G * θ θ/G dRBG BG * BG \array{ G && \longrightarrow && \ast \\ & \searrow^{\mathrlap{\theta}} && && \searrow^{\mathrlap{\theta/G}} \\ \downarrow && \flat_{dR}\mathbf{B}G && \longrightarrow && \flat \mathbf{B}G \\ & \swarrow && && \swarrow \\ \ast && \longrightarrow && \mathbf{B}G }
G Z * θ x x:Xθ x θ/G dRBG BG * x X 1 g BG \array{ G && \longrightarrow && Z && \longrightarrow && \ast \\ & \searrow^{\mathrlap{\theta_x}} && && \searrow^{\mathrlap{\underset{x\colon X}{\sum} \theta_x}} && && \searrow^{\mathrlap{\theta/G}} \\ \downarrow && \flat_{dR}\mathbf{B}G && \longrightarrow && && \longrightarrow && \flat \mathbf{B}G \\ & \swarrow && && \swarrow && && \swarrow \\ \ast && \stackrel{x}{\longrightarrow} && X_1 && \stackrel{g}{\longrightarrow} && \mathbf{B}G }

Somebody writes:

I want to touch two points concerning Lawvere’s concept of Aufhebung:

Let us briefly recall the definition given in the entry. Given a topos \mathcal{B} define a level ii as an essential subtopos that is i !i *i *:𝒜i_!\dashv i^ *\dashv i_*:\mathcal{A}\to\mathcal{B} with i *i_* fully faithful. This yields an adjoint modality i i\lozenge_i\dashv \Box_i via i:=i !i *\lozenge_i := i_!i^* and i:=i *i *\Box_i :=i_*i^*. A level jj is higher iff

j j i i \array{ \lozenge_j & \dashv & \Box_j\\ \cup & & \cup\\ \lozenge_i & \dashv & \Box_i }

jj resolves ii if furthermore j i= i \Box_j\lozenge_i = \lozenge_i and jj is the Aufhebung (term taken from Hegel) of jj if every other level kk that resolves ii is higher than jj.

  1. The first point which suggests that there might be something wrong here (at least in my understanding) is that this appears to imply that a quintessential localization aka quality type where i !i *i_!\simeq i_* is his own Aufhebung as then i i\lozenge_i\simeq\Box_i whence i i= i i= i\Box_i\lozenge_i=\lozenge_i\lozenge_i=\lozenge_i.

    This is somewhat puzzling because Lawvere in (1991, p.10) asks for the Aufhebung of a level that is an quality type and also the Aufhebung operator computed in Kennett-Riehl-Roy-Zaks for some homotopical cats does have no fix points implying e.g. that sSet has no quintessential localizations. So there probably is a bug here!?

  2. The second point is related to Mike’s quest in a recent thread to deconstruct the modalities internally. Taking a look at Urs’ table for differential cohesion and rearranging a bit :

    inf inf * \array{ \Re&\dashv&\int_{inf}&\dashv&\flat_{inf} & & \\ & & \cup & &\cup & & \\ & &\int &\dashv & \flat &\dashv &\sharp \\ & & & & \cup & & \cup \\ & & & & \empty & \dashv & \ast }

    This suggests to read the \cup as Aufhebung with the left-peripheral ,\Re ,\int just there to ensure that the Aufhebung exists at the preceding level so that we start with *\empty\dashv\ast at the bottom, lift with carry-over \int, lift once more with final carry-over \Re. As Aufhebung consists intuitively basically in shifting the ‘negative’ part \lozenge to the ‘positive’ right \Box at a higher level (in particular, the Aufhebung of a \lozenge can be viewed as a stand-in for a left adjoint which transforms \lozenge\to\Box' already at the same level) this could be viewed as a repeated rectification of \empty - non-being. So the idea would be to put aside the base and the thickened topos and try to characterize the adjunctions in the table via the existence of the appropriate Aufhebungen. In order to find out whether this story holds water it would be necessary to check that the \cup are indeed Aufhebungen and then that the carry-overs actually can be connected to these Aufhebungen, also the implicit quantification over all levels is potentially nasty from an internal point of view.

Revised on November 25, 2014 21:20:03 by Urs Schreiber (