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    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • Added some references on the categorical point of view on AECs. I think ideally this page would be rewritten to take that perspective from the start, in line with the nPOV, but I certainly don’t have the time (or background) to do that myself.

      diff, v9, current

    • I’ve added to Eilenberg-Moore category an explicit definition of EM objects in a 2-category and some other universal properties of EM categories, including Linton’s construction of the EM category as a subcategory of the presheaves on the Kleisli category.

      Question: can anyone tell me what Street–Walters mean when they say that this construction (and their generalised one, in a 2-category with a Yoneda structure) exhibits the EM category as the ‘category of sheaves for a certain generalised topology on’ the Kleisli category?

    • Am starting a write-up (here) of how (programming languages for) quantum circuits “with classical control and/by measurement” have a rather natural and elegant formulation within the linear homotopy type theory of Riley 2022.

      Aspects of this have a resemblance to some constructions considered in/with “Quipper”, but maybe it helps clarify some issues there, such as that of “dynamic lifting”.

      The entry is currently written without TOC and without Idea-section etc, but rather as a single top-level section that could be !include-ed into relevant entries (such as at quantum circuit and at dependent linear type theory). But for the moment I haven’t included it anywhere yet, and maybe I’ll eventually change my mind about it.

      v1, current

    • Page created, but author did not leave any comments.

      Anonymous

      v1, current

    • Page created, but author did not leave any comments.

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • I’ll try to start add some actual content to the entries classical mechanics, quantum mechanics, etc. For the time being I added a simple but good definition to classical mechanics. Of course this must eventually go with more discussion to show any value. I hope to be able to use some nice lecture notes from Igir Khavkine for this eventually.

      For the time being, notice there was this old discussion box, which I am herby mving to the forum here:

      –

      +–{.query} Edit: I changed the above text, incorporating a part of the discussion (Zoran).

      Zoran: I disagree. Classical mechanics is classical mechanics of anything: point particles, rigid bodies (the latter I already included), infinite systems (mechanics of strings, membranes, springs, elastic media, classical fields). It includes statics, not only dynamics. The standard textbooks like Goldstein take it exactly in that generality.

      One could even count the simplified beginning part of the specialized branches like aerodynamics and hydrodynamics (ideal liquids for example), which are usually studied in separate courses and which in full formulation are not just mechanical systems, as the thermodynamics also affects the dynamics. There are also mechanical models of dissipative systems, where the dissipative part is taken only phenomenologically, e.g. as friction terms. Hydrodynamics can also be considered as a part of rheology.

      Toby: I take your point that ’dynamics’ was not the right word. But do you draw any distinction between ’classical mechanics’ and ’classical physics’? Conversely, what word would you use to restrict attention to particles instead of fields, if not ’mechanics’? (Incidentally, I would take point particles as possibly spinning, although I agree that I should not assume that the particle are points anyway.)

      Zoran: you see, in classical mechanics you express all you have by attaching mass, position, velocity etc. to the parfts of mechanical systems. Not all classical physics belongs to this kind of description. The thermodynamical quantities may influence the motion of the systemm, but their description is out of the frame of classical mechanics. If you study liquids you have to take into account both the classical mechanics of the liquid continuum but also variations of its temperature, entropy and so on, which are not expressable within the variables of mechanics. Formally speaking of course, the thermodynamics has very similar formal structure as mechanics, for example Gibbs and Helmholtz free energies and enthalpy are like Lagrangean, the quantities which are extremized when certain theremodynamical quantities are kept constant. To answer the terminological question, there is a classical mechanics of point particles and it is called classical mechanics of point particles, there is also cm of fields and cm of rigid bodies.

      Toby: So ’mechanics’ for you means ‹not taking into account thermal physics›? That's not the way that I learned it! But I admit that I do not have a slick phrase for that (any more than you have a slick phrase for ‹mechanics of point particles›), so I will try to ascertain how the term is usually used and defer to that. =–

    • added pointer to p. 245 of Sets for Mathematics for the idea of

      UX:comodalX ⟶ X ⟶ modalX opposite1 unity opposite2 U X \;\colon\; \array{ comodal X &\longrightarrow& X &\longrightarrow& modal X \\ opposite\;1 && unity && opposite\;2 }

      diff, v60, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • I corrected an apparent typo:

