CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
The Sierpinski topos is the arrow category of Set.
Equivalently, this is the category of presheaves over the interval category $\Delta[1] := \mathbf{2} = \{0 \to 1\}$, or equivalently the category of sheaves over the Sierpinski space $Sierp$
Yet another description is that it is the Freyd cover of Set.
Similarly, the Sierpinski (∞,1)-topos is the arrow category $\infty Grpd^{\Delta[1]}$ of ∞Grpd.
Equivalently this is the (∞,1)-category of (∞,1)-presheaves on $\Delta[1]$ and equivalently the (∞,1)-category of (∞,1)-sheaves on $Sierp$:
Being a (∞,1)-category of (∞,1)-functors, the Sierpinski (∞,1)-topos is presented by any of the model structure on simplicial presheaves $[\Delta[1], sSet]$.
Specifically the Reedy model structure of simplicial presheaves on the interval category $[\Delta[1], sSet]_{Reedy}$ provides a univalent model for homotopy type theory in the Sierpinski $(\infty,1)$-topos (Shulman)
We discuss the connectedness, locality and cohesion of the Sierpinski topos. We do so relative to an arbitrary base topos/base (∞,1)-topos $\mathbf{H}$, hence regard the global section geometric morphism
The Sierpinski topos is a cohesive topos.
The Sierpinski $(\infty,1)$-topos is a cohesive (∞,1)-topos.
For the first statement, see the detailed discussion at cohesive topos here.
For the second statement, see the discussion at cohesive (∞,1)-topos here.
The fact that the Sierpienski $(\infty,1)$-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 $Sierp$, in turn,
which implies the corresponding three properties of the Sierpinski $\infty$-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 $(\infty,1)$-topos may be thought of as ∞-groupoids (relative to $\mathbf{H}$) equipped with the notion of cohesion modeled on this: they are bundles $[P \to X]$ of ∞-groupoids whose fibers are regarded as being geometrically contractible, in that
and so in particular
Hence these objects are discrete ∞-groupoids $X$, to each of whose points $x : * \to X$ may be attached a contractible cohesive blob with inner structure given by the $\infty$-groupoid $P_x := P \times_X \{x\}$.
Accordingly, the underlying $\infty$-groupoid of such a bundle $[P \to X]$ is the union
of the discrete base space and the inner structure of the fibers.
The discrete object in the Sierpinski $(\infty,1)$-topos on an object $X \in \mathbf{H}$ is the bundle
which is $X$ with “no cohesive blobs attached”.
Finally the codiscrete object in the Sierpinski $(\infty,1)$-topos on an object $X \in \mathbf{H}$ is
the structure where all of $X$ is regarded as one single contractible cohesive ball.
The $(\Pi \dashv Disc)$-adjunction unit
on $[P \to X]$ is
The $(Disc \dashv \Gamma)$-counit $Disc \Gamma \to id$ on $[P \to X]$ is
Hence the canonical natural transformation
from “points to pieces” is on $[P \to X]$ simply the morphism $P \to X$ itself
Therefore
the full sub-(∞,1)-category on those objects in $\mathbf{H}^I$ for which “pieces have points”, hence those for which $\Gamma \to \Pi$ is an effective epimorphism, is the $(\infty,1)$-category of effective epimorphisms in the ambient $(\infty,1)$-topos, hence the $(\infty,1)$-category of groupoid objects in the ambient $(\infty,1)$-topos;
the full sub-$(\infty,1)$-category on the objects with “one point per piece” is the ambient $(\infty,1)$-topos itself.
We unwind what some of the canonical structures in a cohesive (∞,1)-topos are when realized in the Sierpinski $(\infty,1)$-topos.
A group object $\mathbf{B}[\hat G \to G]$ in $\mathbf{H}^I$ is a morphism in $\mathbf{H}$ of the form $= \mathbf{B}\hat G \to \mathbf{B}G$.
