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Giulio Lo Monaco: Vopěnka’s principle in ∞-categories

24th June, 2021

Vopěnka's principle has arisen as a model-theoretical statement, provably independent of ZFC set theory. However, there are a number of categorical ways of formulating it, preventing the existence of proper classes of objects with some conditions in presentable categories, and these are what our attention will be focused on. In particular, we will look at analogous statements in the context of ∞-categories and we will ask how these new statements interact with the older ones. Moreover, some of the consequences of Vopěnka's principle on classes of subcategories of presentable categories are investigated and to some extent generalized to ∞-categories. A parallel discussion is undertaken about the similar but weaker statement known as weak Vopěnka's principle.

Leonardo Larizza: Lax factorization systems and categories of partial maps

17th June, 2021

Factorization systems describe morphisms in a category by factorizing them into pairs of composable morphisms. Their definition depends on a kind of orthogonality relation between morphisms, which entails the existence of some diagonal morphisms for certain squares. In this seminar we present the new notion of lax weak orthogonality between morphisms, which involves lax squares and the factorization systems it generates. Then we will introduce lax versions of functorial and algebraic weak factorization systems and some of their properties. These lax factorization systems are discussed, keeping the theory of ordinary factorization systems as a blueprint and providing useful properties. An overview of the examples of such lax factorization systems is presented in the context of partial maps. We conclude with a discussion of general constructions of these examples and their description in the particular case of sets with partial maps.

Raffael Stenzel: Proof relevance in higher topos theory

27th May, 2021

In the short course of its definition and early exploration, the theory of higher toposes (by which I specifically mean (∞,1)-toposes) has been found to exhibit various traits which appear rather odd from the perspective of ordinary topos theory. Motivated by the fact that the internal language of every higher (Grothendieck) topos is a univalent type theory - and hence is intrinsically 'proof relevant' - we reconsider the basic characteristic notions associated to a higher topos from a purely logical proof relevant point of view.

Given a small ∞-category C, this will motivate the notion of a logical structure sheaf on C whose ideals correspond exactly to the left exact localizations of the infinity-category [Cop, S] of presheaves over C. This in turn will naturally lead to a corresponding notion of generalized Grothendieck topologies on C which, first, capture all higher toposes embedded in [Cop, S], and second, correspond exactly to the classical notion of Grothendieck topologies in the monic (i.e. the proof irrelevant) context. We will see that these notions induce a Kripke-Joyal semantics valued in spaces (rather than in the classical subobject classifier) in obvious fashion as well. In the end of the talk we will take a look at a few examples of such topologies and, if time permits (which it rarely ever does, time appears to be pretty absolute when it comes to this), end with a discussion of some open questions.

Tashi Walde: Higher Segal spaces via Higher Excision

20th May, 2021

Starting from the classical Segal spaces, Dyckerhoff and Kapranov introduced a hierarchy of what they 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 higher levels and the hierarchy as a whole.

In 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 manifold calculus' on the simplex category and on the cyclic category, for which the polynomial functors are precisely the higher Segal objects. Furthermore, this perspective yields intrinsic categorical characterizations of higher Segal objects in the spirit of higher excision.

Hoang Kim Nguyen: Contravariant homotopy theories and Quillen’s Theorem A

29th April, 2021

In this talk I will show how to construct a model structure on a locally presentable category with a suitable cylinder object such that the model structure behaves in a 'covariant' or 'contraviariant' way with respect to the cylinder. Examples of such model structures include the covariant and contravariant model structures on simplicial sets and the cocartesian and cartesian model structures on marked simplicial sets modelling presheaves with values in ∞-groupoids and ∞-categories respectively.

The model structures come with an abstract notion of cofinal functor which recovers the usual definition of cofinal functor for ∞-categories when applied to the covariant and contravariant model structures on simplicial sets. When applied yo presheaves valued in n-types, one obtains a version of Quillen’s Theorem A for n-categories.

Michael Ching: Tangent ∞-categories and Goodwillie calculus

8th April, 2021

In 1984 Rosický introduced tangent categories in order to capture axiomatically some properties of the tangent bundle functor on the category of smooth manifolds and smooth maps. Starting in 2014 Cockett and Cruttwell have developed this theory in more detail to emphasize connections with cartesian differential categories and other contexts arising from computer science and logic.

