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Bryce Clarke (Nov 09 2023 at 06:11):Today at 12:00 UTC I am giving a talk in Tallinn titled Lifting twisted coreflections against delta lenses. The talk will be streamed on Zoom, so if you would like a link, send me a message.
Abstract: Delta lenses are functors equipped with a functorial choice of lifts, and generalise the notion of a split opfibration. It is known that every functor factors (via a comma category) into a split coreflection followed by a split opfibration, and that split coreflections lift against split opfibrations --- this is the main example of a lifting awfs recently introduced by Bourke [1]. In this talk, I will introduce the notion of twisted coreflection as a split coreflection equipped with certain additional structure. We will then see that every functor factors into a twisted coreflection followed by a delta lens, and that twisted coreflections lift against delta lenses. The main theorem is that the double categories of twisted coreflections and delta lenses form a lifting awfs in the sense of Bourke. This result revolves the question of characterising the L-coalgebras for the algebraic weak factorisation system for delta lenses [2], and establishes a deeper understanding on the differences between delta lenses and split opfibrations. Future work will be discussed if time allows.
References:
[1] John Bourke, An orthogonal approach to algebraic weak factorisation systems, Journal of Pure and Applied Algebra, Vol 227 (2023). [doi:10.1016/j.jpaa.2022.107294, arXiv:2204.09584]
[2] Bryce Clarke, The Algebraic Weak Factorisation System for Delta Lenses, preprint (2023). [arXiv:2305.02732]
Bryce Clarke (Nov 09 2023 at 06:55):Fixed the time: 12:00 UTC not 00:00 UTC.
Tom Hirschowitz (Nov 09 2023 at 08:20):I wasn't aware of John's presentation of awfs, it looks really cute!
Bryce Clarke (Nov 09 2023 at 10:27):The slides are now available on my website here.
Bryce Clarke (Dec 05 2023 at 07:51):Yesterday I gave a talk at UCLouvain in Belgium titled The AWFS of twisted coreflections and delta lenses. The talk is about the same topic as previously, but with slightly different emphasis. In particular, I resolved a conjecture at the end of my previous talk concerning a "diagrammatic" characterisation of twisted coreflections, so I can now use this to give a nice construction of the lifting. The abstract and slides for the talk are available below.
Abstract: In a lextensive category, each morphism factorises into a coproduct injection followed by a split epimorphism --- this is one of the main examples of an algebraic weak factorisation system (AWFS) introduced by Grandis and Tholen. Another well-known example of an AWFS arises from factorising each functor into a split coreflection followed by a split opfibration. Delta lenses, first introduced in computer science, generalise the notion of a split opfibration and are a focus of ongoing research in (applied) category theory. In this talk, I will introduce the notion of a twisted coreflection as a split coreflection with a certain property, and construct an AWFS that factorises each functor into a twisted coreflection followed by a delta lens. Examples of twisted coreflections will be explored, and the close connections with the previously stated examples of AWFS will be discussed. The talk will emphasise a double categorical approach to AWFS recently introduced by Bourke, however no prior knowledge of double categories will be assumed.
Slides are available on my website here.
Bryce Clarke (Jan 31 2024 at 07:06):Today I have a new paper on the arXiv! Check it out here: https://arxiv.org/abs/2401.17250
Title: Lifting twisted coreflections against delta lenses
Abstract: Delta lenses are functors equipped with a suitable choice of lifts, generalising the notion of split opfibration. In recent work, delta lenses were characterised as the right class of an algebraic weak factorisation system. In this paper, we show that this algebraic weak factorisation system is cofibrantly generated by a small double category, and characterise the left class as split coreflections with a certain property; we call these twisted coreflections. We demonstrate that every twisted coreflection arises as a pushout of an initial functor from a discrete category along a bijective-on-objects functor. Throughout the article, we take advantage of a reformulation of algebraic weak factorisation systems, due to Bourke, based on double-categorical lifting operations.
Bryce Clarke (May 06 2024 at 08:00):Today I am happy to be starting a new postdoc position at Tallinn University of Technology, in the Logic and Semantics Group led by Tarmo Uustalu. I am sharing an office with @Nathanael Arkor .
Bryce Clarke (May 06 2024 at 08:05):I will also be delivering two talks at conferences in June:
Morgan Rogers (he/him) (May 06 2024 at 09:21):Congratulations!!
