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    • Added to commutativity of limits and colimits the case of coproducts commuting with connected limits in a topos, and the generalization to higher topoi. This particular instance of commutativity is not mentioned very often, probably because it’s not very impressive in Set, but its generalization to higher topoi (for which I couldn’t find a reference) is more interesting. For instance, cofiltered limits commute with taking quotients by an ∞-group in an ∞-topos.

    • I have added the actual general definition of the Cech groupoid as presheaf of groupoids, and headlined the definition previously offered here as “Idea”. Then I added detailed statement and proof, that the Cech-groupoid co-represents sets of matching families for set-valued presheaves (now this prop.)

      diff, v7, current

    • This entry is currently undecided as to whether “full subcategory” inclusion requires the functor to be an injection on objects. It begins by pointing to subcategory which does require this, but before long it speaks about fully faitful functors being full subcategory inclusions.

      This will be confusing to newcomers. There should be at least some comments about invariance under equivalence of categories.

      Ah, now I see that at subcategory there is such a discussion (here). Hm, there is some room for cleaning-up here.

      diff, v10, current

    • Created page, with a brief definition of the rules, and a remark that the naive formula for cofree comonoids always satisfies the laws of the soft exponential (is this well-known?)

      v1, current

    • worked on space and quantity a bit

      • tried to polish the introduction and the Examples-section a bit

      • added a section on the adjunction with a detailed end/coend computation of the fact that it is an adjunction.

    • “small site” used to redirect to “little site”. Despite the warning there, this doesn’t seem helpful, and so I created a little disambiguation entry

      v1, current

    • I have expanded a little at sifted category: added the example of the reflexive-coequalizer diagram, added the counter-examples of the non-reflexive coequalizer diagram, added a references.


      Suggests that Stone, Gelfand, … duality are special cases of the adjunction between CoPresheaves and Presheaves. A similar question is raised here.

      However, this paper

      seems to use another definition. Could someone please clarify?

    • I have created an entry spectral symmetric algebra with some basics, and with pointers to Strickland-Turner’s Hopf ring spectra and Charles Rezk’s power operations.

      In particular I have added amplification that even the case that comes out fairly trivial in ordinary algebra, namely Sym RRSym_R R is interesting here in stable homotopy theory, and similarly Sym R(Σ nR)Sym_R (\Sigma^n R).

      I am wondering about the following:

      In view of the discussion at spectral super scheme, then for RR an even periodic ring spectrum, the superpoint over RR has to be

      R 0|1=Spec(Sym RΣR)Spec(R(nBΣ(n) n) +). R^{0 \vert 1} \;=\; Spec(Sym_R \Sigma R) \simeq Spec\left( R \wedge \left( \underset{n \in \mathbb{N}}{\coprod} B\Sigma(n)^{\mathbb{R}^n} \right)_+ \right) \,.

      This of course is just the base change/extension of scalars under Spec of the “absolute superpoint”

      𝕊 0|1Spec(Sym 𝕊(Σ𝕊)) \mathbb{S}^{0\vert 1} \simeq Spec(Sym_{\mathbb{S}} (\Sigma \mathbb{S}))

      (which might deserve this notation even though the sphere spectrum is of course not even periodic).

      This looks like a plausible answer to the quest that David C. and myself were on in another thread, to find a plausible candidate in spectral geometry of the ordinary superpoint 0|1\mathbb{R}^{0 \vert 1}, regarded as the base of the brane bouquet.

    • Started this page normal form, but I see there might be a difference between the no-further-rewrites idea and the designated set of normal terms idea (as in disjunctive normal form).

      v1, current

    • I am changing the page title – this used to be “A first idea of quantum field theory”, which of course still redirects. The “A first idea…” seemed a good title for when this was an ongoing lecture that was being posted to PhysicsForums. I enjoyed the double meaning one could read into it, but it’s a bad idea to carve such jokes into stone. And now that the material takes its place among the other chapters of geometry of physics, with the web of cross-links becoming thicker, the canonical page name clearly is “perturbative quantum field theory”.

      diff, v187, current

    • Made a remark, to fill in a gap in the constructive proof that group monomorphisms are regular.

      diff, v11, current

    • I have fleshed out (and corrected) and then spelled out the proof of the statement (here) that Kan extension of an adjoint pair is an adjoint quadruple:

      For 𝒱\mathcal{V} a symmetric closed monoidal category with all limits and colimits, let 𝒞\mathcal{C}, 𝒟\mathcal{D} be two small 𝒱\mathcal{V}-enriched categoriesand let

      𝒞pq𝒟 \mathcal{C} \underoverset {\underset{p}{\longrightarrow}} {\overset{q}{\longleftarrow}} {\bot} \mathcal{D}

      be a 𝒱\mathcal{V}-enriched adjunction. Then there are 𝒱\mathcal{V}-enriched natural isomorphisms

      (q op) *Lan p op:[𝒞 op,𝒱][𝒟 op,𝒱] (q^{op})^\ast \;\simeq\; Lan_{p^{op}} \;\colon\; [\mathcal{C}^{op},\mathcal{V}] \longrightarrow [\mathcal{D}^{op},\mathcal{V}] (p op) *Ran q op:[𝒟 op,𝒱][𝒞 op,𝒱] (p^{op})^\ast \;\simeq\; Ran_{q^{op}} \;\colon\; [\mathcal{D}^{op},\mathcal{V}] \longrightarrow [\mathcal{C}^{op},\mathcal{V}]

      between the precomposition on enriched presheaves with one functor and the left/right Kan extension of the other.

      By essential uniqueness of adjoint functors, this means that the two Kan extension adjoint triples of qq and pp

      Lan q op (q op) * Ran q op Lan p op (p op) * Ran p op \array{ Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& Ran_{q^{op}} \\ && Lan_{p^{op}} &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}} }

      merge into an adjoint quadruple

      Lan q op (q op) * (p op) * Ran p op:[𝒞 op,𝒱][𝒟 op,𝒱] \array{ Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}} } \;\colon\; [\mathcal{C}^{op},\mathcal{V}] \leftrightarrow [\mathcal{D}^{op}, \mathcal{V}]

      diff, v8, current