Todd Trimble
cartologies

Introduction

This is to be an extension of the pseudogroup approach to defining general notions of manifold. The pseudogroup approach makes clear how to define the groupoid of manifolds, but as far as I know does not cover the general notion of morphism between manifolds. To rectify this, a new notion (cartology) is introduced, providing an abstract environment in which pseudogroups make sense and also one where morphisms may be naturally defined. Manifolds are seen as akin to torsors in this approach.

Bicategories of partial maps

We recall some ideas (due to Carboni) on bicategories of partial maps, close in spirit to the Carboni-Walters notions of bicategories of relations and cartesian bicategories. (Note: compositions are given in traditional order, where the composite of r:XYr: X \to Y and s:YZs: Y \to Z is denoted sr:XZs r: X \to Z.)

Definition

(Carboni) A structure of bicategory of partial maps is a symmetric monoidal locally posetal bicategory in which

  1. Each object XX comes equipped with a (unique) cocommutative comonoid structure (δ X:XXX,ε X:X1)(\delta_X: X \to X \otimes X, \varepsilon_X: X \to 1),

  2. Every morphism r:abr \colon a \to b is compatible with the comultiplication in the strict sense and with the counit in the lax sense:

    δ br=(rr)δ a;ε brε A\delta_b \circ r = (r \otimes r) \circ \delta_a; \qquad \varepsilon_b \circ r \leq \varepsilon_A

Given a 1-cell r:XYr: X \to Y in such a structure, the domain d(r)d(r) is the composite

Xδ XXXr1 XYXε Y1 X1XXX \stackrel{\delta_X}{\to} X \otimes X \stackrel{r \otimes 1_X}{\to} Y \otimes X \stackrel{\varepsilon_Y \otimes 1_X}{\to} 1 \otimes X \cong X

where the last isomorphism in the left unit constraint. We say rr is total if d(r)=1 Xd(r) = 1_X. The following lemma is easily proven using the method of string diagrams:

Lemma

(Carboni) In a structure of bicategory of partial maps B\mathbf{B}, we have that

  1. For any map r:XYr: X \to Y, the arrow d(r):XXd(r): X \to X is coreflexive: d(r)1 Xd(r) \leq 1_X.

  2. d(sr)=d(d(s)r)d(s r) = d(d(s)r). In particular, d(s)=d(d(s))d(s) = d(d(s)).

  3. rd(r)=rrd(r) = r.

  4. d(s)d(r)=d(r)d(s)=d(d(r)d(s))d(s)d(r) = d(r)d(s) = d(d(r)d(s)).

  5. d(r)d(r)=d(r)d(r)d(r) = d(r).

  6. r:XYr: X \to Y is total if and only if ε Yr=ε X\varepsilon_Y r = \varepsilon_X.

  7. Total 1-cells (as composed in B\mathbf{B}) are the morphisms of a category, denoted Tot(B)Tot(\mathbf{B}). Furthermore, if srs r is total, so is rr.

Finally, the monoidal product \otimes restricts to a cartesian monoidal product on Tot(B)Tot(B). The comultiplications δ X:XXX\delta_X: X \to X \otimes X are the diagonal maps in Tot(B)Tot(\mathbf{B}), and 11 is terminal via ε X:X1\varepsilon_X: X \to 1.

Definition

A domainal bicategory is a bicategory admitting a structure of bicategory of partial maps, such that

  • Axiom (D): all coreflexive maps rr satisfy the domain equation r=d(r)r = d(r) and split as idempotents.

(This is a form of functional completeness.)

We can show

Proposition

Assuming that all domains d(r)d(r) split, every morphism rr is of the form r=fi *r = f i^\ast, where ii *i \dashv i^\ast and frif \coloneqq r i is total.

Proof

Split d(r)d(r) as ii *i i^\ast, and put f=rif = r i. We have

r=rd(r)=rii *=fi *r = r d(r) = r i i^\ast = f i^\ast

The following equations show ff is total:

(ε Y1 U)(ri1 U)δ U = (ε Yi *i)(r1 U)(i1 U)δ U = (ε Yi *)(1 Yi)(r1 U)(i1 U)δ U = i *(ε Y1 X)(r1 X)(ii)δ U = i *(ε Y1 X)(r1 X)δ Xi = i *d(r)i = i *ii *i = 1 U\array{ (\varepsilon_Y \otimes 1_U)(r i \otimes 1_U)\delta_U & = & (\varepsilon_Y \otimes i^\ast i)(r \otimes 1_U)(i \otimes 1_U)\delta_U \\ & = & (\varepsilon_Y \otimes i^\ast)(1_Y \otimes i)(r \otimes 1_U)(i \otimes 1_U)\delta_U \\ & = & i^\ast (\varepsilon_Y \otimes 1_X)(r \otimes 1_X)(i \otimes i)\delta_U \\ & = & i^\ast(\varepsilon_Y \otimes 1_X)(r \otimes 1_X)\delta_X i \\ & = & i^\ast d(r) i \\ & = & i^\ast i i^\ast i \\ & = & 1_U }

(Call this fi *f i^\ast the canonical factorization of a partial map rr.)

