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A subcategory C of an accessible category D is accessible if C is an accessible category and the inclusion functor C→D is an accessible functor.
Some authors, e.g., Lurie in Higher Topos Theory and Adámek–Rosický, require accessible subcategories to be full subcategory.
Some authors, e.g., Adámek–Rosický in Locally Presentable and Accessible Categories merely require C to be accessible, referring to the stronger notion as an accessibly embedded accessible subcategory.
Accessible subcategories are idempotent complete and are closed under set-indexed intersections.
See, for example, Definition 5.4.7.8 in
Fixed a hyperlink to Jardine’s lectures. Removed a query box:
+– {: .query} Can any of you size-issue experts help to clarify this?
Mike: I wish. I added some stuff, but I still don’t really understand this business. In particular I don’t really know what is meant by “inessential.” It certainly seems unlikely that you would get equivalent homotopy theories, but it does seem likely that you would get similar behavior no matter where you draw the line. And if all you care about is, say, having a good category of sheaves in which you can embed any particular space or manifold you happen to care about, then that may be good enough. But I don’t really know what the goal is of considering such large sites. =–
in reply to discussion on the blog I
added more details to Lie algebroid
added a reference by Courant to Lie algebroid, Poisson Lie algebroid and tangent Lie algebroid
created Legendre transformation as a placeholder that currently just serves to keep some references on Legendre transformation from the point of view of Lie algebroid theory.
Added:
A bijective correspondence between Lie algebroid structures, homological vector fields of degree 1, and odd linear Poisson structures is established in the paper
I started a page logicality and invariance. In Bristol the other day, Steve Awodey was promoting the thought that HoTT is a realisation of that thrust to understand logic as maximally invariant.
What would it be to take that seriously? If invariants are picked up by dependent product in some BAut context, could there be a useful context BAut(𝒰) for the universe 𝒰?
felt like the nLab should have an entry fraction
I created an entry on Larry Lambe. I included a link to some (on line) notes of his on Symbolic Computation which includes discussion of the perturbation lemma from homological perturbation theory.
Unfortunately, there are two entries on the same topic, both created by Urs: quantum Hall effect (redirecting also fractional quantum Hall effect what should eventually split off) with some substance, and the microstub quantum hall effect. I would like to create quantum spin Hall effect and I think I should rename/reclaim the stub quantum hall effect for this. Do others agree ? Urs ?
As the action is now delayed I record here the reference which I wanted to put there
Somewhat surprisingly, the authors and roughly this work of them are mentioned (though not in the list of references) in a paper in algebraic geometry
which considers the mirror symmetry and topological states of matters (topological insulators in particular) as main applications.
am starting curvature characteristic form and Chern-Simons form.
But still working…
started some bare minimum at Spin Chern-Simons theory
pointer, concerning the so-called “negative branes” of e.g. DHJV ’18
I was involved in some discussion about where the word “intensional” as in “intensional equality” comes from and how it really differs from “intenTional” and what the point is of having such a trap of terms.
Somebody dug out Martin-Löf’s lecture notes “Intuitionistic type theory” from 1980 to check. Having it in front of me and so before I forget, I have now briefly made a note on some aspects at equality in the section Different kinds of equalits (below the first paragraph which was there before I arrived.)
Anyway, on p. 31 Martin-Löf has
intensional (sameness of meaning)
I have to say that the difference between “sameness of meaning” and “sameness of intenTion”, if that really is the difference one wants to make, is at best subtle.
added pointer to section 7.5 of
in analogy to what I just did at classical mechanics, I have now added some basic but central content to quantum mechanics:
Quantum mechanical systems
States and observables
Spaces of states
Flows and time evolution
Still incomplete and rough. But I have to quit now.
for discussion at geometry of physics I needed to be able to point to principle of extremal action, so I created a little entry.
There used to be a stub entry Lagrangian quantum field theory. I have now given it a bit more of an Idea-section.
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.
a stub, for the moment just so as to make links work at differential category
added pointer to:
there is some confusion on this MO thread about sheafification, with the nLab entry sheafification somehow involved. I had a look at the entry and find that it can do with lots of polishing, but that the statement discussed over there is clearly right. (the misleading answer on MO that seems to claim a problem on the nLab page gets twice as many votes as the good answer by Clark Barwick, which confirms the statement) I have tried to edit it a bit to make things clearer, but don’t have the leisure for that now.
Given the recent success with the polishing of the entry on geometric realization, maybe I should announce that sheafification is going to be submitted for nJournal peer-review soon, so that everybody here will jump on it to brush it up ;-)
edited the entry orthogonality a bit, for instance indicated that there are other meanings of orthogonality. This should really be a disambiguation page.
And what makes the category-theoretic notion of orthogonality not be merged with weak factorization system? And why is orthogonal factorization system the first example at orthogonality if in fact that imposes unique lifts, while in orthogonality only existence of lifts is required?
I think the entry-situation here deserves to be further harmonized.
Added a section with a collection of references on intersection laws for black D-branes ending on black NS5-branes. Then I spelled out the case of D6-branes in some detail, collecting the relevant diagrams from the reference EGKRS00, and used this to identify the corresponding M-theory lift by the M5-brane near horizon ADE-orbifolds of the 4-sphere.
added pointer to:
a complete stub entry, for the moment just to make the link work (which has long been requested universality class)
I gave the book
a category:reference entry and linked to it from a few relevant entries.
(permutations of) this title used to redirect to closed monoidal category, but its useful to have a direct pointer for the symmetric monoidal closed case, and so I am splitting this off as a little entry
Mike Stay kindly added the standard QM story to path integral.
I changed the section titles a bit and added the reference to the Baer-Pfaeffle article on the QM path integral. Probably the best reference there is on this matter.
I created a short page for chord diagram, and also added a bit of relevant information to Vassiliev invariant.
I have added a reference to Cheng-Gurski-Riehl to two-variable adjunction, and some comments about the cyclic action.
Just a definition (hope I got it right) and a couple properties. I wasn’t sure how to set up the redirects; currently “modest set” redirects here while “PER” redirects to partial equivalence relation, but other suggestions are welcome.
Added a mention of the category of PERs.
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.