varkor

Category theorist and type theorist. I make Rust compiler team alumnus.

[varkor @varkora]

quiver (https://t.co/90Jksq1LWh) is a new commutative diagram editor for the web. I've written a short blog post describing the motivation behind it, as well as some of its most useful features.

https://t.co/RbrxC4FEpW

[arXiv math.CT Category Theory @mathCTbot]

Nathanael Arkor, Dylan McDermott: Presheaves and cocompletions in formal category theory https://t.co/2DGACmfY0z https://t.co/9tgE86NCor https://t.co/vdvT1hZ5ma

[varkor @varkora]

@RadishHarmers @evanewashington @q_uiver_app I have some good news :)

[quiver @q_uiver_app]

quiver version 1.6.0 is now available: https://t.co/LHCthx6KX4
Includes:
- Typst support
- Option to produce standalone LaTeX documents
Fixes:
- Several issues on Safari

Thanks to Théophile Cailliau for implementing Typst support and to b-reinke for the standalone option!

[quiver @q_uiver_app]

quiver version 1.5.5 is now available: https://t.co/LHCthx6KX4
Includes:
- Several new arrow body styles
- Improved LaTeX output for shortened arrows
- Improved LaTeX output for barred arrows

You will need to update quiver.sty to make use of the improved shortened arrows.

[varkor @varkora]

@khoiiiind @radokirov Relative monads can be seen as monoids in categories of (non-endo) functors. However, it's much more subtle than for monads, as one needs to consider monoids in skew-monoidal categories. See §3 of "Monads need not be endofunctors" (https://t.co/Y3iWADXPFg), for instance.

[varkor @varkora]

@radokirov @khoiiiind Although T is not defined to be a functor, it becomes a functor automatically, and crucially this makes use of the functor structure on J. E.g. see §2.1 of "Monads need not be endofunctors" (https://t.co/Y3iWADXPFg).

[quiver @q_uiver_app]

quiver version 1.5.3 is now available: https://t.co/LHCthx6KX4
Includes:
- UX improvements (including improved arrow offset heuristic)
- Simplified LaTeX output
- Several bug fixes

[varkor @varkora]

@tangled_zans @mattecapu To express general (co)ends you need a symmetric closed monoidal category, though you may be able to get away with less for specific (co)ends.

[varkor @varkora]

@tangled_zans @mattecapu It is possible to define pointwise extensions using (co)ends with a nice enough base of enrichment, but not in general. If you want to work with enrichment over an arbitrary monoidal category, it's not possible to define extensions this way. So it depends on your assumptions.

[varkor @varkora]

@mattecapu The definitions are essentially present in §8 of Koudenburg's "Formal category theory in augmented virtual double categories" (https://t.co/QU1p3OWsGH), though Koudenburg does not define ends/coends.

[varkor @varkora]

@mattecapu However, most of the theorems in category theory for which ends and coends are typically used may alternatively be proven formally using other techniques, which require far less structure on an equipment, e.g. the calculus of lifts and extensions.

[varkor @varkora]

@mattecapu Ends and coends may be easily defined as hom-weighted limits and colimits in an equipment, but do require more structure on the equipment than is necessary for most formal category theory: namely closed monoidal structure and a symmetry/duality.

[quiver @q_uiver_app]

quiver version 1.5.0 is now available: https://t.co/LHCthx6d7w
Includes:
- Loops

[varkor @varkora]

@YogoYogoY Garner–Shulman's "Enriched categories as a free cocompletion" is another good reference for categories enriched in the double categories of categories, profunctors, functors, and natural transformations. For Span-enrichment, the work of Betti and Walters is relevant.

[varkor @varkora]

@deinemerle In my experience, it is less common than "if and only if", but still relatively common. It doesn't sound unnatural to me as a native English speaker.

[varkor @varkora]

@deinemerle I personally would avoid saying something like "this is known" or "this is folklore" unless you have heard others mention or allude to the result. Just because you think a result is simple doesn't mean it is certainly known.

[varkor @varkora]

@deinemerle No-one will mind if you state a lemma that may not be original if you're not claiming it as a main result. If someone points out the result is known, then you can add a reference. If you want to be careful about references, then you could ask for one on MathOverflow.

[varkor @varkora]

@tangled_zans @jjcarett2 @PTOOP @Iceland_jack @kmett @myers_jaz @StringDiagram This may be more convenient for capturing universal properties, as it is more symmetric than the kind/type/term interpretation. However, both approaches should generalise to arbitrary dimensions (either by giving each kind a type, or by introducing rewrites between rewrites).

[varkor @varkora]

@tangled_zans @jjcarett2 @PTOOP @Iceland_jack @kmett @myers_jaz @StringDiagram There's another 2-dimensional approach to type theory, in which we view objects as types, 1-cells as terms, and 2-cells as rewrites between terms (cf. Seely's "Modelling computations: a 2-categorical framework").

[varkor @varkora]

@tangled_zans @jjcarett2 @PTOOP @Iceland_jack @kmett @myers_jaz @StringDiagram This is one of the motivations for formal category theory: by working in a more abstract setting, we can prove more general theorems, with proofs that are often simpler, because we don't have to worry about (enriched) functoriality conditions, or naturality conditions, etc. (2/2)

[varkor @varkora]

@tangled_zans @jjcarett2 @PTOOP @Iceland_jack @kmett @myers_jaz @StringDiagram This reply is a little late, but I presume you're referring to what are usually called "monads in bicategories" and "enriched monads" respectively? If so, I agree that the former concept is simpler. Furthermore, it subsumes the latter. (1/2)