Bryce Clarke (May 31 2024 at 09:55):Yesterday I gave a talk in Tallinn titled A new perspective on comodules of polynomial comonads. The talk was on the whiteboard, but I made some handwritten notes on which the talk was based available on my website here.
Abstract: Categories, functors, profunctors, and natural transformations form the building blocks of category theory. Recently in the seminar, Priyaa spoke about viewing categories as polynomial comonads, however the notion of comonad morphism and comodule we obtain are quite different to functors and profunctors. Morphisms of polynomial comonads are spans of an identity-on-objects functor and a discrete opfibration, while comodules of polynomial comonads are spans of a profunctor and a discrete opfibration. In this talk, I will explain these notions, provide examples, and demonstrate a new characterisation of comodules of polynomial comonads as functors .
fosco (Jun 04 2024 at 10:28):how can you write so well omg
Bryce Clarke (Jun 21 2024 at 10:58):Today I gave my talk at the ACT 2024 conference on the Grothendieck construction for delta lenses. The slides are now available here.
Bryce Clarke (Jun 29 2024 at 07:24):Yesterday I gave my talk at the CT 2024 conference on The algebraic weak factorisation system of twisted coreflections and delta lenses. The slides are available here: https://bryceclarke.github.io/talk-slides/2024-06-CT2024.pdf
Tim Hosgood (Jun 30 2024 at 20:41):only looked through these briefly, so maybe the answer is "it's in the talk/paper/elsewhere in the slides", but the diagrams on page 9 make me wonder if the factorisation system (twisted coreflection, delta lens) can somehow be lifted from/induced by the factorisation system (initial functor, discrete opfibration) ?
Tim Hosgood (Jun 30 2024 at 20:42):maybe this is actually how you prove it (looking at p.10) but I'm wondering if there's some more general trick going on here
Bryce Clarke (Jul 03 2024 at 08:51):Hi @Tim Hosgood, this is a great question! (I also received this question during the talk).
The setting should be that one starts with a category equipped with a OFS , like the (initial functor, discrete opfibration) OFS on , together with an idempotent comonad , like the discrete category comonad on . One also needs to ask that has enough pushouts (pushouts of morphisms in along the counit components of ), and that preserves these pushouts. Let denote the wide subcategory of containing the morphisms which are inverted by (this includes the counit components) --- in our example, this is the bijective-on-objects functors.
Then one can construct a double category whose vertical morphisms are pushout squares of counit components along morphisms in (i.e. the twisted coreflections) and a double category whose vertical morphisms are composable pairs of a morphism in followed by a morphism in , whose composite lies in R (i.e. the delta lenses). We can show that the cospan admits a canonically induced lifting operation, and that there is a canonically induced factorisation of each morphism in into a morphism followed by a morphism. The only axiom on an AWFS which does not seems to follow immediately is showing that and --- this is the hard part.
Tim Hosgood (Jul 03 2024 at 11:25):great, thanks!
Bryce Clarke (Aug 09 2024 at 11:59):My paper Lifting twisted coreflections against delta lenses is now published in TAC: http://www.tac.mta.ca/tac/volumes/41/26/41-26abs.html
Bryce Clarke (Oct 03 2024 at 14:32):Yesterday, I spoke (online) at the Edinburgh Category Theory Seminar. The title of the talk was Reflecting on lenses and split opfibrations and the slides are now available here: https://bryceclarke.github.io/talk-slides/2024-10-Edinburgh-seminar.pdf
Bryce Clarke (Oct 03 2024 at 14:34):(Note that the description on the on slide 4 is not correct).
Bryce Clarke (Mar 03 2025 at 11:01):Today I have a new paper on the arXiv! Check it out here: https://arxiv.org/abs/2502.21288
Title: The Grothendieck construction for delta lenses
Abstract: Delta lenses are functors equipped with a functorial choice of lifts, generalising the notion of split opfibration. In this paper, we introduce a Grothendieck construction (or category of elements) for delta lenses, thus demonstrating a correspondence between delta lenses and certain lax double functors into the double category of sets, functions, and split multivalued functions. We show that the double category of split multivalued functions admits a universal property as a certain kind of limit, and inherits many nice properties from the double category of spans. Applications of this construction to the theory of delta lenses are explored in detail.
