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Tuesday, June 12, 2018

Enriched and straightened categories

Definition: A category C is monoidal if it is accompanied by
  • a functor :C×CC,
  • an object 1Obj(C), and
  • isomorphisms
    • αX,Y,ZHomC((XY)Z,X(YZ)),
    • λXHomC(1X,X), and
    • ρXHomC(X1,X),
for all X,Y,Z,WObj(C), such that is unital and α is associative over . That is, the diagrams below commute.

Definition: Let C be monoidal as above. A category D is enriched over C if it is accompanied by
  • an object D(P,Q)Obj(C) for every P,QObj(D), and
  • morphisms
    • γP,Q,RHomC(D(Q,R)D(P,Q),D(P,R)), and
    • iPHomC(1,D(P,P)),
for all P,Q,R,SObj(D), such that γ is unital and associative over . The category D is weakly enriched over C if γ is unital and associative over up to homotopy. That is, the diagrams below commute for D enriched, and commute up to homotopy for D weakly enriched.

Definition: A topological space X is compactly generated if its basis of topology of closed sets is given by continuous images of compact Hausdorff spaces K whose preimages are closed in K. A topological space is weakly Hausdorff if the continous image of every compact Hausdorff space is closed in X.

We write CG for the category of compactly generated and weakly Hausdorff spaces. This is a monoidal category with the usual product of topological spaces.

Example: Here are some examples of enriched categories.
  • A topological category is a category enriched over CG.
  • A bicategory, or weak 2-category, is a category weakly enriched over Cat, the category of small categories.

Definition: Let C,D be bicategories. An assignment F:CD is a pseudofunctor when it has
  • an object F(X)Obj(D),
  • a functor F(X,Y):C(X,Y)D(F(X),F(Y)), and
  • invertible 2-morphisms
    • F(idX):idXF(X,X)(idX), and
    • F(X,Y,Z)(f,g):F(Y,Z)(g)F(X,Y)(f)F(X,Z)(gf),
for all X,Y,ZObj(C), such that F(X,Y) is unital and associative over composition. The assignment F is a lax functor when the last two morphisms are not necessarily invertible.

Definition: Let C,D be categories and F:CD a functor. A morphism fHomC(A,B) is F-cartesian if

commutes for some unique gHomC(A,Y) (all the vertical arrows are F).

This definition can be rephrased in the language of simplicial sets: the morphism f is F-cartesian if whenever Ff=d1Δ2 for some Δ2D2, then every Λ2C with Λ21=f and FΛ20=d0Δ2 can be filled in by g with Fg=d2Δ2.

Definition: Let f:CD be a functor.
  • The category C is F-fibered over D if for every morphism hHomD(U,V) and every BObj(C) with F(B)=V, there is some F-cartesian fHomC(,B) with Ff=h.
  • A cleavage of an F-fibered category C is a class of cartesian morphisms K in C such that for every morphism hHomD(U,V) and every BObj(C) with F(B)=V, there is a unique F-cartesian fK with Ff=h.
  • A cleavage of C is a splitting if it contains all the the identity morphisms and is closed under composition.

If C is F-fibered over D and C is F-fibered over D, then a functor F:CC is a \emph{morphism of fibered categories} if F=FF and Ff is F-cartesian whenever f is F-cartesian.

Theorem: Let C be F-fibered over D.
  • Every cleavage of C defines a pseudofunctor DCat.
  • Every pseudofunctor DCat defines an F-fibered category C with a cleavage over D.

The above result follows from sections 3.1.2 and 3.1.3 of Vistoli. Theorem 2.2.1.2 of Lurie generalizes this and provides an equivalence between the category of fibered simplicial sets over SsSet and the category of functors sCatsSet. The forward direction is called straightening and he backward direction is called unstraightening.

References: nLab (articles "Monoidal category," "enriched category," and "pseudofunctor."), Strickland (The category of CGWH spaces), Vistoli (Notes on Grothendieck topologies, Chapter 3), Noohi (A quick introduction), Lurie (Higher Topos Theory, Section 2.2)

Monday, June 4, 2018

Integral transforms

Let X,Y be topological spaces.

Definition: A set UX is constructible if it is a finite union of locally closed sets. A function f:XY is constructible if f1(y)X is constructible for all yY.

Write CF(X) for the set of constructible functions f:XZ. Recall if UX is constructible, it is triangulable.

Definition: Let XRN be constructible and {Xr}rR a filtration of X by constructible sets Xr. The kth persistence diagram of X is the set PD(Xr,k)={(a,b)(R{±})2:a<b}, where each element represents the longest sequence of identity morphisms in the decomposition of the image of the kth persistent homology functor PH(X_r,k)\colon (\R,\leqslant )\to Vect to each component.

Write D for the set of all persistence diagrams.

Definition: Let X,Y\subseteq \R^N be constructible, S\subseteq X\times Y also constructible with \pi_1,\pi_2 the natural projections, and \sigma a simplex in a triangulation of X. The Euler integral of elements of CF(X) is the assignment \begin{array}{r c l} \displaystyle \int_X\ \cdot\ d\chi\colon CF(X) & \to & \Z, \\ \mathbf{1}_\sigma & \mapsto & (-1)^{\dim(\sigma)}. \end{array} The Radon transform of elements of CF(X) is the assignment \begin{array}{r c l} \mathcal R_S \colon CF(X) & \to & CF(Y), \\ (x\mapsto h(x)) & \mapsto & \left(y\mapsto \displaystyle \int_{\pi_2^{-1}(y)} \pi_1^*h\ d\chi\right). \end{array} The persistent homology transform of X is the assignment \begin{array}{r c l} PHT_X \colon S^{N-1} & \to & D^N, \\ v & \mapsto & \left\{PD(\{x\in X:x\cdot v\leqslant r\},0),\dots,PD(\{x\in X : x\cdot v\leqslant r\},N-1)\right\} \end{array}

The Euler integral is also called the Euler transform, or the Euler charateristic transform. The Radon transform has a weighted version, where every simplex in S is assigned a weight.

References: Schapira (Tomography of constructible functions), Baryhsnikov, Ghrist, Lipsky (Inversion of Euler integral transforms), Turner, Mukherjee, Boyer (Persistent homology transform).