      A 2-monad TT as above is lax-idempotent if and only if for any TT-algebra a:TA→Aa \colon T A \to A there is a 2-cell θ a:1⇒η∘a\theta_a \colon 1 \Rightarrow \eta \circ a

      to

      A 2-monad TT as above is lax-idempotent if and only if for any TT-algebra a:TA→Aa \colon T A \to A there is a 2-cell θ a:1⇒η A∘a\theta_a \colon 1 \Rightarrow \eta_A \circ a

      It might be nice to say η A\eta_A is the unit of the algebra….

      diff, v22, current

    • I’ve wondered for a while whether there is a notion of lax-idempotent 2-adjunction, but for some reason until now I’d never thought to try the obvious route of simply generalizing the conditions defining an idempotent adjunction. Haven’t had time to cross-link it yet.

    • brief category:people-entry for hyoerlinking references

      v1, current

    • I rescued combinatory logic from being a “my first slide” spam and gave it some content, mainly to record the fact (which I just learned) that under propositions as types, combinatory logic corresponds to a Hilbert system.

      I feel like there should be something semantic to say here too, like λ\lambda-calculus corresponding to a “closed, unital, cartesian multicategory” (a cartesian multicategory that is “closed and unital” as in the second example here) and combinatory logic corresponding to a closed category that is also “cartesian” in some sense. Has anyone defined such a sense?

      Relatedly, is there a notion of “linear combinatory logic” that would correspond to ordinary (symmetric) closed categories? My best guess is that instead of SS and KK you would have combinators with the following types:

      (B→C)→(A→B)→(A→C) (B\to C) \to (A\to B) \to (A\to C) (A→(B→C))→B→A→C (A \to (B\to C)) \to B \to A\to C

      coming from the two ways to eliminate a dependency in SS to make it linear (KK is irreducibly nonlinear). These are of course the ways that you express composition and symmetry in a closed category.

    • Doi link for Scott 1976, and more details for Hyland 2017 (arxiv, doi, journal ref)

      diff, v18, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • a bare list of references, to be !include-ed into the References-list of relevant entries

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • starting something – so far an Idea-section, references, and some quotes on and around the question “Why did this take us 60+ years?”

      next I’ll produce a tikz-diagram showing and explaining the “protocol”.

      v1, current

    • Added some references to Bob Walter’s work, with links.

      diff, v6, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • Nestruev only established the equivalence for vector bundles over a connected manifold.

      Anonymous

      diff, v5, current

    • This is intended to continue the issues discussed in the Lafforgue thread!

      I have added an idea section to Morita equivalence where I sketch what I perceive to be the overarching pattern stressing in particular the two completion processes involved. I worked with ’hyphens’ there but judging from a look in Street’s quantum group book the pattern can be spelled out exactly at a bicategorical level.

      I might occasionally add further material on the Morita theory for algebraic theories where especially the book by Adamek-Rosicky-Vitale (pdf-draft) contains a general 2-categorical theorem for algebraic theories.

      Another thing that always intrigued me is the connection with shape theory where there is a result from Betti that the endomorphism module involved in ring Morita theory occurs as the shape category of a ring morphism in the sense of Bourn-Cordier. Another thing worth mentioning on the page is that the Cauchy completion of a ring in the enriched sense is actually its cat of modules (this is in Borceux-Dejean) - this brings out the parallel between Morita for cats and rings.

    • brief category:people-entry in hyperlinkung references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • expanded concrete sheaf: added the precise definition and some important properties.

    • Page created, but author did not leave any comments.

      Tom Mainiero

      v1, current

    • the entry braid group said what a braid is, but forgot to say what the braid group is; I added in a sentence, right at the beginning (and fixed some other minor things).

    • I have added to homotopy group a very brief pointer to Mike’s HoTT formalization of π 1(S 1)\pi_1(S^1).

      Eventually I would like to have by default our nnLab entries be equipped with detailed pointers to which aspects have been formalized in HoTT (if they have), and in which .v-file precisely.

    • following discussion here I am starting an entry with a bare list of references (sub-sectioned), to be !include-ed into the References sections of relevant entries (mainly at homotopy theory and at algebraic topology) for ease of updating and syncing these lists.

      The organization of the subsections and their items here needs work, this is just a start. Let’s work on it.

      I’ll just check now that I have all items copied, and then I will !include this entry here into homotopy theory and algebraic topology. It may best be viewed withing these entries, because there – but not here – will there be a table of contents showing the subsections here.

      v1, current

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