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    • CommentRowNumber1.
    • CommentAuthorDexter Chua
    • CommentTimeAug 6th 2016
    • (edited Aug 6th 2016)

    Restructured the manifold entry to avoid duplication with pseudogroup, and moved the section on the tangent bundle to tangent bundle

    • CommentRowNumber2.
    • CommentAuthorJoshua Meyers
    • CommentTimeAug 31st 2019
    • (edited Aug 31st 2019)

    The definition via pseudogroups is supposed to have the following problem:

    There is in general no notion of morphisms between manifolds. At best, we can only talk about isomorphisms of manifolds.

    However, couldn’t this problem be solved by using a “pseudomonoid” instead of a pseudogroup? We can define a pseudomonoid MM on the topological space XX as a wide subcategory of Op(X)\text{Op}(X) such that for all U,V,U αU, V, U_\alpha open in XX, f:UVf: U\rightarrow V such that (U α)(U_\alpha) covers UU, we have the condition fM(α)f| U α:UU αVf(U α)Mf\in M \iff (\forall \alpha) f|_{U_\alpha}:U\cap U_\alpha\rightarrow V\cap f(U_\alpha) \in M.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 31st 2019

    I’m not sure right away. I’d have to think about it.

    • CommentRowNumber4.
    • CommentAuthorJoshua Meyers
    • CommentTimeSep 2nd 2019
    • (edited Sep 2nd 2019)

    Here’s a new way of characterizing manifolds that I like better.

    Let XX be a topological space and let NN be the monoid of partial continuous functions XXX\rightarrow X (any element of NN has open domain by continuity), ordered by inclusion of graphs. Then define a pseudomonoid on XX to be a submonoid SS of NN such that

    • If f,f αNf, f_\alpha\in N and f= αf αf=\cup_\alpha f_\alpha, then fSf\in S iff (α)(f αS)(\forall \alpha) (f_\alpha\in S). (Equivalently, SS is a lower set of NN and the inclusion SNS\rightarrow N lifts joins.)

    (This condition implies that SS is closed under open restriction, and, being a submonoid of NN, SS contains the identity map id X\text{id}_X, so SS contains all coreflexive elements of NN.)

    Let GG be the inverse semigroup GG of all elements of SS that are bijective and whose opposite (as a relation) is also in SS.

    Now let MM be a set. A chart on MM is a partial bijection MXM\rightarrow X. Two charts a,ba,b are SS-compatible iff b opaGb^{\text{op}}a\in G (or equivalently, b opa,a opbSb^{\text{op}}a, a^{\text{op}}b \in S). (A chart is compatible with itself iff its image is open.)

    An SS-atlas on MM is a collection of pairwise SS-compatible maps whose domains cover MM.

    An SS-manifold is a set MM equipped with an SS-atlas. A map ff between SS-manifolds MM, NN with SS-atlases AA, BB is a morphism iff there exists a collection of pairs (ϕ α,ψ α)A×B(\phi_\alpha, \psi_\alpha)\in A\times B such that the collection {domϕ α×domψ α}\{\text{dom} \phi_\alpha \times \text\{dom} \psi_\alpha\} covers the graph of ff and for all α\alpha, ψ αfϕ α opS\psi_\alpha f \phi_\alpha^{\text{op}}\in S.

    By using partial functions we don’t need to worry about having all different domains involved and keeping track of restrictions and such.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 2nd 2019

    Joshua, is this in the literature someplace?

    • CommentRowNumber6.
    • CommentAuthorJoshua Meyers
    • CommentTimeSep 2nd 2019
    • (edited Sep 3rd 2019)

    Not that I know of, but I haven’t looked very hard.

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