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Joanna Ko

Publications and source records attributed to Joanna Ko.

7 recordsLinked to original sources
arXiv2026

Enhanced $2$-categories of models of sketches as enhanced $2$-categories of algebras over monads

We establish the equivalence between models of enhanced $2$-sketches and algebras over monads, including the (co)lax morphisms. More precisely, for any enhanced limit $2$-sketch $\mathbb{T}$ with tight cones, the enhanced $2$-category $\mathbb{M}\mathrm{od}_{s, w}(\mathbb{T}, \mathbb{K})$ of models of $\mathbb{T}$ in a locally presentable enhanced $2$-category $\mathbb{K}$, in which the tight and the loose morphisms are the $\mathscr{F}$-natural transformations and the loose $w$-natural transformations, respectively, is equivalent to the enhanced $2$-category ${\mathrm{T}\text{-}\mathbb{A}\mathrm{lg}}_{s, w}$ of algebras over an enhanced $2$-monad $T$ on the models $\mathbb{M}\mathrm{od}(\mathcal{T}_τ, \mathbb{K})$ restricted to the tights with strict $T$-morphisms and $w$-$T$-morphisms. As a consequence, we explore the limits in the enhanced $2$-category $\mathbb{M}\mathrm{od}_{s, w}(\mathbb{T}, \mathbb{K})$ of models with loose $w$-natural transformations, and conclude that $\mathbb{M}\mathrm{od}_{s, w}(\mathbb{T}, \mathbb{K})$ inherits all $w$-rigged limits. Along the way, we establish an enriched analogue of the Orthogonal Sub-category Theorem, and generalise results on the reflectivity and the monadicity of models of enriched limit sketches in the base of enrichment to any arbitrary locally presentable enriched category.

math.CT↗
arXiv2026

Dotted $2$-limits

Marked limits, or Cartesian quasi-limits introduced by Gray, give an alternative approach to $\mathbf{Cat}$-weighted limits in $2$-category theory. This was first established by Street, and we aim to give a new approach to this result using marked codescent objects of marked coherence data which we introduce in this article. We then propose the notion of dotted $2$-limits, which is a natural generalisation of marked limits to the enhanced $2$-categorical setting. We establish that dotted $2$-limits and $\mathscr{F}$-weighted limits both have the same expressive power.

math.CT↗
arXiv2025

Limits of $(\infty, 1)$-categories with structure and their lax morphisms

Riehl and Verity have established that for a quasi-category $A$ that admits limits, and a homotopy coherent monad on $A$ which does not preserve limits, the Eilenberg-Moore object still admits limits; this can be interpreted as a completeness result involving lax morphisms. We generalise their result to different models for $(\infty, 1)$-categories, with an abundant variety of structures. For instance, $(\infty, 1)$-categories with limits, Cartesian fibrations between $(\infty, 1)$-categories, and adjunctions between $(\infty, 1)$-categories. In addition, we show that these $(\infty, 1)$-categories with structure in fact possess an important class of limits of lax morphisms, including $\infty$-categorical versions of inserters and equifiers, when only one morphism in the diagram is required to be structure-preserving. Our approach provides a minimal requirement and a transparent explanation for several kinds of limits of $(\infty, 1)$-categories and their lax morphisms to exist.

math.CT↗
arXiv2024

Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation

Many structures of interest in two-dimensional category theory have aspects that are inherently strict. This strictness is not a limitation, but rather plays a fundamental role in the theory of such structures. For instance, a monoidal fibration is - crucially - a strict monoidal functor, rather than a pseudo or lax monoidal functor. Other examples include monoidal double categories, double fibrations, and intercategories. We provide an explanation for this phenomenon from the perspective of enhanced 2-categories, which are 2-categories having a distinguished subclass of 1-cells representing the strict morphisms. As part of our development, we introduce enhanced 2-categorical limit sketches and explain how this setting addresses shortcomings in the theory of 2-categorical limit sketches. In particular, we establish the symmetry of internalisation for such structures, entailing, for instance, that a monoidal double category is equivalently a pseudomonoid in an enhanced 2-category of double categories, or a pseudocategory in an enhanced 2-category of monoidal categories.

math.CT↗
arXiv2022

Monadic forgetful functors and (non-)presentability for $C^*$- and $W^*$-algebras

We prove that the forgetful functors from the categories of $C^*$- and $W^*$-algebras to Banach $*$-algebras, Banach algebras or Banach spaces are all monadic, answering a question of J.Rosický, and that the categories of unital (commutative) $C^*$-algebras are not locally-isometry $\aleph_0$-generated either as plain or as metric-enriched categories, answering a question of I. Di Liberti and Rosický. We also prove a number of negative presentability results for the category of von Neumann algebras: not only is that category not locally presentable, but in fact its only presentable objects are the two algebras of dimension $\le 1$. For the same reason, for a locally compact abelian group $\mathbb{G}$ the category of $\mathbb{G}$-graded von Neumann algebras is not locally presentable.

math.OA↗
arXiv2022

Existence of groupoid models for diagrams of groupoid correspondences

This article continues the study of diagrams in the bicategory of étale groupoid correspondences. We prove that any such diagram has a groupoid model and that the groupoid model is a locally compact étale groupoid if the diagram is locally compact and proper. A key tool for this is the relative Stone-Čech compactification for spaces over a locally compact Hausdorff space.

math.CT↗
arXiv2021

The bicategory of groupoid correspondences

We define a bicategory with étale, locally compact groupoids as objects and suitable correspondences, that is, spaces with two commuting actions as arrows; the 2-arrows are injective, equivariant continuous maps. We prove that the usual recipe for composition makes this a bicategory, carefully treating also non-Hausdorff groupoids and correspondences. We extend the groupoid C*-algebra construction to a homomorphism from this bicategory to that of C*-algebra correspondences. We describe the C*-algebras of self-similar groups, higher-rank graphs, and discrete Conduché fibrations in our setup.

math.OA↗