arXiv subjects

Johnathon Taylor

Publications and source records attributed to Johnathon Taylor.

6 recordsLinked to original sources

arXiv2026

Tensor products, internal homs, and model structures in two dimensional category theory

In this paper, we introduce a new symmetric monoidal structure on $\mathbf{Cat}$, called the \emph{graph tensor product}, with unit given by the terminal category. This tensor product falls in the middle of a factorization between the funny tensor product and the Cartesian product, giving a factorization connecting these two classical monoidal structures. We extend this construction to a symmetric monoidal structure on $2\mathbf{Cat}$, again with unit $D^0$, which provides an analogous factorization between the funny tensor product and the Cartesian product of $2$-categories. Using the $(\mathrm{bo},\mathrm{lff})$ factorization system on $2\mathbf{Cat}$, we construct a new symmetric monoidal closed model structure on $2\mathbf{Cat}$ whose tensor product restricts to the Cartesian product on the subcategory of flexible $2$-categories. Finally, we prove that this symmetric monoidal model structure fits into a square of weak symmetric monoidal Quillen equivalences relating the Gray tensor product and the flexible tensor product.

math.CT↗

arXiv2026

Algebraic coherators, controlled theories, and Grothendieck realizations

We introduce a construction of algebraic coherators for Grothendieck $\infty$-groupoids using the algebraic small object argument, replacing previous approaches we have used based on distributive series of monads with a more direct method for freely adjoining coherence data. Given a controlled theory, we define unreduced and reduced Grothendieck realizations, producing $\infty$-Lawvere theories and extending this construction functorially to connected diagrams of controlled theories. We apply this framework to construct globular models for monoidal $\infty$-groupoids, symmetric monoidal $\infty$-groupoids, coherent $\infty$-groups, and Picard $\infty$-groupoids. We define canonical semi-model structures on categories of models over $\infty$-Lawvere theories and formulate a generalized pushout conjecture that implies the existence of these semi-model structures and the Homotopy Hypothesis for Grothendieck $\infty$-groupoids.

math.CT↗

arXiv2026

Controlled theories, categorification, and homotopification

In this paper, we introduce the notion of a controlled theory, originally developed in the author's thesis, as a structural tool for the study of higher categorical algebra. We define a notion of deformation for pros and controlled theories in a cartesian closed category. Furthermore, we show that deformations of controlled theories naturally produce Lawvere theories enriched over the same base category. We construct functorial one-dimensional categorifications and homotopifications of controlled theories, yielding Lawvere $2$-theories and Lawvere theories enriched in simplicial sets, respectively. As an application, we obtain a new model for $\infty$-groups and construct a model of coherent group-like $E_\infty$-spaces, which we will show in future work models infinite loop spaces.

math.CT↗

arXiv2026

An Inductive Strategy Towards a Solution to the Generalized Homotopy Hypothesis

Using the theory of distributive series of monads, we construct an $(\infty,0)$-coherator called the \emph{inductive coherator}. The category of models out of the inductive coherator serve as a model for $\infty$-groupoids that possess an underlying globular set. Once we establish the construction for the inductive coherator, we provide the framework for an inductive strategy to prove the Generalized Homotopy Hypothesis obtained by transferring model structure off of the category of $n$-groupoids onto the category of $(n+1)$-groupoids. Moreover, we provide a necessary and sufficient condition for the transfer of model structure to be successful. We conclude by showing if the transfer of model structure may be completed successively, then the Generalized Homotopy Hypothesis is true.

math.CT↗

arXiv2025

Limit Sketches and the Universal Realization of a Limit Sketch

We construct the universal realized limit sketch associated to a given limit sketch. The construction uses factorization systems to organize the classical argument of [2], yielding a streamlined and conceptually unified formulation of the technical steps. This provides a structured framework for understanding realizations of limit sketches in terms of factorization-theoretic data.

math.CT↗

arXiv2025

The Inductive Coherator For Grothendieck Infinity Groupoids

We extend the theory of distributive series of monads of \cite{EC1} by extending the definition to include an $\bN$-indexed collection of monads. Under certain conditions, distributive series of monads will have a colimit in the category of pointed endofunctors. We define a \emph{completable} distributive series of monads to be a distributive series of monads whose induced pointed endofunctor, if it exists, lifts to a monad. We then construct factorization systems used to generate monads on the category of theories over $\Theta_0^\op$, in order to form two \emph{completable} distributive series of monads. The first completable distributive series of monads induces a monad that sends the identity theory over $\Theta_0^\op$ to an $(\infty,0)$-coherator whose inductive construction mimics inductive weak enrichment. The second completable distributive series of monads induces a monad that sends the identity theory over $\Theta_0^\op$ to a theory for strict $\infty$-groupoids.

math.CT↗