In this talk I will discuss joint work with Kristine Bauer and Matthew Burke which extends the notion of tangent category to ∞-categories. To make this generalization we use a characterization by Leung of tangent categories as modules over a symmetric monoidal category of Weil-algebras and algebra homomorphisms. Our main example of a tangent ∞-category is based on Lurie's model for the tangent bundle to an ∞-category itself. Thus we show that there is a tangent structure on the ∞-category of (differentiable) ∞-categories. This tangent structure encodes all the higher derivative information in Goodwillie’s calculus of functors, and sets the scene for further applications of ideas from differential geometry to higher category theory.

Jiří Adámek: C-Varieties of Ordered and Quantitative Algebras

1st April, 2021

Mardare, Panangaden and Plotkin introduced C-varieties of algbebras on metric spaces. These are categories of metric-enriched algebras specified by equations in a context. A context puts restrictions on the distances of variables one uses. We prove that C-varieties are precisely the monadic categories over Met for countably accessible enriched monads preserving epimorphisms.

We analogously introduce C-varieties of ordered algebras as categories specified by inequalities in a context. Which means that conditions on inequalities between variables are imposed. We prove that C-varieties precisely correspond to enriched finitary monads on Pos preserving epimorphisms.

Karol Szumilo: ∞-groupoids in lextensive categories

18th March, 2021

I will discuss a construction of a new model structure on simplicial objects in a countably lextensive category (i.e., a category with well-behaved finite limits and countable coproducts). This builds on previous work on a constructive model structure on simplicial sets, originally motivated by modelling Homotopy Type Theory, but now applicable in a much wider context.

Paolo Perrone: Kan extensions are partial colimits

11th February, 2021

One way of interpreting a left Kan extension is as taking a kind of 'partial colimit', where one replaces parts of a diagram by their colimits. We make this intuition precise by means of the 'partial evaluations' sitting in the so-called bar construction of monads. The (pseudo)monads of interest for forming colimits are the monad of diagrams and the monad of small presheaves, both on the category CAT of locally small categories. We also define a morphism of monads between them, which we call 'image', and which takes the 'free colimit' of a diagram. This morphism allows us in particular to generalize the idea of 'cofinal functors', i.e. of functors which leave colimits invariant in an absolute way. This generalization includes the concept of absolute colimit as a special case. The main result of this work says that a pointwise left Kan extension of a diagram corresponds precisely to a partial evaluation of its colimit. This categorical result is analogous to what happens in the case of probability monads, where a conditional expectation of a random variable corresponds to a partial evaluation of its centre of mass.

Jiří Rosický: Metric monads

4th February, 2021

We develop universal algebra over an enriched category and relate it to finitary enriched monads. Using it, we deduce recent results about ordered universal algebra where inequations are used instead of equations. Then we apply it to metric universal algebra where quantitative equations are used instead of equations. This contributes to understanding of finitary monads on the category of metric spaces.

John Bourke: Accessible ∞-cosmoi

10th December, 2020

Riehl and Verity introduced ∞-cosmoi - certain simplicially enriched categories - as a framework in which to give a model-independent approach to ∞-categories. For instance, there is an infinity cosmos of ∞-categories with finite limits or colimits, or of cartesian fibrations. In this talk, I will introduce the notion
of an accessible ∞-cosmos and explain that most, if not all, ∞-cosmoi arising in practice are accessible. Applying results of earlier work, it follows that accessible ∞-cosmoi have homotopy weighted colimits and admit a broadly applicable homotopical adjoint functor theorem.

Martin Bidlingmaier: Model categories of lcc categories and the gros model of dependent type theory.

26th November, 2020

In this talk we discuss various model categories of locally cartesian closed (lcc) categories and their relevance to coherence problems, in particular the coherence problem of categorical semantics of dependent type theory. We begin with Lcc, the model category of locally cartesian closed (lcc) sketches. Its fibrant objects are precisely the lcc categories, though without assigned choices of universal objects. We then obtain a Quillen equivalent model category sLcc of strict lcc categories as the category of algebraically fibrant objects of Lcc. Strict lcc categories are categories with assigned choice of lcc structure, and their morphisms preserve these choices on the nose. Conjecturally, sLcc is precisely Lack’s model category of algebras for a 2-monad T , where T is instantiated with the free lcc category functor on Cat. We then discuss the category of algebraically cofibrant objects of sLcc and show how it can serve as a "gros" model of dependent type theory.

