Tootfinder

Symmetric monoidal extensions and graph cobordisms between finite sets
Andrea Bianchi
https://arxiv.org/abs/2509.22575 https://arxiv.org/pdf/2509.22575

Symmetric monoidal extensions and graph cobordisms between finite sets
Given a symmetric monoidal $(\infty,n)$-category $\mathcal{C}$ and a space $X$, we address the problem of explicitly describing the symmetric monoidal $(\infty,n)$-category freely obtained from $\mathcal{C}$ by adjoining $X$ new $n$-morphisms with prescribed sources and targets. We develop an apparatus of tools that allow one to detect in concrete situations such a free symmetric monoidal extension. As motivating application, we introduce a symmetric monoidal $(\infty,2)$-category ${\mathbb{G}\…

Monoidal model structures over infinite groups
Ioannis Emmanouil, Olympia Talelli
https://arxiv.org/abs/2510.08195 https://arxiv.org/pdf/2510.08195

Monoidal model structures over infinite groups
The stable category of modules over the algebra of a finite group with coefficients in a field is a compactly generated tensor triangulated category, that has been studied extensively in representation theory. In this paper, we provide a plethora of infinite groups G, for which the category of kG-modules (where k is a commutative coherent ring of finite global dimension) admits a monoidal model structure, in the sense of Hovey, whose associated homotopy category is a compactly generated tensor …

Geometric Construction of Quiver Tensor Products
Daigo Ito, John S. Nolan
https://arxiv.org/abs/2510.05277 https://arxiv.org/pdf/2510.05277

Geometric Construction of Quiver Tensor Products
By a classic theorem of Beilinson, the perfect derived category $\operatorname{Perf}(\mathbb{P}^n)$ of projective space is equivalent to the category of derived representations of a certain quiver with relations. The vertex-wise tensor product of quiver representations corresponds to a symmetric monoidal structure $\otimes_{\mathsf{Q}}$ on $\operatorname{Perf}(\mathbb{P}^n)$. We prove that, for a certain choice of equivalence, the symmetric monoidal structure $\otimes_{\mathsf{Q}}$ may be descr…

The Morita $(\infty,2)$-category of a monoidal category as a $2$-complicial set
Arghan Dutta, Stefano Luneia, Martina Rovelli, Sam Silver
https://arxiv.org/abs/2509.21472 https:…

The Morita $(\infty,2)$-category of a monoidal category as a $2$-complicial set
We provide an explicit and elementary construction of the Morita $(\infty,2)$-category of a monoidal category which satisfies minimal conditions. We construct it as a $3$-coskeletal $2$-complicial set, in which the vertices encode the monoids, the edges encode the bimodules, the triangles encode the bimodule maps out of a balanced tensor product, and tetrahedra encode composition of bimodule maps. The marked edges encode invertible bimodules, and the marked triangles encode bimodule isomorphism…

Monoid Structures on Indexed Containers
Michele De Pascalis, Tarmo Uustalu, Niccol\`o Veltr\`i
https://arxiv.org/abs/2509.25879 https://arxiv.org/pdf/2509.…

Monoid Structures on Indexed Containers
Containers represent a wide class of type constructions relevant for functional programming and (co)inductive reasoning. Indexed containers generalize this notion to better fit the scope of dependently typed programming. When interpreting types to be sets, a container describes an endofunctor on the category of sets while an I-indexed container describes an endofunctor on the category Set^I of I-indexed families of sets. We consider the monoidal structure on the category of I-indexed containe…

An algebra modality admitting countably many deriving transformations
Jean-Baptiste Vienney
https://arxiv.org/abs/2510.03953 https://arxiv.org/pdf/2510.039…

An algebra modality admitting countably many deriving transformations
A differential category is an additive symmetric monoidal category, that is, a symmetric monoidal category enriched over commutative monoids, with an algebra modality, axiomatizing smooth functions, and a deriving transformation on this algebra modality, axiomatizing differentiation. Lemay proved that a comonoidal algebra modality has at most one deriving transformation, thus differentiation is unique in models of differential linear logic. It was then an open problem whether this result extend…

Hopf bimodules for bialgebroids
Sophie Chemla, Niels Kowalzig
https://arxiv.org/abs/2510.06417 https://arxiv.org/pdf/2510.06417

Hopf bimodules for bialgebroids
Generalising a result for Hopf algebras, we not only define the four possible types of Hopf modules in the bialgebroid setting but also yield the notion of two-sided two-cosided Hopf modules, also known as Hopf bimodules or tetramodules, in this realm. By explicitly formulating a fundamental theorem for Hopf modules via the concept of Hopf-Galois comodules, we prove that the category of Hopf bimodules can be endowed with the structure of a (pre-)braided monoidal category in two different ways, …

Closed exact categories of modules over generalized adic rings. Part 1: The bounded case
Francesco Baldassarri
https://arxiv.org/abs/2510.00848 https://arx…

Closed exact categories of modules over generalized adic rings. Part 1: The bounded case
We develop general foundations of topological algebra over a linearly topologized ring k in a format applicable to both formal schemes and analytic adic spaces. We are especially interested in determining quasi-abelian categories of complete linearly topologized k-modules, which are also closed symmetric monoidal for a suitable choice of tensor product and internal Hom, and have enough projectives or injectives. For k a suitably generalized adic ring, we describe here a few examples of such cat…

Crosslisted article(s) found for math.AT. https://arxiv.org/list/math.AT/new
[1/1]:
- The Morita $(\infty,2)$-category of a monoidal category as a $2$-complicial set
Arghan Dutta, Stefano Luneia, Martina Rovelli, Sam Silver

Notions of Cauchy completeness for normed categories
Dirk Hofmann, Walter Tholen
https://arxiv.org/abs/2510.00912 https://arxiv.org/pdf/2510.00912

Notions of Cauchy completeness for normed categories
As already mentioned by Lawvere in his 1973 paper, the characterisation of Cauchy completeness of metric spaces in terms of representability of adjoint distributors amounts to the idempotent-split property of an ordinary category when the governing symmetric monoidal-closed category is changed from the extended real half-line to the category of sets. In this paper, for any commutative quantale \(\mathcal{V}\), we extend these two characterisations of Lawvere-style completeness to \(\mathcal{V}\…