5d = 'p&#101;t&#101;r.b&#117;b&#101;n&#105;k' + '&#64;' + '&#117;fl' + '&#46;' + '&#101;d&#117;';document.getElementById('cloakc70d1904774dea7d3d1a344f15187f5d').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyc70d1904774dea7d3d1a344f15187f5d + '\'>'+addy_textc70d1904774dea7d3d1a344f15187f5d+'<\/a>';

Persistent homology is an important tool in topological data analysis, whose goal is to quantify and learn from the shape' of data. First we encode scientific data as a diagram of spaces and then we apply a homology functor to obtain a diagram in an abelian category. In nice cases, this diagram can be represented by a formal sum in a pointed metric space. I will show how categorical constructions give us a family of distances on these formal sums.

Joint work with Alex Elchesen (University of Florida, USA).

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Wednesday, July 21, 19:00 ~ 19:30 UTC-3

## The homotopy theory of simplicial Beck modules

### Martin Frankland

#### University of Regina, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloake1bec9da58ba6b77249c2083f4b57725').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addye1bec9da58ba6b77249c2083f4b57725 = 'M&#97;rt&#105;n.Fr&#97;nkl&#97;nd' + '&#64;'; addye1bec9da58ba6b77249c2083f4b57725 = addye1bec9da58ba6b77249c2083f4b57725 + '&#117;r&#101;g&#105;n&#97;' + '&#46;' + 'c&#97;'; var addy_texte1bec9da58ba6b77249c2083f4b57725 = 'M&#97;rt&#105;n.Fr&#97;nkl&#97;nd' + '&#64;' + '&#117;r&#101;g&#105;n&#97;' + '&#46;' + 'c&#97;';document.getElementById('cloake1bec9da58ba6b77249c2083f4b57725').innerHTML += '<a ' + path + '\'' + prefix + ':' + addye1bec9da58ba6b77249c2083f4b57725 + '\'>'+addy_texte1bec9da58ba6b77249c2083f4b57725+'<\/a>';

In his 1967 thesis, Beck proposed a notion of module over an object in a category C. This provided a natural notion of coefficient module for André-Quillen (co)homology of any algebraic structure, generalizing the original case of commutative rings. As one varies the object in C, the categories of Beck modules over different objects assemble into a fibered category over C, sometimes called the tangent category of C. Motivated by Quillen (co)homology, I will discuss the homotopy theory of simplicial Beck modules over simplicial objects, generalizing some work of Quillen on simplicial commutative rings.

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Wednesday, July 21, 19:30 ~ 20:00 UTC-3

## No set of spaces detects isomorphisms in the homotopy category

### J. Daniel Christensen

#### University of Western Ontario, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak44b879295691701c7268476f083310e2').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy44b879295691701c7268476f083310e2 = 'jdc' + '&#64;'; addy44b879295691701c7268476f083310e2 = addy44b879295691701c7268476f083310e2 + '&#117;w&#111;' + '&#46;' + 'c&#97;'; var addy_text44b879295691701c7268476f083310e2 = 'jdc' + '&#64;' + '&#117;w&#111;' + '&#46;' + 'c&#97;';document.getElementById('cloak44b879295691701c7268476f083310e2').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy44b879295691701c7268476f083310e2 + '\'>'+addy_text44b879295691701c7268476f083310e2+'<\/a>';

Whitehead's theorem says that a map of pointed, connected CW complexes is a homotopy equivalence if and only if it induces an isomorphism on homotopy groups.

In the unpointed setting, one can ask whether there is a set $\mathcal{S}$ of spaces such that a map $f : X \to Y$ between connected CW complexes is a homotopy equivalence if and only if it induces bijections $[A, X] \to [A, Y]$ for all $A$ in $\mathcal{S}$. Heller claimed that there is no such set $\mathcal{S}$, but his argument relied on an "obvious" statement about weak colimits in the homotopy category of spaces. We show that this obvious statement is false, thus reopening the question above. We then show that Heller was in fact correct that no such set $\mathcal{S}$ exists, using a different, more direct method.

This talk is based on the material in arXiv:1910.04141.

Joint work with Kevin Arlin.

