N∞ operads over a group G encode homotopy commutative operations together with a class of equivariant transfer (or norm) maps. Their homotopy theory is given by transfer systems, which are certain discrete objects that have a rich combinatorial structure defined in terms of the subgroup lattice of G. In this talk, we will show that when G is finite Abelian, transfer systems are in bijection with weak factorization systems on the poset category of subgroups of G. This leads to an involution on the lattice of transfer systems, generalizing the work of Balchin-Bearup-Pech-Roitzheim for cyclic groups of squarefree order. This is joint work with Evan Franchere, Usman Hafeez, Peter Marcus, Kyle Ormsby, Weihang Qin, and Riley Waugh.

Together with Imma Galvez-Carillo and Andy Tonks, we constructed Bi and Hopf-algebras from a three-fold hierarchy, simplicial objects, Co-operads with multiplication and Feynman categories. The common theme is that the co-product is a dualized composition product and the product iis an extra structure, which is free in the most prominent examples. These are the Hopf algebra of Baues in Topology, those of Goncharov and Brown in number theory, and those of Connes and Kreimer in mathematical physics. The natural structures are bi-algebras and the Hopf algebras appear as a connected quotient.

Together with Yang Mo, we introduced the notion of a path-like bi-algebra in order to better understand the relationship between the bialgebras and the Hopf algebras, which allowed us to complete the picture. The usual Quillen condition of being connected is replaced by the so-called Quillen-Takeushi filtration being exhaustive. This allows us to reduce the obstructions to having an antipode to (semi)-grouplike elements. This theory comprises all the examples above as well as those of May as special cases. Finally, quotients now appear naturally as the universal quotient spaces through which characters -with particular properties- factor. The characters take values in Rota-Baxter-algebras which are at the heart of the renormalization theory. This brings the theory full circle.

Decomposition spaces (also known as 2-Segal spaces) and the machinery of homotopy linear algebra allow us to realise objective combinatorial bialgebra structures, rendering classical combinatorial bialgebras once cardinalites are taken. In many cases such bialgebras are related, through CULF functors given by base change, Galois connection, duality, that descend to their numerical counterparts but provide deeper understanding of phenomena at the objective level. We will present a few important cases of such constructions, in particular relating to the bialgebras of Malvenuto-Reutenauner and of symmetric functions. Joint work in progress with Joachim Kock and Andrew Tonks.

Overlapping cubical decompositions admit hodge star dualities and analogues of Grassman algebra for differential forms and multivector fields with their exterior d and the divergence operator differentials del. For these operators there are hierarchies of deformation corrections to the first order derivation structure of exterior d and the second order derivation structure of the divergence operator. These suggest computer codes with surprising properties. There is a joint paper in the Atiyah Memorial Volume 2021 with Ruth Lawrence and Nissim Ranade describing this discretization with explicit calculations of the deformations.The computer studies are in progress.

In this talk, I shall present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations of objects restricted to certain subcategories. Several applications will be given in the settings of relative Gorenstein homological algebra and chain complexes, as well as to characterize some important results on the Finitistic Dimension Conjecture, the existence of right adjoints of quotient functors by Serre subcategories, and the description of cotorsion pairs in triangulated categories as co-t-structures.

This is a joint work with Mindy Huerta and Octavio Mendoza (Instituto de Matemáticas - Universidad Nacional Autónoma de México).

The S-construction, first defined in the setting of cofibration categories by Waldhausen, gives a way to define the algebraic K-theory associated to certain kinds of categorical input. It was proved by Galvez-Carrillo, Kock, and Tonks that the result of applying this construction to an exact category is a decomposition space, also called a 2-Segal space, and Dyckerhoff and Kapranov independently proved the same result for the slightly more general input of proto-exact categories. In joint work with Osorno, Ozornova, Rovelli, and Scheimbauer, we proved that these results can be maximally generalized to the input of augmented stable double Segal spaces, so that the S-construction defines an equivalence of homotopy theories. In this talk, we'll review the S-construction and the reasoning behind these stages of generalization. Time permitting, we'll discuss attempts to characterize those augmented stable double Segal spaces that correspond to cyclic spaces, which is work in progress with Walker Stern.

Lie groupoids up to Morita equivalences serve as models for differentiable stacks, categorified spaces that generalize manifolds and orbifolds, and which are useful when dealing with singular quotients. Vector bundles over Lie groupoids have the tangent and cotangent constructions as prominent examples, and they admit a nice interpretation in terms of representations up to homotopy. In this talk, based on joint works with C. Ortiz, D. Stefani, and J. Desimoni, I will first discuss the Morita invariance of vector bundles, then describe the general linear 2-groupoid, and finally present a classification of stacky vector bundles by the resulting 2-Grassmannian.