In this talk, I will discuss recent work with Philippe Malbos, Eric Goubault and Georg Struth, concerning the algebraic formalisation of coherence and confluence. In the context of abstract rewriting systems (ARSs), the Church-Rosser theorem states that every branching can be completed with a confluence if, and only if, every zig-zag can be completed with a confluence. The relational formulation of this theorem has been abstracted to the setting of Kleene algebra, in which a proof is accomplished by means of a formal calculation. Furthermore, rewriting properties may be defined in Kleene algebra using simple inequalities, and a rich modal structure, defined via domain and codomain operators, has been developed. Polygraphs have equally been applied to ARSs. Higher dimensional cells are used to define coherence properties of ARSs and reduction/normalisation strategies. These constructions are linked to the notion of cofibrant replacement in the folk model structure on omega-categories, in the sense that a coherent presentation of a category by a polygraph constitutes its cofibrant replacement. A coherent formulation of the Church-Rosser theorem is expressed in the language of polygraphs. We combine these two approaches by introducing the structure of (globular) modal n-Kleene algebra, thereby providing a natural setting for the formalisation of coherent confluence. We formulate and prove both the Church-Rosser theorem and Newman's lemma in this setting, and provide an explicit link to polygraphs: the power set of the set of $n$-cells of the free category generated by a polygraph is a globular modal $n$-Kleene algebra with source, target and composition maps lifted to the set-level. In the talk, I will present the main definitions and examples, and briefly sketch a proof of the coherent Church-Rosser theorem in globular modal n-Kleene algebras.

We work in the setting of ∞-categories and use the terminology “category” = “∞-category” = “(∞,1)-category”, “topos” = “∞-topos”, “presheaf” = “presheaf of ∞-groupoids”. All our results hold for 1-categories and 1-topoi as well.

The small object argument of Quillen is a well-known construction of the weak factorisation system generated “on the left” by a small set of arrows of a category.

We recall a variant of the small object argument, essentially due to Kelly, that constructs the unique factorisation system (^\bot(W^\bot), W^\bot) generated by a small diagram W of morphisms of a locally presentable category C. Our main result shows that, given sufficient conditions on W (called a “pre-modulator”), Kelly's construction simplifies so that the unique factorisation of any morphism is given by iterating a “plus construction” generalising the one known from sheafification. Further, any small diagram can be replaced with a pre-modulator that generates the same unique factorisation system. Thus we show that every accessible reflective localisation of a locally presentable category can be calculated as a transfinite iteration of a plus construction. The classical plus construction for Grothendieck sites is a particular case, given by the pre-modulator (in fact a lex modulator) corresponding to the Grothendieck topology (seen as a diagram of sub-representables).

We also define “modulators” (resp. “lex modulators”) and prove that their corresponding unique factorisation systems are modalities (resp. lex modalities). It makes sense to see lex modulators as the correct generalisation of the notion of Grothendieck topology from 1-categories to ∞-categories, since every left-exact localisation (topological or not) of an ∞-topos can be obtained from a lex modulator. We show moreover that the plus construction associated to any lex modulator on an ∞-topos converges after (n+2) iterations when applied to an n-truncated object. This explains why the usual plus construction for 1-topoi converges after 2 iterations. The talk will present the main results and examples, without going into much detail of the demonstrations.

Categorical abstraction is a powerful organizing principle in computer algebra. In this talk, we explain the concept of constructive category theory and how we implement this concept in our software project CAP-Categories, algorithms, programming. In CAP it is possible to implement higher algorithms and data structures using basic categorical operations as primitives, which in turn often rely on classical algorithms in computer algebra like the computation of Gröbner bases. As an example, we show how our categorical framework can be used for computing with finitely presented functors.

The notion of operadic category appeared in the work of Batanin and Markl as a tool for working with various operad like structures. Each operadic category O has an associated category of O-operads with values in an arbitrary symmetric monoidal category. Operadic categories and operadic functors form a category.

In my talk I will show that the category of symmetric operads in Set (variation of colours is allowed) is a reflective subcategory of the category of operadic categories. The inclusion is given by (operadic) Grothendieck construction and the reflection is given by evaluation of the left Kan extension along arity functor on the terminal operad. Thus the notion of operadic category can be considered as a flexible extension of the notion of symmetric operad.

Moreover, there is yet another functor from operadic categories to symmetric operads which sends an operadic category O to a symmetric operad in Set whose algebras are exactly O-operads. These three functors (Grothendieck construction, its left adjoint and free operad functor) fit in a nice picture with a universal property. In particular, they various composites generate the Baez-Dolan +-constructions for both symmetric operads and operadic categories.

An (oo,1)-category is a weak notion of a category where composition and associativity is only defined up to higher coherences. For that reason it is often impossible to directly define functors and we thus use fibrations instead.

For quasi-categories, a popular model of (oo,1)-categories, fibrations have been studied carefully by Joyal and Lurie and are commonly used in all kinds of categorical constructions.

In this talk we define and study left fibrations for another model of (oo,1)-categories, namely complete Segal spaces. We will show that these fibrations come with a model structure and that we can characterize the fibrant objects and the equivalences of the model structure, which allows us to prove many strong results about such fibrations without ever translating to functors.

If time permits we will discuss one particular strength of this approach to the theory of fibrations, namely how it can be generalized to fibrations for the (oo,n)-categorical analogue of complete Segal spaces: n-fold Segal spaces.

Les algèbres double Poisson sont une version non commutative des algèbre de Poisson et apparaissent aujourd'hui dans de nombreux domaines mathématiques. Afin de déterminer une version à homotopie près de cette structure algébrique, il est nécessaire de déterminer une résolution cofibrante de la propérade qui l'encode. Après avoir rappeler la notion de propérade, je présenterai la stratégie afin de démontrer la koszulité de cette propérade. Celle-ci fait notamment intervenir un nouveau type de monoïde, appelé protopérade, qui permet de ramener ce problème difficile de koszulité à un autre, plus simple, permettant ainsi d'utiliser des arguments de réécriture pour conclure.