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    • I have added to monoidal model category statement and proof (here) of the basic statement:


      Let (𝒞,)(\mathcal{C}, \otimes) be a monoidal model category. Then 1) the left derived functor of the tensor product exsists and makes the homotopy category into a monoidal category (Ho(𝒞), L,γ(I))(Ho(\mathcal{C}), \otimes^L, \gamma(I)). If in in addition (𝒞,)(\mathcal{C}, \otimes) satisfies the monoid axiom, then 2) the localization functor γ:𝒞Ho(𝒞)\gamma\colon \mathcal{C}\to Ho(\mathcal{C}) carries the structure of a lax monoidal functor

      γ:(𝒞,,I)(Ho(𝒞), L,γ(I)). \gamma \;\colon\; (\mathcal{C}, \otimes, I) \longrightarrow (Ho(\mathcal{C}), \otimes^L , \gamma(I)) \,.

      The first part is immediate and is what all authors mention. But this is useful in practice typically only with the second part.

    • This entry didn’t have any really pertinent references yet.

      On the equivalence between flat connections and reps of the fundamental group I have now added pointer to:

      diff, v18, current

    • Updating reference to cubical type theory. This page need more work.

      diff, v55, current

    • Added direct descriptions of the various universal fibrations.

      diff, v12, current

    • starting stub article on synthetic (,1)(\infty,1)-category

      Anonymous

      v1, current

    • starting stub article on enriched types in homotopy type theory, which are a synthetic version of enriched infinity-groupoids in higher category theory.

      Anonymous

      v1, current

    • I have considerably expanded the entry sigma-model and will probably continue to do so in small steps in the nearer future (with interruptions). This goes in parallel with a discussion we are having on the nnCafé here.

    • starting stub on simplicial type theory

      Anonymous

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • Created:

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

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

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

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

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

      where B(X)B(X) denotes the semigroup of bounded operators XXX\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) t0(U_t)_{t\ge0} of operators in a Hilbert space HH are in bijection with self-adjoint unbounded operators AA on HH

      The bijection sends

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

      The operator AA is bounded if and only if UU is norm-continuous.

      Hille–Yosida theorem

      Strongly continuous one-parameter semigroups TT of bounded operators on a Banach space XX (alias C 0C_0-semigroups) satisfying T(t)Mexp(ωt)\|T(t)\|\le M\exp(\omega t) are in bijection with closed operators A:XXA\colon X\to X with dense domain such that any λ>ω\lambda\gt \omega belongs to the resolvent set of AA and for any λ>ω\lambda\gt\omega we have

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

      References

      […]

      v1, current

    • 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.

      v1, current

    • 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.

      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.

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

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

      v1, current

    • @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

      v1, current

    • The induced map most likely isn’t a homeomorphism when X,YX, 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,YX, Y are compact Hausdorff though.

      Adam

      diff, v13, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • 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

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

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

      Anonymous

      v1, current

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

      Anonymous

      v1, current

    • 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 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 E_\infty 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.)

      v1, current

    • 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 SetTopSet \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?

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

      Antonin Delpeuch

      v1, current

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

      diff, v32, current

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

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

      Anonymous

      v1, current

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

      Anonymous

      v1, current

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

      Anonymous

      v1, current

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

      Anonymous

      v1, current

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

      Anonymous

      v1, current

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

      Anonymous

      v1, current

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

      Anonymous

      v1, current

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

      Anonymous

      v1, current