[Paper Review] Limits and colimits of quantaloid-enriched categories and their distributors

Lili Shen, Walter Tholen|arXiv (Cornell University)|Apr 13, 2015

Homotopy and Cohomology in Algebraic Topology29 references5 citations

TL;DR

This paper establishes that the category of small Q-categories and Q-functors, as well as the category of Q-distributors and Q-Chu transforms, are both total and cototal for any small quantaloid Q. By proving the existence of a generating set in the category of Q-Chu spaces, the authors demonstrate that both categories admit all small limits and colimits, including those indexed by large diagrams, thereby extending the categorical completeness properties of quantaloid-enriched structures beyond classical smallness constraints.

ABSTRACT

It is shown that, for a small quantaloid Q, the category of small Q-categories and Q-functors is total and cototal, and so is the category of Q-distributors and Q-Chu transforms.

Motivation & Objective

• To investigate the existence of limits and colimits in categories enriched over small quantaloids.
• To establish that the category of Q-categories and Q-functors is total and cototal.
• To show that the category of Q-distributors (Q-Chu spaces) with Q-Chu transforms also admits all small limits and colimits.
• To construct a generating set in Q-Chu, enabling proof of total completeness and cocompleteness.

Proposed method

• Prove that Q-Cat is topological over Set/ob Q, enabling explicit construction of small limits and colimits.
• Use initial liftings of structured cones to describe limits and colimits in Q-Chu over small diagrams.
• Construct a generating set in Q-Chu using the objects ηs: ∅ →◦ Ds and λt: {t} →◦ bC for s,t ∈ ob Q.
• Leverage the existence of a generating set to prove total completeness and total cocompleteness of Q-Chu via Proposition 2.6.
• Verify that monomorphisms and epimorphisms in Q-Chu correspond to injective and surjective Q-functors in Q-Cat, ensuring wellpoweredness.
• Demonstrate that Q-Chu is hypercomplete and hypercocomplete by establishing both total completeness and cocompleteness.

Experimental results

Research questions

• RQ1 Does the category Q-Cat of small Q-categories and Q-functors admit all small limits and colimits for a small quantaloid Q?
• RQ2 Can the category Q-Chu of Q-distributors and Q-Chu transforms be shown to be total and cototal?
• RQ3 What is the structure of limits and colimits in Q-Chu, and how can they be explicitly described?
• RQ4 Does Q-Chu possess a generating set, and if so, how does this imply total completeness and cocompleteness?
• RQ5 How do the categorical properties of Q-Chu compare to those of Q-Cat in terms of completeness and cocompleteness?

Key findings

• The category Q-Cat is topological over Set/ob Q, which implies that it admits all small limits and colimits.
• The category Q-Chu is total and cototal, meaning it admits all small limits and colimits, including those indexed by large diagrams.
• A generating set in Q-Chu is constructed as {ηs: ∅ →◦ Ds | s ∈ ob Q} ∪ {λt: {t} →◦ bC | t ∈ ob Q}, where Ds is from a generating set of Q-Cat and bC is a coproduct of presheaf categories.
• The existence of a generating set in Q-Chu implies that Q-Chu is both totally complete and totally cocomplete.
• Consequently, Q-Chu is hypercomplete and hypercocomplete, meaning it admits all small and large limits and colimits.
• The results generalize classical completeness properties of categories such as Ord and Met to the broader setting of quantaloid-enriched categories and their distributors.