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## The Grothendieck Construction in the Lax Setting of Bicategory Theory

### Pablo Bustillo Vazquez

#### Ecole Normale Supérieure, France   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak8c7acb74b3b6b02e49f999dbd42e8029').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy8c7acb74b3b6b02e49f999dbd42e8029 = 'pb&#117;st&#105;ll&#111;05' + '&#64;'; addy8c7acb74b3b6b02e49f999dbd42e8029 = addy8c7acb74b3b6b02e49f999dbd42e8029 + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_text8c7acb74b3b6b02e49f999dbd42e8029 = 'pb&#117;st&#105;ll&#111;05' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloak8c7acb74b3b6b02e49f999dbd42e8029').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy8c7acb74b3b6b02e49f999dbd42e8029 + '\'>'+addy_text8c7acb74b3b6b02e49f999dbd42e8029+'<\/a>';

The Grothendieck construction associates a fibration $\int F$ over $C$ to each pseudofunctor $F\colon C^{op} \to \textbf{Cat}$. A classic result that can be found in [SGA] is that the pseudocolimit of $F$ can be computed by localizing this Fibration at the cartesian arrows, and that when $C$ is (pseudo)Filtered this localization can be constructed using a calculus of Fractions. These notions (Fibrations, Fractions and Filteredness) have been generalized to the context of bicategories [Buckley,Pronk,PS,DS], and we show how they can be used, in a similar fashion to the one in [SGA], to compute (higher dimensional) colimits of diagrams of bicategories, indexed by bicategories.

The Grothendieck construction can be naturally generalized to dimension 2 [Hermida,Bakovic,Buckley]. In its most general studied case, for $\mathcal{B}$ a bicategory, it's a trihomomorphism $\int\colon [\mathcal{B}, \textbf{Bicat}]_{\text{weak}} \longrightarrow (\textbf{Bicat}/\mathcal{B})^{\text{co}}$ from the tricategory of trihomomorphisms, trinatural transformations, trimodifications and perturbations, to the strict slice tricategory of bicategories over $\mathcal{B}$ (we use the covariant case throughout this work). Looking carefully at the construction, one can notice that it still makes sense when some of the considered data is lax. We use the construction described by Buckley [Buckley] to reverse engineer a definition of lax functor and lax natural transformation in dimension 3. We obtain very similar definitions to those introduced in [GPS] or [Gurski], but with some subtle changes.

This idea of reverse engineering definitions of lax structures in dimension 3, out of the 2-dimensional Grothendieck construction, gives good motivations for our definitions. It also highlights links between the definitions of: bicategories and lax functors between tricategories; pseudo-functors between bicategories and lax natural transformations between lax functors between tricategories, and so on... by exploiting the dimensional shift inherent to the construction.

The functor $\int$ above is known to be a local biequivalence [Buckley], and we have extended this result to this larger lax setting. We obtain in particular that $\int\colon [\mathcal{B}, \textbf{Bicat}]_{\text{lax}}(F,G) \longrightarrow (\textbf{Bicat}/\mathcal{B})^{\text{co}}\left(\int F,\int G\right)$ is a biequivalence whenever $G$ is, for example, a trihomomorphism.

We then use this result to construct all types of bicategory-indexed colimits in $\textbf{Bicat}$ (and hence in $\textbf{Cat}$: by applying $\pi_0$, the left adjoint to the inclusion of categories into bicategories). Note that we allow our diagrams to be lax, so when the indexing bicategory is discrete we recover in particular the results from [CCG] with a different, conceptual proof. Using the key observation that $\int \Delta(\mathcal{X}) \simeq \mathcal{X} \times \mathcal{B}$, when $\Delta(\mathcal{X})$ is the constant strict functor at the bicategory $\mathcal{X}$, we apply this result to: compute conical pseudo-colimits directly with Buckley's result, compute conical lax-colimits directly with our generalization, compute conical $\sigma$-colimits [DDS], which gives us any type of limits in $\textbf{Cat}$, with a more precise fine-tuning of the result.

Using the other key observation that $\int \textbf{Cat}(F(-),\mathcal{X}) \simeq (F\downarrow \mathcal{X})$, where $F\colon \mathcal{B} \to \textbf{Cat}$ is a pseudo-functor, $\mathcal{X}$ is a category and $(F\downarrow \mathcal{X})$ is a form of lax comma, we can also directly compute weighted pseudo-colimits in $\textbf{Cat}$ (we could extend this to $\textbf{Bicat}$ with substantially more tricategory theory).