Maru Sarazola: The stable homotopy hypothesis

19th November, 2020

The homotopy hypothesis is a well-known bridge between topology and category theory. Its most general formulation, due to Grothendieck, asserts that topological spaces should be "the same" as infinity-groupoids. In the stable version of the homotopy hypothesis, topological spaces are replaced with spectra.

In this talk we will review the classical homotopy hypothesis, and then focus on the stable version. After discussing what the stable homotopy hypothesis should look like on the categorical side, we will use the Tamsamani model of higher categories to provide a proof.

Ivan Di Liberti: Enriched Locally Generated Categories

5th November, 2020

We introduce the notion of M-locally generated category for a factorization system (E,M) and study its properties. We offer a Gabriel-Ulmer duality for these categories, introducing the notion of nest. We develop this theory also from an enriched point of view. We apply this technology to Banach spaces showing that it is equivalent to the category of models of the nest of finite-dimensional Banach spaces.

Taichi Uemura: The universal exponentiable arrow

15th October, 2020

Cartmell showed that the category of generalized algebraic theories is equivalent to the category of contextual categories. This implies that the theory of generalized algebraic theories is essentially algebraic. We characterize the essentially algebraic theory of generalized algebraic theories as the free category with finite limits and with an exponentiable arrow.

Christina Vasilakopoulou: Oplax Hopf Algebras

8th October, 2020

Hopf categories were introduced by Batista, Caenepeel and Vercruysse in 2016, as a many-object generalization of Hopf algebras linked to other notions like multiplier Hopf algebras and with applications to categorical Galois theory. What is of particular interest is that the multiplication and comultiplication appear to make use of different monoidal products: Gabriella Böhm in subsequent work expressed Hopf categories as specific opmonoidal monads. In our work, we follow a different direction of generalizing Hopf monoids in a braided monoidal bicategory, that allows us to realize Hopf categories as Hopf-type objects over the same monoidal product, restoring in a sense the self-dual feature of classical Hopf algebras. In this talk, we introduce oplax bimonoids and oplax Hopf monoids in an arbitrary braided monoidal bicategory, we study their main properties and we exemplify such structures in a Span-type bicategory where they return semi-Hopf and Hopf categories.

Raffael Stenzel: (∞,1)-comprehension schemes

24th September, 2020

Comprehension schemes arose as crucial notions in the early work on the foundations of set theory, and hence they found expression in a considerable variety of foundational settings for mathematics. Particularly, they have been introduced to the context of categorical logic first by Lawvere and then by Benabou in the 1970s.

In this talk we define and study a theory of comprehension schemes for fibered ∞-categories, generalizing Johnstone’s respective notion for ordinary categories. This includes natural generalizations of all the fundamental instances originally defined by Benabou, and their application to Jacob's comprehension categories. Thereby, we can characterize

  • numerous categorical structures arising in higher topos theory,
  • the notion of univalence,
  • internal ∞-categories,

in terms of comprehension schemes, while some of the 1-categorical counterparts fail to hold in ordinary category theory. As an application, we can show that the universal cartesian fibration is represented via externalization by the "freely walking chain" in the ∞-category of small ∞-categories.

In the end, if my time management permits, we take a look at the externalization construction of internal ∞-categories from a model categorical perspective and review some examples from the literature in this light.

Edoardo Lanari: Gray tensor products and lax functors of (∞,2)-categories

10th September, 2020

We give a definition of the Gray tensor product in the setting of scaled simplicial sets which is associative and forms a left Quillen bi-functor with respect to the bicategorical model structure of Lurie. We then introduce a notion of oplax functor in this setting, and use it in order to characterize the Gray tensor product by means of a universal property. A similar characterization was used by Gaitsgory and Rozenblyum in their definition of the Gray product, thus giving a promising lead for comparing the two settings.