The Sierpinski topos is a cohesive topos.
The Sierpinski -topos is a cohesive (∞,1)-topos.
The fact that the Sierpienski -topos is, therefore, in particular
all follow directly from the fact that it is the image, under localic reflection, of the Sierpinski space (hence that it is 0-localic, its (-1)-truncation being the frame of opens of the Sierpinski space).
That space , in turn,
which implies the corresponding three properties of the Sierpinski -topos above.
By the discussion at cohesive (∞,1)-topos every such may be thought of as a fat point, the abstract cohesive blob. In this case, this fat point is the Sierpinski space. This space can be thought of as being the abstract “point with open neighbourhood”.
Accordingly, the objects of the Sierpinski -topos may be thought of as ∞-groupoids (relative to ) equipped with the notion of cohesion modeled on this: they are bundles of ∞-groupoids whose fibers are regarded as being geometrically contractible, in that
and so in particular
Hence these objects are discrete ∞-groupoids , to each of whose points may be attached a contractible cohesive blob with inner structure given by the -groupoid .
Accordingly, the underlying -groupoid of such a bundle is the union
of the discrete base space and the inner structure of the fibers.
The discrete object in the Sierpinski -topos on an object is the bundle
which is with “no cohesive blobs attached”.
Finally the codiscrete object in the Sierpinski -topos on an object is
the structure where all of is regarded as one single contractible cohesive ball.
The -adjunction unit
The -counit on is
Hence the canonical natural transformation
from “points to pieces” is on simply the morphism itself
the full sub-(∞,1)-category on those objects in for which “pieces have points”, hence those for which is an effective epimorphism, is the -category of effective epimorphisms in the ambient -topos, hence the -category of groupoid objects in the ambient -topos;
the full sub--category on the objects with “one point per piece” is the ambient -topos itself.
We unwind what some of the canonical structures in a cohesive (∞,1)-topos are when realized in the Sierpinski -topos.
A group object in is a morphism in of the form .
The corresponding flat coefficient object is
Hence the corresponding de Rham coefficient object is
where exhibits has an -group extension of by in .
The corresponding Maurer-Cartan form
exhibiting the -cocycle that classifies the extension .
The above discussion generalizes immediately as follows.
Conversely, we may think of as being an “infinitesimal thickening” of , as formalized at infinitesimal cohesion, where we regard
as exhibiting as an infinitesimal cohesive neighbourhood of (here denotes the enpoint inclusions, following the notation here).
(See also the corresponding examples at Q-category.)
We have for all that
For all in we have
which is indeed naturally equivalent to
Therefore an object of given by a morphism in is regarded by the infinitesimal cohesion as being an infinitesimal thickening of by the fibers of : where before we just had that the fibers of are “contractible cohesive thickenings” of the discrete object , now is “discrete relative to ” (hence not necessarily discrete in ) and the fibers are in addition regarded as being infinitesimal.
This is of course a very crude notion of infinitesimal extension. Notice for instance the following
With respect to the above infinitesimal cohesion , every morphism in is a formally étale morphism.
By definition, given a morphism , it is formally étale precisely if
is an (∞,1)-pullback.
By prop. 2 the above square diagram in is
Since -pullbacks of -presheaves are computed objectwise, this is an -pullback in precisely if the “back and front sides”
are -pullbacks in . This is clearly always the case.
The Sierpinski topos is mentioned around remark B3.2.11 in
The homotopy type theory of the Sierpinski -topos is discussed in
Cohesion of the Sierpinski -topos is discussed in section 2.2.4 of