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Ross Street

Publications and source records attributed to Ross Street.

At least 19 recordsLinked to original sources
arXiv2026

Absolute colimits

In the context of enriched category theory, we give necessary and sufficient conditions for a module morphism $\alpha \dd M \to \CC(F,Z)$ to exhibit a functor $Z\dd \CA\to \CC$ as an absolute $M$-weighted colimit of a functor ${F\dd \CB\to \CC}$. We also review, with short proofs, the various criteria for the weight $M$ itself to be absolute, in the sense that any $M$-weighted colimit is absolute. Finally, we prove that any absolute $M$-weighted colimit can be viewed as a colimit weighted by an absolute weight $M'$.

math.CT↗
arXiv2026

Homodular pseudofunctors and bicategories of modules

The universal property for the B\'enabou bicategory of distributors (although we call them "modules") presented here is somewhat implicitly spread over a series of papers and yet, to my knowledge, does not appear in print. The inclusion of a bicategory $\mathscr{W}$ into the bicategory $\mathscr{W}\text{-}\mathrm{Mod}$ of $\mathscr{W}$-enriched categories and modules between them does have a completion property with respect to freely adjoining lax colimits (collages). Here we are interested in the universal property of the construction of $\mathscr{W}\text{-}\mathrm{Mod}$ from $\mathscr{W}\text{-}\mathrm{Cat}$. What we have in mind is an objective version of the notion of {\em homological functor} used by Andr\'e Joyal in 1985.

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arXiv2025

Objective Mackey and Tambara functors via parametrized categories

The first word in the title is intended in a sense suggested by Lawvere and Schanuel whereby finite sets are objective natural numbers. At the objective level, the axioms defining abstract Mackey and Tambara functors are categorically familiar. The first step was taken by Harald Lindner in 1976 when he recognized that Mackey functors, defined as pairs of functors, were single functors with domain a category of spans. We define objective Mackey and objective Tambara functors as parametrized categories which have local finite products and satisfy some parametrized completeness and cocompleteness restriction. However, we can replace the original parametrizing base for objective Mackey functors by a bicategory of spans while the replacement for objective Tambara functors is a bicategory obtained by iterating the span construction; these iterated spans are polynomials. There is an objective Mackey functor of ordinary Mackey functors. We show that there is a distributive law relating objective Mackey functors to objective Tambara functors analogous to the distributive law relating abelian groups to commutative rings. We remark on hom enrichment matters involving the 2-category $\mathrm{Cat}_{+}$ of categories admitting finite coproducts and functors preserving them, both as a closed base and as a skew-closed base.

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arXiv2023

Wood fusion for magmal comonads

The goal is to show how a 1978 paper of Richard Wood on monoidal comonads and exponentiation relates to more recent publications such as Pastro-Street (2009) and Brugui\'eres-Lack-Virelizier (2011). In the process, we mildly extend the ideas to procomonads in a magmal setting and suggest it also works for algebras for any club in the sense of Max Kelly (1972).

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arXiv2022

The core groupoid can suffice

This work results from a study of Nicholas Kuhn's paper entitled "Generic representation theory of finite fields in nondescribing characteristic". Our goal is to abstract the categorical structure required to obtain an equivalence between functor categories $[\mathscr{F},\mathscr{V}]$ and $[\mathscr{G},\mathscr{V}]$ where $\mathscr{G}$ is the core groupoid of the category $\mathscr{F}$ and $\mathscr{V}$ is a category of modules over a commutative ring.

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arXiv2021

Variation on a comprehensive theme

The main result concerns a bicategorical factorization system on the bicategory $\mathrm{Cat}$ of categories and functors. Each functor $A\xra{f} B$ factors up to isomorphism as $A\xra{j}E\xra{p}B$ where $j$ is what we call an ultimate functor and $p$ is what we call a groupoid fibration. Every right adjoint functor is ultimate. Functors whose ultimate factor is a right adjoint are shown to have bearing on the theory of polynomial functors.

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arXiv2020

Cauchy completeness for DG-categories

We go back to the roots of enriched category theory and study categories enriched in chain complexes; that is, we deal with differential graded categories (DG-categories for short). In particular, we recall weighted colimits and provide examples. We solve the 50 year old question of how to characterize Cauchy complete DG-categories in terms of existence of some specific finite absolute colimits. As well as the interactions between absolute weighted colimits, we also examine the total complex of a chain complex in a DG-category as a non-absolute weighted colimit.

