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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• starting stub on simplicial type theory

Anonymous

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A one-parameter group (of unitary operators in a Hilbert space) is a homomorphism of groups

$\mathbf{R} \to U(H),$

where $H$ is a Hilbert spaces and $U(H)$ denotes its group of unitary operators.

More generally, one can define one-parameter semigroups of operators in a Banach space $X$ as homomomorphisms of groups

$\mathbf{R} \to B(X),$

where $B(X)$ denotes the semigroup of bounded operators $X\to X$.

Typically, we also require a continuity condition such as continuity in the strong topology.

## Stone theorem

Strongly continuous one-parameter unitary groups $(U_t)_{t\ge0}$ of operators in a Hilbert space $H$ are in bijection with self-adjoint unbounded operators $A$ on $H$

The bijection sends

$A\mapsto (t\mapsto \exp(itA)).$

The operator $A$ is bounded if and only if $U$ is norm-continuous.

## Hille–Yosida theorem

Strongly continuous one-parameter semigroups $T$ of bounded operators on a Banach space $X$ (alias $C_0$-semigroups) satisfying $\|T(t)\|\le M\exp(\omega t)$ are in bijection with closed operators $A\colon X\to X$ with dense domain such that any $\lambda\gt \omega$ belongs to the resolvent set of $A$ and for any $\lambda\gt\omega$ we have

$\|(\lambda I-A)^{-n}\|\le M (\lambda-\omega)^{-n}.$

## References

[…]

• 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.

## Selected writings

On the Feynman–Kac formula on smooth manifolds:

• James R. Norris, A complete differential formalism for stochastic calculus in manifolds, Séminaire de Probabilités XXVI, Lecture Notes in Mathematics (1992), 189–209. doi.
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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.

## Selected writings

On the Feynman–Kac formula:

• Mark Kac, On distributions of certain Wiener functionals, Transactions of the American Mathematical Society 65:1 (1949), 1–13. doi.
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## Idea

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.

## References

The original reference is

• Mark Kac, On distributions of certain Wiener functionals, Transactions of the American Mathematical Society 65:1 (1949), 1–13. doi.

The case of smooth manifolds is treated in

• James R. Norris, A complete differential formalism for stochastic calculus in manifolds, Séminaire de Probabilités XXVI, Lecture Notes in Mathematics (1992), 189–209. doi.
• brief category:people-entry for hyperlinking references

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

• link to PDF for reference

• @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

• brief category:people-entry for hyperlinking references

• 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 $X, Y$ 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 $X, Y$ are compact Hausdorff though.

• brief category:people-entry for hyperlinking references

• 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.

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• This article came from the HoTT wiki, I am not sure how accurate the contents on here are.

Anonymous

• Corrected the first name to have a é

• starting discussion page

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Anonymous

• 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 “$E_\infty$ 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 $\infty$-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 $E_\infty$ monoid, which makes the description of being “maximally abelian” misleading as well.

• 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 $Set \to Top$ (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?

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Antonin Delpeuch

• Added the contents of the canonical isomorphism induced by some non-canonical isomorphism as coming from Lack’s proof.

• I have briefly fixed the clause for topological spaces at contractible space, making manifest the distinction between contractible and weakly contractible.

Anonymous

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Anonymous

• 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 $Equiv(X,Y)$. 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…

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Anonymous

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Anonymous

• added missing cross-links (without commentary, for the moment, am running out of steam on this cross-link quest):

• now that Mike announced a proof, and hearing Steve’s comment, I felt it would be nice to have a name for conjecture (partially) proven thereby, for ease of communiucating it to the rest of the world. Just a start, please edit the entry as need be.

• copied from HoTT wiki

Anonymous

• Well I made a start, basing the entry on Urs’s FOM comment.

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Anonymous

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• stub