  1. All pseudo-invertible maps rr can be written in the form ji *j i^\ast.

  2. For any pseudo-invertible r=ji *r = j i^\ast, we have d(r)=ij *d(r) = i j^\ast j i^\ast$.

  3. If i:VYi: V \to Y and j:UYj: U \to Y are left adjoints, and we write i *j:UVi^\ast j: U \to V as kh *k h^\ast where h:WUh: W \to U is a left adjoint and kk is total, then we can show ik=jhi k = j h directly (I hope). Then we may calculate that h *j *ih^\ast j^\ast i is right adjoint to kk. It follows (from a lemma below) that i^\ast j = k h^\ast$ is pseudo-invertible, and now we conclude that pseudo-invertible maps are closed under composition.

  4. In a domainal bicategory of partial maps, the subbicategory whose 1-cells are total is locally discrete.

Proposition

Referring to 3. above, we may conclude that UhWkVU \stackrel{h}{\leftarrow} W \stackrel{k}{\to} V is the pullback of UjYiVU \stackrel{j}{\to} Y \stackrel{i}{\leftarrow} V.

Proof

The arrow j *ij^\ast i is pseudo-invertible, and thus can be written as hk *h k^\ast where hh *h \dashv h^\ast and kk *k \dashv k^\ast. Taking pseudo-inverses of both sides of j *i=hk *j^\ast i = h k^\ast, we have i *j=kh *i^\ast j = k h^\ast. Now suppose r:AU,s:AVr: A \to U, s: A \to V are any arrows such that we have a commutative square jr=isj r = i s. From j *j=1 Uj^\ast j = 1_U and i *i=1 Vi^\ast i = 1_V, we derive

r=j *is=hk *s,kh *r=s.r = j^\ast i s = h k^\ast s, \qquad k h^\ast r = s.

From these equations, it is easy to deduce that k *sh *rk^\ast s \leq h^\ast r and h *rk *sh^\ast r \leq k^\ast s, so k *g=h *f:AWk^\ast g = h^\ast f: A \to W. Call this p:AWp: A \to W. We then have hp=hk *g=fh p = h k^\ast g = f and kp=kh *r=sk p = k h^\ast r = s. Thus (h:WU,k:WV)(h: W \to U, k: W \to V) is the pullback.

Observation: given a domainal bicategory of partial maps B\mathbf{B}, the bicategory obtained by splitting all coreflexives is another domainal bicategory of partial maps.

Can we go through Carboni’s development and show that there is at most one such structure in the domainal case?

Definition

A regional bicategory is a domainal bicategory such that the join αf α\bigvee_\alpha f_\alpha exists for a family {f α:XY} αF\{f_\alpha: X \to Y\}_{\alpha \in F} whenever the sheaf condition

f α[d(f α)d(f β)]=f β[d(f α)d(f β)]f_\alpha \circ [d(f_\alpha) \cap d(f_\beta)] = f_\beta \circ [d(f_\alpha) \cap d(f_\beta)]

holds for all α,βF\alpha, \beta \in F.

In addition to the axioms above, two more axioms are useful to consider:

We say that a bicategory of partial maps is functionally complete if it satisfies these two axioms. In this case, it may be shown that the category of objects and maps (i.e., left adjoints) is a left exact category CC, and the bicategory is in fact the bicategory of partial maps in CC.

Cartologies

Definition

The bicategory of regions is the locally posetal bicategory where

  • Objects are topological spaces (or locales if you prefer);

  • Morphisms are partial functions with open domain, that is spans

    XiUfYX \overset{i}{\leftarrow} U \overset{f}{\to} Y

    where ff is continuous and ii is an open embedding. These are clearly closed under span composition.

  • Such spans of type hom(X,Y)\hom(X, Y) are locally ordered by inclusion.

This is a functionally complete bicategory of partial maps, denoted Reg\mathbf{Reg}.