The corresponding flat coefficient object $\mathbf{\flat} \mathbf{B}[\hat G \to G] \to \mathbf{B}[\hat G \to G]$ is
Hence the corresponding de Rham coefficient object is
where $A \to \hat G \to G$ exhibits $\hat G$ has an $\infty$-group extension of $G$ by $A$ in $\mathbf{H}$.
The corresponding Maurer-Cartan form
is
exhibiting the $A$-cocycle that classifies the extension $\hat G \to G$.
The above discussion generalizes immediately as follows.
For $\mathbf{H}$ any cohesive (∞,1)-topos, we have the “Sierpinski $(\infty,1)$-topos relative to $\mathbf{H}$” given by the arrow category $\mathbf{H}^{\Delta[1]}$, whose geometric morphism to the base topos is the domain cofibration
Conversely, we may think of $\mathbf{H}^{\Delta[1]}$ as being an “infinitesimal thickening” of $\mathbf{H}$, as formalized at infinitesimal cohesion, where we regard
as exhibiting $\mathbf{H}^{\Delta[1]}$ as an infinitesimal cohesive neighbourhood of $\mathbf{H}$ (here $(\bot, \top) : \Delta[0] \coprod \Delta[0] \to \Delta[1]$ denotes the enpoint inclusions, following the notation here).
(See also the corresponding examples at Q-category.)
We have for all $X \in \mathbf{H}$ that
For all $[A \to B]$ in $\mathbf{H}^{\Delta[1]}$ we have
which is indeed naturally equivalent to
Therefore an object of $\mathbf{H}^{\Delta[1]}$ given by a morphism $[P \to X]$ in $\mathbf{H}$ is regarded by the infinitesimal cohesion $i : \mathbf{H} \hookrightarrow \mathbf{H}^{\Delta[1]}$ as being an infinitesimal thickening of $X$ by the fibers of $P$: where before we just had that the fibers of $P$ are “contractible cohesive thickenings” of the discrete object $X$, now $X$ is “discrete relative to $\mathbf{H}$” (hence not necessarily discrete in $\mathbf{H}$) 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 $i : \mathbf{H} \hookrightarrow \mathbf{H}^{\Delta[1]}$, every morphism in $\mathbf{H}$ is a formally étale morphism.
By definition, given a morphism $f : X \to Y$, it is formally étale precisely if
is an (∞,1)-pullback.
By prop. 2 the above square diagram in $\mathbf{H}^{\Delta[1]}$ is
Since $(\infty,1)$-pullbacks of $(\infty,1)$-presheaves are computed objectwise, this is an $(\infty,1)$-pullback in $\mathbf{H}^{\Delta[1]}$ precisely if the “back and front sides”
and
are $(\infty,1)$-pullbacks in $\mathbf{H}$. This is clearly always the case.
The Sierpinski topos is the classifying topos for subterminal objects in toposes (see e.g. Johnstone 77, p. 117).
The generic subterminal inclusion in the Sierpinski topos is the unique inclusion of $[\emptyset \to \ast]$ into $[\ast \to \ast]$.
The (non-full) inclusion
induces via restriction and right Kan extension an essential geometric morphism
of the Sierpinski topos into the classifying topos for pointed objects, $\mathbf{H}[X_\ast]$. (This becomes a geometric embedding if we refined the Sierpinski topos of bundles to the topos of bundles with sections).
The pointed object in $\mathbf{H}^{\Delta^1}$ which is classified by this geometric morphism is $[\ast \to S^0]$ (with $S^0 = \ast \coprod \ast$ the 0-sphere) with its canonical map from $[\ast \to \ast]$.
In summary, the generic subterminal object and the generic pointed object fit into a sequence of the form
The Sierpinski topos is mentioned around remarks A2.1.12, B3.2.11 (p.83, p.387f) in
See also
The homotopy type theory of the Sierpinski $(\infty,1)$-topos is discussed in
Cohesion of the Sierpinski $\infty$-topos is discussed in section 2.2.4 of