Tootfinder

Existence Theorem for Cumulative Universe Towers and Its Applications
Higuchi Joaquim Reizi
https://arxiv.org/abs/2506.21653 https://…

Existence Theorem for Cumulative Universe Towers and Its Applications
This paper builds a cumulative tower of Grothendieck universes that provides a precise size discipline for higher type theory. Starting from an increasing sequence of inaccessible cardinals, we give an inductive-recursive definition of universe codes, their decoding functor, and a rank that controls the growth of each code. For every level of the tower we prove closure under dependent products, dependent sums, identity types, and all finite limits and colimits; the colimit part is obtained with…

Thomason's colimit theorem for the double category of elements
Andrew Gill, Maru Sarazola
https://arxiv.org/abs/2506.08246 https://

Thomason's colimit theorem for the double category of elements
We show that, for any 2-category $C$ and 2-functor $F\colon C \to Cat$, the double category of elements $\iint_C F$ introduced by Grandis and Paré satisfies a version of Thomason's colimit theorem; that is, there is a weak homotopy equivalence $B hocolim F\simeq B(\iint_C F)$.

A Dwyer-Rezk classification for polynomial functors in Weiss calculus
David Barnes, Magdalena K\k{e}dziorek, Niall Taggart
https://arxiv.org/abs/2508.03808 https://

A Dwyer-Rezk classification for polynomial functors in Weiss calculus
In Goodwillie calculus, unpublished work of Dwyer and Rezk provides a classification of reduced filtered colimit preserving $d$-excisive functors from pointed spaces to spectra as spectrum-valued functors on the category of finite sets of cardinality at most $d$ and epimorphisms. We prove through different methods the analogous result in Weiss calculus: $d$-polynomial functors are equivalent to spectrum-valued functors on the category of finite-dimensional inner product spaces of dimension at m…

Compositional Control-Driven Boolean Circuits
Damian Arellanes
https://arxiv.org/abs/2506.22687 https://arxiv.org/pdf/2506.22687

Compositional Control-Driven Boolean Circuits
Boolean circuits abstract away from physical details to focus on the logical structure and computational behaviour of digital components. Despite they have been studied for many decades, compositionality has been widely ignored or examined in an informal manner, which is a property for combining circuits without delving into their internal structure, while supporting modularity and formal reasoning. In this paper, we address this longstanding theoretical gap by proposing colimit-based operators…

Filter Quotient Model Structures
Nima Rasekh
https://arxiv.org/abs/2508.07735 https://arxiv.org/pdf/2508.07735

Filter Quotient Model Structures
The filter quotient construction is a particular instance of a filtered colimit of categories. It has primarily been considered in the context of categorical logic, where it has been used effectively to construct non-trivial models, for example new models of set theory. In this work we prove that given a model category and a suitable notion of filter of subterminal objects, the filter quotient construction will preserve the model structure. We also show that this new model structure inherits …

Semi-infinite parabolic IC-sheaf II: the Ran space version
Gurbir Dhillon, Sergey Lysenko
https://arxiv.org/abs/2508.01527 https://arxiv.org/pdf/2508.01527…

Semi-infinite parabolic IC-sheaf II: the Ran space version
This paper is a sequel to arXiv:2310.0638. Let G be a split reductive group. We define the parabolic semi-infinite category on the Ran version of the affine Grassmanian Gr_{G,Ran}. We study the semi-infinite parabolic IC-sheaf in this category. We provide several descriptions of this object, one as certain colimit, another as an intermediate extension in certain category. We relate the global and local semi-infinite categories of sheaves. We also relate the intersection cohomology sheaf of the …