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Stub Frobenius reciprocity.
I finally created an entry internal category in homotopy type theory.
There is old discussion of this topic which I had once written at category object in an (infinity,1)-category in the sub-section HoTT formulation, but it’s probably good to give this a stand-alone entry, for ease of linking (such as from equivalence of categories now).
Created:
A one-parameter group (of unitary operators in a Hilbert space) is a homomorphism of groups
where is a Hilbert spaces and denotes its group of unitary operators.
More generally, one can define one-parameter semigroups of operators in a Banach space as homomomorphisms of groups
where denotes the semigroup of bounded operators .
Typically, we also require a continuity condition such as continuity in the strong topology.
Strongly continuous one-parameter unitary groups of operators in a Hilbert space are in bijection with self-adjoint unbounded operators on
The bijection sends
The operator is bounded if and only if is norm-continuous.
Strongly continuous one-parameter semigroups of bounded operators on a Banach space (alias -semigroups) satisfying are in bijection with closed operators with dense domain such that any belongs to the resolvent set of and for any we have
[…]
Created:
Hugo Dyonizy Steinhaus was a mathematician at the University of Lwów and the University of Wrocław.
He get his PhD in 1911 from the University of Göttingen, advised by David Hilbert.
His PhD students include Mark Kac.
James Ritchie Norris is a mathematician at the University of Cambridge.
He got his PhD in 1985 from the University of Oxford, advised by David Edwards.
On the Feynman–Kac formula on smooth manifolds:
Created:
Mark Kac was a mathematician at Cornell University and Rockefeller University.
He got his PhD from the University of Lwów in 1937, advised by Hugo Steinhaus.
On the Feynman–Kac formula:
Created:
The Feynman–Kac formula expresses the integral kernel of the one-parameter semigroup generated by a Laplacian on a smooth manifold as the path integral of the parallel transport map associated to the given connection with respect to all paths of a given length connecting the two given points.
The original reference is
The case of smooth manifolds is treated in
@Todd. Thanks for correcting my atrocious English!
Does anyone have any ideas as to how we could provide a bit more for this entry?
felt like adding a handful of basic properties to epimorphism
this is a bare list of references, to be !included into the lists of references of relevant entries (such as at compactly generated topological space, parameterized homotopy theory, exponential law for spaces)
there used to be, all along, a section titled “Derived adjunction”, which however fell short of saying anything about the derived adjunction as such.
Have added the statement now, with pointer to a new stand-alone entry derived adjunction.
The induced map most likely isn’t a homeomorphism when are locally compact Hausdorff.
The original statement was in monograph by Postnikov without proof.
Not only that, in the current form it couldn’t possibly be true, since the map could lack to be bijective.
For more details see here: https://math.stackexchange.com/questions/3934265/adjunction-of-pointed-maps-is-a-homeomorphism .
I’ve added a reference in the case when are compact Hausdorff though.
Adam
reformatted the bib-item, added link to the publisher page, and cross-link with Topos Theory
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.
While working at geometry of physics on the next chapter Differentiation I am naturally led back to think again about how to best expose/introduce infinitesimal cohesion. To the reader but also, eventually, to Coq.
First the trivial bit, concerning terminology: I am now tending to want to call it differential cohesion, and differential cohesive homotopy type theory. What do you think?
Secondly, I have come to think that the extra right adjoint in an infinitesimally cohesive neighbourhood need not be part of the axioms (although it happens to be there for ).
So I am now tending to say
Definition. A differential structure on a cohesive topos is an ∞-connected and locally ∞-connected geometric embedding into another cohesive topos.
And that’s it. This induces a homotopy cofiber sequence
Certainly that alone is enough axioms to say in the model of smooth cohesion all of the following:
So that seems to be plenty of justification for these axioms.
We should, I think, decide which name is best (“differential cohesion”?, “infinitesimal cohesion”?) and then get serious about the “differential cohesive homotopy type theory” or “infinitesimal cohesive homotopy type theory” or maybe just “differential homotopy type theory” respectively.
I left a counter-query underneath Zoran’s query at compactly generated space. It may be time for a clean-up of this article; the query boxes have been left dangling and unanswered for quite some time. Either proofs or references to detailed proofs would be welcome.
Should this topic be renamed to something like “ group” or some similar thing? I haven’t seen “abelian” used elsewhere to describe this notion.
IMO that choice of name is potentially misleading. For example, it could also refer to a model of the usual finite product theory of abelian groups: i.e. an object of the -category of by connective chain complexes of abelian groups modulo quasi-isomorphism. This is actually specifically what I would have expected from the term.
This example is, in some sense, also “more commutative” than being a grouplike monoid, which makes the description of being “maximally abelian” misleading as well.
Link to topic: abelian infinity-group
This content used to be sitting inside decidability, and “type checking” was redirecting to there. But clearly type checking deserves its own entry (though currently it remains a stub.)
created a stub for decidability, mainly only so that the mainy pointers to it do point somewhere
I added a bunch of things to connected space: stuff on the path components functor, an example of a countable connected Hausdorff space, and the observation that the quasi-components functor is left adjoint to the discrete space functor (Wikipedia reports that the connected components functor is left adjoint to the discrete space functor, but that’s wrong).
This bit about quasi-components functor had never occurred to me before, although it seems to be true. I’m having difficulty getting much information on this functor. For example, does it preserve finite products? I don’t know, but I doubt it. Does anyone reading this know?
I have briefly fixed the clause for topological spaces at contractible space, making manifest the distinction between contractible and weakly contractible.
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.
I had set out to add to the entry equivalence in homotopy type theory a detailed derivation of the categorical semantics of . But then I ended up getting distracted by various editorial work in other entries and for the moment I only have this puny remark added, expanding on the previous discussion there.
Maybe more later…
motivated by the blog discussion I added to rational homotopy theory a section Differential forms on topological spaces
added references to Lean
added missing cross-links (without commentary, for the moment, am running out of steam on this cross-link quest):
Well I made a start, basing the entry on Urs’s FOM comment.
added the original reference for the term q-bit: