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Stream: community: our work

Topic: Bryce Clarke

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:

• The Grothendieck construction for delta lenses at ACT 2024 (abstract here).
• The algebraic weak factorisation system of twisted coreflections and delta lenses at CT 2024 (abstract here).

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 $A^{\mathrm{op}} \times B \rightarrow \mathrm{\mathcal{P}oly}$.

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 $\mathbf{C}$ equipped with a OFS $(L, R)$, like the (initial functor, discrete opfibration) OFS on $\mathbf{Cat}$, together with an idempotent comonad $(S, \varepsilon)$, like the discrete category comonad on $\mathbf{Cat}$. One also needs to ask that $\mathbf{C}$ has enough pushouts (pushouts of morphisms in $L$ along the counit components of $S$), and that $S$ preserves these pushouts. Let $\mathbf{W}$ denote the wide subcategory of $\mathbf{C}$ containing the morphisms which are inverted by $S$ (this includes the counit components) --- in our example, this is the bijective-on-objects functors.

Then one can construct a double category $\mathbb{L}'$ whose vertical morphisms are pushout squares of counit components along morphisms in $L$ (i.e. the twisted coreflections) and a double category $\mathbb{R}'$ whose vertical morphisms are composable pairs of a morphism in $\mathbf{W}$ followed by a morphism in $\mathbf{C}$, whose composite lies in R (i.e. the delta lenses). We can show that the cospan $\mathbb{L}' \rightarrow \mathbb{S}\mathrm{q}(\mathbf{C}) \leftarrow \mathbb{R}'$ admits a canonically induced lifting operation, and that there is a canonically induced factorisation of each morphism in $\mathbf{C}$ into a $\mathbb{L}'$ morphism followed by a $\mathbb{R}'$ morphism. The only axiom on an AWFS which does not seems to follow immediately is showing that $\mathbb{L}' = LLP(\mathbb{R}')$ and $\mathbb{R}' = RLP(\mathbb{L}')$ --- 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 $L(C, x)$ on slide 4 is not correct).