These local posets of Reg\mathbf{Reg} are not cocomplete, but they admit certain obvious joins: given a family of regional maps

(U α,f α):XY(U_\alpha, f_\alpha): X \to Y

the join α(U α,f α)\bigvee_\alpha (U_\alpha, f_\alpha) exists iff we have local compatibility:

f α| U αU β=f β| U αU βf_{\alpha}|_{U_\alpha \cap U_\beta} = f_{\beta}|_{U_\alpha \cap U_\beta}

for all α,β\alpha, \beta. Composition on either side with a 11-cell preserves any local joins which exist.

Every coreflexive morphism r1 Xr \leq 1_X in Reg\mathbf{Reg} splits: there is an object dom(r)\dom(r) and morphisms i:dom(r)Xi \colon \dom(r) \to X, j:Xdom(r)j \colon X \to \dom(r) such that the equations

ji=1 dom(r)ij=rj \circ i = 1_{\dom(r)} \qquad i \circ j = r

hold. The object dom(r)\dom(r) may be called the domain of rr.

Remark

In any category, splittings of idempotents (of coreflexives in particular, for poset-enriched categories) are unique up to isomorphism. For splitting pairs (i,j)(i, j) of coreflexives, we have iji \vdash j.

Definition

A cartology is a (locally full) subbicategory i:CRegi: C \hookrightarrow \mathbf{Reg} such that

  • (Closure under open subspaces) If XOb(C)X \in Ob(C) and r1 Xr \leq 1_X in Reg\mathbf{Reg}, then rr belongs to CC and splits in CC.

  • (Closure under joins) If f j:XYf_j \colon X \to Y are maps of CC and the join of i(f j)i(f_j) exists in Reg(X,Y)\mathbf{Reg}(X, Y), then this join belongs to CC (i.e., is of the form i(f)i(f) for some ff in C(X,Y)C(X, Y)).

The notion of cartology makes sense for any bicategory of partial maps in place of RegReg, and we will develop the theory with that generalization in mind.

Examples

Intended examples include the case where the objects of CC are Euclidean spaces n\mathbb{R}^n, and morphisms are spans

(U,f): m n(U, f): \mathbb{R}^m \to \mathbb{R}^n

where ff is smooth (or PL, etc.).

Pseudogroups

Definition

Let CC be a cartology. A morphism r:XYr \colon X \to Y in CC is pseudo-invertible if there exists s:YXs \colon Y \to X such that

  • sr:XXs r: X \to X and rs:YYr s: Y \to Y are coreflexive,

  • r=rsrr = r s r and s=srss = s r s.

The morphism ss is called a pseudo-inverse of rr.

Proposition

A pseudo-inverse is unique when it exists.

Proof

Suppose ss and tt are pseudo-inverse to rr. We have

rt=rsrtrs1 Y=rsr t = r s r t \leq r s \cdot 1_Y = r s

and similarly rsrtr s \leq r t, so rs=rtr s = r t. By a similar argument, sr=trs r = t r. In that case

s=srs=trs=trt=ts = s r s = t r s = t r t = t

as claimed.

Lemma

Suppose r:XYr \colon X \to Y and s:YXs \colon Y \to X are such that rsr s and srs r are coreflexive, and (i:UX,i *:XU)(i \colon U \to X, i^\ast \colon X \to U) splits srs r and (j:VY,j *:YV)(j \colon V \to Y, j^\ast \colon Y \to V) splits rsr s. Then j *ri:UVj^\ast r i \colon U \to V is inverse to i *sj:VUi^\ast s j \colon V \to U.

Proof

We calculate

j *rii *sj=j *rsrsj=j *rsj=j *jj *j=1 V1 V=1 Vj^\ast r i i^\ast s j = j^\ast r s r s j = j^\ast r s j = j^\ast j j^\ast j = 1_V 1_V = 1_V

where the second equation may be inferred from idempotence of the coreflexive morphism rsr s. Similarly i *sjj *ri=1 Ui^\ast s j j^\ast r i = 1_U.

Proposition

An arrow r:XYr: X \to Y is pseudo-invertible iff there are left adjoints i:UXi: U \to X, j:UYj: U \to Y such that r=ji *r = j i^\ast.

Proof

For the “if” clause, we have (in a bicategory of partial maps) that i *i=1 U=j *ji^\ast i = 1_U = j^\ast j. In that case, ji *j i^\ast has a pseudo-inverse ij *i j^\ast, since we have

ij *ji *=ii *1 X,ji *ij *=jj *1 Yi j^\ast j i^\ast = i i^\ast \leq 1_X, \qquad j i^\ast i j^\ast = j j^\ast \leq 1_Y

and

ij *ji *ij *=i(i *i)j *=ij *,ji *ij *ji *=j(j *j)i *=ji *.i j^\ast j i^\ast i j^\ast = i(i^\ast i) j^\ast = i j^\ast, \qquad j i^\ast i j^\ast j i^\ast = j(j^\ast j) i^\ast = j i^\ast.