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arXiv2020

Monoidal centres and groupoid-graded categories

We denote the monoidal bicategory of two-sided modules (also called profunctors, bimodules and distributors) between categories by $\mathrm{Mod}$; the tensor product is cartesian product of categories. For a groupoid $\scr{G}$, we study the monoidal centre $\mathrm{ZPs}(\scr{G},\mathrm{Mod}^{\mathrm{op}})$ of the monoidal bicategory $\mathrm{Ps}(\scr{G},\mathrm{Mod}^{\mathrm{op}})$ of pseudofunctors and pseudonatural transformations; the tensor product is pointwise. Alexei Davydov defined the full centre of a monoid in a monoidal category. We define a higher dimensional version: the full monoidal centre of a monoidale (= pseudomonoid) in a monoidal bicategory $\scr{M}$, and it is a braided monoidale in the monoidal centre $\mathrm{Z}\scr{M}$ of $\scr{M}$. Each fibration $\pi : \scr{H} \to \scr{G}$ between groupoids provides an example of a full monoidal centre of a monoidale in $\mathrm{Ps}(\scr{G},\mathrm{Mod}^{\mathrm{op}})$. For a group $G$, we explain how the $G$-graded categorical structures, as considered by Turaev and Virelizier in order to construct topological invariants, fit into this monoidal bicategory context. We see that their structures are monoidales in the monoidal centre of the monoidal bicategory of $k$-linear categories on which $G$ acts.

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arXiv2020

Freely adjoining monoidal duals

Given a monoidal category $\mathscr{C}$ with an object $J$, we construct a monoidal category $\mathscr{C}[J^{\vee}]$ by freely adjoining a right dual $J^{\vee}$ to $J$. We show that the canonical strong monoidal functor $\Omega : \mathscr{C}\to \mathscr{C}[J^{\vee}]$ provides the unit for a biadjunction with the forgetful 2-functor from the 2-category of monoidal categories with a distinguished dual pair to the 2-category of monoidal categories with a distinguished object. We show that $\Omega : \mathscr{C}\to \mathscr{C}[J^{\vee}]$ is fully faithful and provide coend formulas for homs of the form $\mathscr{C}[J^{\vee}](U,\Omega A)$ and $\mathscr{C}[J^{\vee}](\Omega A,U)$ for $A\in \mathscr{C}$ and $U\in \mathscr{C}[J^{\vee}]$. If $\mathbb{N}$ denotes the free strict monoidal category on a single generating object $1$ then $\mathbb{N}[1^{\vee}]$ is the free monoidal category $\mathrm{Dpr}$ containing a dual pair $- \dashv +$ of objects. As we have the monoidal pseudopushout $\mathscr{C}[J^{\vee}] \simeq \mathrm{Dpr} +_{\mathbb{N}} \mathscr{C}$, it is of interest to have an explicit model of $\mathrm{Dpr}$: we provide both geometric and combinatorial models. We show that the (algebraist's) simplicial category $\Delta$ is a monoidal full subcategory of $\mathrm{Dpr}$ and explain the relationship with the free 2-category $\mathrm{Adj}$ containing an adjunction. We describe a generalization of $\mathrm{Dpr}$ which includes, for example, a combinatorial model $\mathrm{Dseq}$ for the free monoidal category containing a duality sequence $X_0\dashv X_1\dashv X_2 \dashv \dots$ of objects. Actually, $\mathrm{Dpr}$ is a monoidal full subcategory of $\mathrm{Dseq}$.

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arXiv2020

Polynomials as spans

The paper defines polynomials in a bicategory $\mathscr{M}$. Polynomials in bicategories $\mathrm{Spn}\mathscr{C} \ $ of spans in a finitely complete category $\mathscr{C} \ $ agree with polynomials in $\mathscr{C} \ $ as defined by Nicola Gambino and Joachim Kock, and by Mark Weber. When $\mathscr{M}$ is \textit{calibrated}, we obtain another bicategory $\mathrm{Poly}\mathscr{M}$. We see that polynomials in $\mathscr{M}$ have representations as pseudofunctors $\mathscr{M}^{\mathrm{op}}\to \mathrm{Cat}$. Calibrations are produced for the bicategory of relations in a regular category and for the bicategory of two-sided modules (distributors) between categories thereby providing new examples of bicategories of "polynomials".

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arXiv2019

Span composition using fake pullbacks

The construction of a category of spans can be made in some categories $\CC$ which do not have pullbacks in the traditional sense. The PROP for monoids is a good example of such a $\CC$. The 2012 book concerning homological algebra by Marco Grandis gives the proof of associativity of relations in a Puppe-exact category based on a 1967 paper of M.Š. Calenko. The proof here is a restructuring of that proof in the spirit of the first sentence of this Abstract. We observe that these relations are spans of EM-spans and that EM-spans admit fake pullbacks so that spans of EM-spans compose. Our setting is more general than Puppe-exact categories.