This lays out a method to formulate the construction of colimits in higher dimensions. Checking the local equivalence of the construction, which is a crucial step, may be very technical but may also be studied more abstractly: we are working on a more fundamental approach by introducing several notions of lax Kan extensions pushing some ideas of Street [Street] in dimension 3. Another crucial step, is that those constructions lead to the required colimits if and only if we know that the localizations exist. This lead us to focusing on a main known case of localization in dimension 2: the calculus of fractions.

We then explore the relations between fibrations, fractions and filteredness in bicategory theory. In a series of lemmas, we generalize some results from ordinary category theory [SGA] to dimension 2: the family of all arrows of a filtered bicategory admits a calculus of fractions, and if $p\colon\mathcal{E} \to \mathcal{B}$ is a fibration and $\mathcal{W}$ admits a calculus of fractions on $\mathcal{B}$, then the family of cartesian arrows above $\mathcal{W}$ admits a calculus of fractions.

Using these results and carefully checking that $\pi_0$ commutes with the bicategorical calculus of fractions [Pronk,PS], we can compute filtered pseudo-colimits in $\textbf{Cat}$ indexed by a bicategory by simply computing the ordinary one-dimensional calculus of fractions. As an application, we get a formula for the hom-categories of a bicategory of fractions [Pronk] as a filtered pseudo-colimit, which establishes another link between filteredness and fractions.

[Bakovic] Igor Bakovic. Fibrations of Bicategories. http://www.irb.hr/korisnici/ibakovic/groth2fib.pdf.

[Buckley] Mitchell Buckley. Fibred 2-categories and bicategories. Journal of Pure and Applied Algebra. Volume 218 Issue 6 (2014) 1034--1074

[CCG] Pilar Carrasco and Antonio M. Cegarra and Antonio R. Garzón. Classifying spaces for braided monoidal categories and lax diagrams of bicategories. arXiv:0907.0930 (2010)

[DS] Eduardo J Dubuc, Ross Street. A construction of $2$-filtered bicolimits of categories. Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 47 (2006) 83--106

[DDS] Maria Emilia Descotte, Eduardo J. Dubuc, and Martin Szyld. Sigma limits in 2-categories and flat pseudofunctors. Advances in Mathematics. Volume 333 (2018) 266--313

[GPS] Robert Gordon, John Power, Ross Street. Coherence for tricategories. American Mathematical Society (1995)

[SGA] Alexander Grothendieck and Jean-Louis Verdier, Conditions de finitude. Exposé VI: Topos et sites fibres. Applications aux questions de passage a la limite. Théorie des Topos et Cohomologie Etale des Schémas. Springer(1972)

[Gurski] Nick Gurski. Coherence in Three-Dimensional Category Theory. Cambridge University Press(2013)

[Hermida] Claudio Hermida. Some properties of Fib as a fibred 2-category. Journal of Pure and Applied Algebra. Volume 134 Issue 1(1999)82--109

[Pronk] Dorette Pronk. Etendues and stacks as bicategories of fractions. Compositio Mathematica, Tome 102 (1996) 243--303

[PS] Dorette Pronk and Laura Scull. Bicategories of fractions revisited: towards small homs and canonical $2$-cells. arXiv:1908.01215 (2018)

[Street] Ross Street. Fibrations and Yoneda's lemma in a 2-category. Kelly G.M. (eds) Category Seminar. Lecture Notes in Mathematics, vol 420. Springer(1974)

Joint work with Dorette Pronk (Dalhousie University, Canada) and Martin Szyld (Dalhousie University, Canada).