Bryce Clarke (Jul 21 2025 at 06:24):Just got back from the International Category Theory conference (CT2025) and had a fabulous time! Close to 200 participants and lots of very interesting work was presented (as you can see in the programme).
I gave a talk on A new framework for limits in double categories -- the abstract is here and the slides are here.
Bryce Clarke (Jul 21 2025 at 06:32):I was very happy to see lots of great research on double categories was presented throughout the conference, including:
Bryce Clarke (Jul 21 2025 at 06:32):(I have linked the abstracts from the conference website, but the slides should also be available later this week I expect).
Bryce Clarke (Jul 21 2025 at 06:41):To cap things off, this week Dorette Pronk is visiting in Tallinn where we will discuss some double categories project(s).
Bryce Clarke (Jul 21 2025 at 06:44):I also organised a photo (taken by @Tim Hosgood ) at the conference dinner of Macquarie category theorists, past and present.
CT2025-Macquarie.JPG
Bryce Clarke (Jul 21 2025 at 06:47):Left-to-right: Paula Verdugo, Giacomo Tendas, Jason Brown, Philip Hackney, JS Lemay, Michael Johnson, Yuki Maehara, me (Bryce Clarke), Simona Paoli, Marcello Lanfranchi, John Bourke, and Soichiro Fujii.
Bryce Clarke (Sep 30 2025 at 08:06):Yesterday I gave a talk on "Limits in double categories, revisited". The slides are available on my website here: https://bryceclarke.github.io/talk-slides/2025-09-VUB.pdf
Evan Patterson (Sep 30 2025 at 15:48):Excited about this one!
James Deikun (Sep 30 2025 at 18:20):Seems like this should generalize nicely to a notion of a limit in a (replete) loose distributor.
Nathanael Arkor (Sep 30 2025 at 18:27):(That's actually the context in which we've been working in our draft.)
Bryce Clarke (Oct 14 2025 at 19:21):I just gave a talk at the Atlantic Category Theory Seminar titled "A second look at limits in double categories". You may find the slides available on my website here: https://bryceclarke.github.io/talk-slides/2025-10-ATCAT.pdf
Adrian Clough (Oct 23 2025 at 08:14):Is the theory of colimits in double categories formally dual to the theory of limits as is the case for categories?
Adrian Clough (Oct 23 2025 at 08:17):Also your slides look really pretty! Do you just write them up by hand on your tablet or is your workflow more involved than this?
Bryce Clarke (Oct 23 2025 at 08:24):Adrian Clough said:
Is the theory of colimits in double categories formally dual to the theory of limits as is the case for categories?
Essentially, yes. One can define a cocone in a dual way to a cone (simply reverse the transformation), and define a colimit as an initial cocone.
Bryce Clarke (Oct 23 2025 at 08:28):In our general theory, we decided to take work with limits whose diagrams are normal lax double functors, but one can take the limit of any sort of double functor (lax/colax/pseudo/strict/etc.) using the same set up. The reason we chose to work with limits of normal lax functors is that stating the result "right adjoints preserve limits" is very easy (as for this to be true, the right adjoint must be a normal lax functor).
Bryce Clarke (Oct 23 2025 at 08:31):In general, we can show that there is a correspondence between lax functors A -> B and normal lax functors A' -> B (using a [[weak morphism classifer]]), so the theory of limits of normal lax functors is no less general.
Bryce Clarke (Oct 23 2025 at 08:32):If we want the general theory to work nicely for colimits, one would work with colimits of normal colax functors, but everything else is essentially unchanged.
Bryce Clarke (Oct 23 2025 at 08:34):Adrian Clough said:
Also your slides look really pretty! Do you just write them up by hand on your tablet or is your workflow more involved than this?
Thank you! I just write them by hand in the Goodnotes app on my iPad. I usually start with a grid template when I am writing, the change to a plain background template when I am done.
Bryce Clarke (Oct 24 2025 at 10:31):Tomorrow I am giving an online talk at Octoberfest 2025 titled Companions and conjoints are (co)limits at .
Abstract: Companions and conjoints are one of the most useful concepts in double category theory as they provide a way of viewing tight morphisms as loose morphisms. For example, in the double category of sets, functions, and relations, each function from A to B can be seen as a relation from A to B (its companion) and as a relation from B to A (its conjoint). Many double categories admit all companions and conjoints of tight morphisms, and their prevalence has led to many names for these double categories such as "framed bicategories" or "fibrant double categories" or "equipments".