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arXiv2019

Combinatorial categorical equivalences of Dold-Kan type

We prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let $\mathscr{X}$ denote an additive category with finite direct sums and split idempotents. The class includes (a) the Dold-Puppe-Kan theorem that simplicial objects in $\mathscr{X}$ are equivalent to chain complexes in $\mathscr{X}$; (b) the observation of Church, Ellenberg and Farb that $\mathscr{X}$-valued species are equivalent to $\mathscr{X}$-valued functors from the category of finite sets and injective partial functions; (c) a result T. Pirashvili calls of "Dold-Kan type"; and so on. When $\mathscr{X}$ is semi-abelian, we prove the adjunction that was an equivalence is now at least monadic, in the spirit of a theorem of D. Bourn.

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arXiv2018

Comonadic base change for enriched categories

For our concepts of change of base and comonadicity, we work in the general context of the tricategory $\mathrm{Caten}$ whose objects are bicategories $\mathscr{V}$ and whose morphisms are categories enriched on two sides. For example, for any monoidal comonad $G$ on a cocomplete closed monoidal category $\mathscr{C}$, the forgetful functor $U : \mathscr{C}^G\to \mathscr{C}$ is comonadic when regarded as a morphism in $\mathrm{Caten}$ between one-object bicategories. We show that the forgetful pseudofunctor $\mathscr{U}:\mathscr{V}^\mathscr{G}\rightarrow \mathscr{V}$ from the bicategory of Eilenberg-Moore coalgebras for a comonad $\mathscr{G}$ on $\mathscr{V}$ in $\mathrm{Caten}$ induces a change of base pseudofunctor $\widetilde{\mathscr{U}}:\mathscr{V}^\mathscr{G}\text{-}\mathrm{Mod}\rightarrow \mathscr{V}\text{-}\mathrm{Mod}$ which is comonadic in a bigger version of $\mathrm{Caten}$. We define Hopfness for such a comonad $\mathscr{G}$ and prove that having that property implies $\mathscr{U}$ creates left (Kan) extensions in the bicategory $\mathscr{V}^\mathscr{G}$. We provide conditions under which Hopfness carries over from $\mathscr{G}$ to the comonad $\widetilde{\mathscr{G}}=\widetilde{\mathscr{U}}\circ \widetilde{\mathscr{R}}$ generated by the adjunction $\widetilde{\mathscr{U}}\dashv \widetilde{\mathscr{R}}$. This has implications for characterizing the absolute colimit completion of $\mathscr{V}^\mathscr{G}$-categories.

math.CT↗
arXiv2018

Hopf rings for grading and differentials

In the category of abelian groups, Pareigis constructed a Hopf ring whose comodules are differential graded abelian groups. We show that this Hopf ring can be obtained by combining grading and differential Hopf rings using semidirect product in fairly general symmetric monoidal additive categories.

math.CT↗
arXiv2018

Generalizations of the Sweedler dual

As left adjoint to the dual algebra functor, Sweedler's finite dual construction is an important tool in the theory of Hopf algebras over a field. We show in this note that the left adjoint to the dual algebra functor, which exists over arbitrary rings, shares a number of properties with the finite dual. Nonetheless the requirement that it should map Hopf algebras to Hopf algebras needs the extra assumption that this left adjoint should map an algebra into its linear dual. We identify a condition guaranteeing that Sweedler's construction works when generalized to noetherian commutative rings. We establish the following two apparently previously unnoticed dual adjunctions: For every commutative ring $R$ the left adjoint of the dual algebra functor on the category of $R$-bialgebras has a right adjoint. This dual adjunction can be restricted to a dual adjunction on the category of Hopf $R$-algebras, provided that $R$ is noetherian and absolutely flat.

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arXiv2018

Real sets

After reviewing a universal characterization of the extended positive real numbers published by Denis Higgs in 1978, we define a category which provides an answer to the questions: \begin{itemize} \item what is a set with half an element? \item what is a set with $π$ elements? \end{itemize} The category of these extended positive real sets is equipped with a countable tensor product. We develop somewhat the theory of categories with countable tensors; we call the commutative such categories {\em series monoidal} and conclude by only briefly mentioning the non-commutative possibility called {\em $ω$-monoidal}. We include some remarks on sets having cardinalities in $[-\infty,\infty]$.

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arXiv2016

What Separable Frobenius Monoidal Functors Preserve

Separable Frobenius monoidal functors were defined and studied under that name by Szlachanyi and by Day and Pastro, and in a more general context by Cockett and Seely. Our purpose here is to develop their theory in a very precise sense. We determine what kinds of equations in monoidal categories they preserve. For example we show they preserve lax (meaning not necessarily invertible) Yang-Baxter operators, weak Yang-Baxter operators in the sense of Alonso Alvarez et al., and (in the braided case) weak bimonoids in the sense of Pastro and Street. In fact, we characterize which monoidal expressions are preserved (or rather, are stable under conjugation in a well-defined sense). We show that every weak Yang-Baxter operator is the image of a genuine Yang-Baxter operator under a separable Frobenius monoidal functor. Prebimonoidal functors are also defined and discussed.

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