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## Persistence Diagrams as Change Action Derivatives

### Deni Salja

#### Dalhousie , Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak3c8793c67b97b5723f1c771ecb9991a3').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy3c8793c67b97b5723f1c771ecb9991a3 = 'dn746393' + '&#64;'; addy3c8793c67b97b5723f1c771ecb9991a3 = addy3c8793c67b97b5723f1c771ecb9991a3 + 'd&#97;l' + '&#46;' + 'c&#97;'; var addy_text3c8793c67b97b5723f1c771ecb9991a3 = 'dn746393' + '&#64;' + 'd&#97;l' + '&#46;' + 'c&#97;';document.getElementById('cloak3c8793c67b97b5723f1c771ecb9991a3').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy3c8793c67b97b5723f1c771ecb9991a3 + '\'>'+addy_text3c8793c67b97b5723f1c771ecb9991a3+'<\/a>';

In 2019 McCleary and Patel proposed a generalization of a persistence diagram for a multi-parameter filtration' of chain complexes of an abelian category as the Möbius inversion of the rank function of the associated `birth-death-(sub)object.' At the 2019 CMS winter meeting Patel spoke at the TDA session about this work and mentioned how he thought of Möbius inversion as a derivative. This poster shows how to (extend and then) view a certain quotient object in their work as a change-action derivative in the sense of Alvarez-Picallo and Lemay.

Joint work with Dr. Kristine Bauer (University of Calgary).

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## Homotopy theory of digraphs: A Categorical viewpoint

### Julio Sampietro

#### Universidad Nacional Autónoma de México, Mexico   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak4978bc638a26bdea536f175d48bd2c43').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy4978bc638a26bdea536f175d48bd2c43 = 'js&#97;mp&#105;&#101;tr&#111;14' + '&#64;'; addy4978bc638a26bdea536f175d48bd2c43 = addy4978bc638a26bdea536f175d48bd2c43 + 'c&#105;&#101;nc&#105;&#97;s' + '&#46;' + '&#117;n&#97;m' + '&#46;' + 'mx'; var addy_text4978bc638a26bdea536f175d48bd2c43 = 'js&#97;mp&#105;&#101;tr&#111;14' + '&#64;' + 'c&#105;&#101;nc&#105;&#97;s' + '&#46;' + '&#117;n&#97;m' + '&#46;' + 'mx';document.getElementById('cloak4978bc638a26bdea536f175d48bd2c43').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy4978bc638a26bdea536f175d48bd2c43 + '\'>'+addy_text4978bc638a26bdea536f175d48bd2c43+'<\/a>';

A new model structure on the category of digraphs is introduced using the language of groupoids. Along the way, a comparison between different homotopies which arise from different products in the category of digraphs is made. We present an overview of how this model structure relates to existing homotopy theories of digraphs in the literature.

Joint work with Carlos Segovia (Instituto de Matemáticas Oaxaca, México).

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## $\mathbb{Z}_k$-Stratifolds

### Arley Fernando Torres Galindo

#### Universidad de Los Andes , Colombia   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakd4ebb7e7ea12c02d00a4a11c5d79a6c6').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyd4ebb7e7ea12c02d00a4a11c5d79a6c6 = '&#97;f.t&#111;rr&#101;s82' + '&#64;'; addyd4ebb7e7ea12c02d00a4a11c5d79a6c6 = addyd4ebb7e7ea12c02d00a4a11c5d79a6c6 + '&#117;n&#105;&#97;nd&#101;s' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'c&#111;'; var addy_textd4ebb7e7ea12c02d00a4a11c5d79a6c6 = '&#97;f.t&#111;rr&#101;s82' + '&#64;' + '&#117;n&#105;&#97;nd&#101;s' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'c&#111;';document.getElementById('cloakd4ebb7e7ea12c02d00a4a11c5d79a6c6').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyd4ebb7e7ea12c02d00a4a11c5d79a6c6 + '\'>'+addy_textd4ebb7e7ea12c02d00a4a11c5d79a6c6+'<\/a>';

The present work brings together two important theories given by the $\mathbb{Z}_k$-manifolds from Sullivan and stratifolds from Kreck. We introduce the bordism theory of $\mathbb{Z}_k$-stratifolds in order to solve Steenrod's problem for $\mathbb{Z}_k$-coefficients in an affirmative way. Finally, we present a geometric interpretation of the Bockstein long exact sequence and the Atiyah-Hirzebruch spectral sequence for $\mathbb{Z}_k$-bordism ($k$ odd).

Joint work with Carlos Segovia Gonzales This email address is being protected from spambots. You need JavaScript enabled to view it. (UNAM-Oaxaca, México) and Angel Cardenas This email address is being protected from spambots. You need JavaScript enabled to view it. (Universidad de Los Andes, Colombia).

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