infinitesimally thickened Sierpinski topos




The infinitesimally thickened Sierpinski topos is the topos of presheaves on the ordinal category 3:{012}\mathbf{3}: \{ 0 \to 1 \to 2 \}. The name emphasizes the view of this topos as exhibiting differential cohesion over the cohesive Sierpinski topos using the coreflective embedding i:23i : \mathbf{2} \hookrightarrow \mathbf{3} sending 00 to 00 and 11 to 22.

The objects of the category are diagrams A 2A 1A 0A_2 \to A_1 \to A_0 in SetSet and arrows are commuting squares.


Differential cohesion

We can view objects of the topos as exhibiting a simple notion of differential cohesion: in A rA infA pA_r \to A_{inf} \to A_p the A rA_r are the “real” points, the A infA_{inf} are the “infinitesimal” points and the A pA_p are the “pieces”, i.e., connected components.

The coreflective embedding above results in an adjoint quadruple

(i !i *i *i !):PSh(3)i !i *i *i !PSh(2). ( i_! \dashv i^* \dashv i_* \dashv i^! ) : PSh(\mathbf{3}) \stackrel{\overset{i_!}{\hookleftarrow}}{\stackrel{\overset{i^*}{\to}}{\stackrel{\overset{i_*}{\hookleftarrow}}{\stackrel{i^!}{\to}}}} PSh(\mathbf{2}) \,.

which are defined explicitly as follows:

i !(A 1A 0)=A 1A 1A 0 i_!(A_1 \to A_0) = A_1 \to A_1 \to A_0
i *(A 2A 1A 0)=A 2A 0i^*(A_2 \to A_1 \to A_0) = A_2 \to A_0
i *(A 1A 0)=A 1A 0A 0i_*(A_1 \to A_0) = A_1 \to A_0 \to A_0
i !(A 2A 1A 0)=A 2A 1 i^!(A_2 \to A_1 \to A_0) = A_2 \to A_1

This gives us elementary characterizations of the various subcategories of Sh(3)Sh(\mathbf{3}) involved in differential cohesion:

and the differential modalities are defined as:

(A 2A 1A 0)=A 2A 2A 0 \Re(A_2 \to A_1 \to A_0) = A_2 \to A_2 \to A_0
(A 2A 1A 0)=A 2A 0A 0 \Im(A_2 \to A_1 \to A_0) = A_2 \to A_0 \to A_0
&(A 2A 1A 0)=A 2A 1A 1 \&(A_2 \to A_1 \to A_0) = A_2 \to A_1 \to A_1

The cohesive structure

(ΠDiscΓcoDisc):Psh(3)coDiscΓDiscΠSet. (\Pi \dashv \Disc \dashv \Gamma \dashv coDisc) : Psh(\mathbf{3}) \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}}} Set \,.

factors through the cohesive structure of the Sierpinski topos and is defined as:

Π(A 2A 1A 0)=A 0\Pi(A_2\to A_1 \to A_0) = A_0
Disc(A)=AAA\Disc(A) = A \to A \to A
Γ(A 2A 1A 0)=A 2\Gamma(A_2 \to A_1 \to A_0) = A_2
coDisc(A)=A11coDisc(A) = A \to 1 \to 1

Giving us two more subcategories:

This is reflective of the general phenomenon that discrete objects are reduced and coreduced, in fact here the discrete objects are exactly the intersection of reduced and coreduced objects.

And the cohesive modalities are defines as:

ʃ(A 2A 1A 0)=A 0A 0A 0 ʃ(A_2 \to A_1 \to A_0) = A_0 \to A_0 \to A_0
(A 2A 1A 0)=A 2A 2A 2 \flat(A_2 \to A_1 \to A_0) = A_2 \to A_2 \to A_2
(A 2A 1A 0)=A 211 \sharp(A_2 \to A_1 \to A_0) = A_2 \to 1 \to 1

Constructions in Differential Cohesion

A morphism f:ABf : A \to B is formally etale when the square

A 1 A 0 f 1 f 0 B 1 B 0 \array{ A_1 &\to& A_0 \\ \downarrow^{\mathrlap{f_1}} && \downarrow^{\mathrlap{f_0}} \\ B_1 &\to& B_0 }

is a pullback. This makes it easy to see explicitly that coreduced objects are exactly the objects with A1A \to 1 formally etale.

The infinitesimal disk bundle of A 2infA 1pcA 0A_2 \overset{inf}{\to}A_1 \overset{pc}{\to} A_0 is the pullback of the unit of \Im along itself. Since pullbacks are computed pointwise in a presheaf topos, this is just:

TA=A 2{(x i,y i)A 1|pc(x i)=pc(y i)}A 0TA = A_2 \to \{(x_i,y_i) \in A_1 | pc(x_i) = pc(y_i) \} \to A_0

with the fiber of some real point x rA 2x_r \in A_2 given by:

{x r}{y iA 1|pc(y i)=pc(inf(x r))}{pc(inf(x r))}\{x_r \} \to \{y_i \in A_1 | pc(y_i) = pc(inf(x_r)) \} \to \{pc(inf(x_r)) \}

which is an infinitesimally thickened point.