For the “only if” clause, let ss be the pseudo-inverse of rr. Let (i,i *)(i, i^\ast) split srs r, let (j,j *)(j, j^\ast) split rsr s, and put ϕ=j *ri\phi = j^\ast r i. By the previous lemma, ϕ\phi is invertible, with ϕ 1=ϕ *\phi^{-1} = \phi^\ast, and so (jϕ,(jϕ) *)(j\phi, (j\phi)^\ast) is a splitting for rsr s. Moreover, we have

ri=rsri=jj *ri=jϕ.r i = r s r i = j j^\ast r i = j\phi.

Thus, if we rename jϕj\phi as jj, we have that ri=jr i = j and (j,j *)(j, j^\ast) splits rsr s. Then

r=rsr=rii *=ji *r = r s r = r i i^\ast = j i^\ast

which completes the proof.

Theorem

Pseudo-invertible morphisms are closed under composition.

Proof

Let r:XYr \colon X \to Y and s:YZs: Y \to Z be pseudo-invertible, and write r=j 0i 0 *r = j_0 i_0^\ast and s=j 1i 1 *s = j_1 i_1^\ast as in the preceding lemma. Also write i 1 *j 0=kh *i_1^\ast j_0 = k h^\ast where kk is total. We may calculate that h *j 0 *i 1h^\ast j_0^\ast i_1 is right adjoint to kk (see above). It follows that

sr=j 1i 1 *j 0i 0 *=(j 1k)(i 0h) *s r = j_1 i_1^\ast j_0 i_0^\ast = (j_1 k)(i_0 h)^\ast

and since j 1kj_1 k and i 0hi_0 h are left adjoints, it follows from the preceding lemma that srs r is pseudo-invertible.

The following definition is well-known; see for example Thurston.

Definition

A pseudogroup on a topological space (or locale) XX is a groupoid GG each of whose objects is an open set of XX, and whose morphisms are homeomorphisms between such open sets, satisfying the following conditions:

  • The objects cover XX. (Equivalently, in light of the last axiom, every open set of XX is an object of GG.)

  • If g:VWg: V \to W belongs to GG and UVU \subseteq V is an open set, then the restriction g| U:Ug(U)g|_U: U \to g(U) belongs to GG. (Equivalently, in light of the other axioms, every inclusion map id| U:UVid|_U: U \to V belongs to GG.)

  • (Sheaf property) If g:UVg: U \to V is a homeomorphism and if there is a covering U αU_\alpha of UU such that the restrictions g| U α:U αg(U α)g|_{U_\alpha}: U_\alpha \to g(U_\alpha) are morphisms of GG, then gg is also morphism of GG.

Lemma

In a cartology CC, the core groupoid of the Cauchy completion of the monoid of pseudo-invertible morphisms from an object XX to itself is a pseudogroup.

Proof

In the first place, every coreflexive endomorphism is pseudo-inverse to itself, and their extensions range over all open subsets of XX. These are the objects; obviously they cover XX.

The notion of a CC-manifold modeled on an object XX of CC is defined just as before, using the pseudogroup on XX implied by the previous lemma. In particular, we have CC-charts of an atlas structure on MM, which are morphisms in Reg\mathbf{Reg}

XiUϕMX \overset{i}{\leftarrow} U \overset{\phi}{\to} M

satisfying the expected properties. We can thus speak of CC-manifolds (or (C,X)(C, X)-manifolds if we want to make explicit the modeling space XX).

Now, given a cartology CC, we define the category of CC-manifolds. Let MM be a (C,X)(C, X)-manifold and NN a (C,Y)(C, Y)-manifold. Then, a CC-morphism from MM to NN is a continuous map f:MNf: M \to N such that the Reg\mathbf{Reg}-composite

U M V i ϕ 1 f ψ j X M N Y\array{ && U &&&& M &&&& V && \\ & i \swarrow && \searrow \phi && 1 \swarrow && \searrow f && \psi \swarrow && \searrow j & \\ X &&&& M &&&& N &&&& Y }

belongs to CC, for every pair of charts (U,ϕ):XM(U, \phi): X \to M and (V,ψ):YN(V, \psi): Y \to N.

These definitions need to be carefully checked against known examples (e.g., the categories TopTop, PLPL, and SmoothSmooth, among others).

References

Created on July 15, 2014 at 04:32:35 by Todd Trimble