In this talk, I will show how companions and conjoints are instances of limits (and colimits!) in a double category. Limits in double categories were introduced by Grandis and Paré in 1999, however companions and conjoints are not examples in their framework. I will introduce a new notion of limit indexed by a span of categories, and show that companions and conjoints, together with parallel limits, generate all limits of this kind.
Bryce Clarke (Oct 25 2025 at 14:37):The slides for the talk are available here: https://bryceclarke.github.io/talk-slides/2025-10-Octoberfest.pdf
Mike Shulman (Oct 27 2025 at 05:29):This is neat! One question: probably you stated this verbally (sorry I wasn't able to attend the talk) but I can't tell from the statement of the theorem on slide 09 whether this is supposed to be two theorems, one about limits and one about colimits, or one theorem with each instance of "(co)limits" meaning "limits and colimits" both?
Nathanael Arkor (Oct 27 2025 at 05:34):There are two theorems, one about limits and one about colimits.
Mike Shulman (Oct 27 2025 at 05:43):Excellent, thanks.
Bryce Clarke (Nov 11 2025 at 11:47):Today I have a new paper on the arXiv with @Noam Zeilberger and Gabriel Scherer! Check it out here: https://arxiv.org/abs/2511.07314
Comments and feedback are very welcome :)
Title: The free bifibration on a functor
Abstract: We consider the problem of constructing the free bifibration generated by a functor of categories . This problem was previously considered by Lamarche, and is closely related to the problem, considered by Dawson, Paré, and Pronk, of "freely adjoining adjoints" to a category. We develop a proof-theoretic approach to the problem, beginning with a construction of the free bifibration in which objects of are formulas of a primitive "bifibrational logic", and arrows are derivations in a cut-free sequent calculus modulo a notion of permutation equivalence. We show that instantiating the construction to the identity functor generates a zigzag double category , which is also the free double category with companions and conjoints (or fibrant double category) on . The approach adapts smoothly to the more general task of building (P,N)-fibrations, where one only asks for pushforwards along arrows in P and pullbacks along arrows in N for some subsets of arrows, which encompasses Kock and Joyal's notion of ambifibration when (P,N) form a factorization system. We establish a series of progressively stronger normal forms, guided by ideas of focusing from proof theory, and obtain a canonicity result under assumption that the base category is factorization preordered relative to P and N. This canonicity result allows us to decide the word problem and to enumerate relative homsets without duplicates. Finally, we describe several examples of a combinatorial nature, including a category of plane trees generated as a free bifibration over , and a category of increasing forests generated as a free ambifibration over , which contains the lattices of noncrossing partitions as quotients of its fibers by the Beck-Chevalley condition.
Bryce Clarke (Nov 15 2025 at 13:14):Applications for the Adjoint School 2026 are now open until 14 December. @Nathanael Arkor and I will be mentoring one of the four projects.
Title: A Double Category Theoretic Perspective on Partiality
Summary: Double category theory provides a convenient and expressive setting in which to study functional and relational structure simultaneously. One kind of relation of particular interest is the notion of partial function, which plays important roles in many areas of mathematics and computer science. It might therefore be expected that double categories are good structures with which to study partiality. An existing category theoretic framework for studying partiality is given by the notion of restriction category, which has seen widespread application in computability theory, automata theory, probability theory, algebraic and differential geometry, logic, algebra, and others. There are multiple aspects of restriction category theory that have a double category theoretic flavour, with some explicit connections having been drawn by Paré, and by Cockett and Garner. The goal of this project is to explore the connections between these areas, demonstrating how phenomena observed in restriction category theory are consequences of general results in double category theory.
Readings:
Bryce Clarke (Nov 15 2025 at 13:17):If this project or any of the other projects looks interesting to you, I highly encourage you to apply! The website linked above has more information and the application form includes the following:
We welcome anyone, from anywhere in the world, who is interested in applying category-theoretic methods to problems outside of pure mathematics. This is emphatically not restricted to math students, but one should be comfortable working with mathematics. In selecting participants we aim to design groups with a mix of people specializing in category theory and in applied fields.
We will consider upper-level undergraduates, Masters students, PhD students, post-docs, as well as people working outside of academia. Members of groups which are underrepresented in the mathematics and computer science communities are especially encouraged to apply.