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39

Suppose $k$ is a field, not necessarily algebraically closed. $\text{Spec } k$ fails to behave like a point in many respects. Most basically, its "finite covers" (Specs of finite etale $k$-algebras) can be interesting, and are controlled by its absolute Galois group / etale fundamental group. For example, $\text{Spec } \mathbb{F}_q$, the Spec of a finite ...

38

Looks like a typo. Condition $(4)$ should say that $B$ is downward closed under $\leq$, not under $\preceq$ (otherwise, $Y_B$ is not defined).

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Since I don't know precisely which parts of Lawvere's article you have difficulties with, this answer is a bit a long and tries to give a bit of context. If you want me to be more specific at some point, just say so. Classically, the spectrum of a ring $A$ can be defined as the set of its prime ideals equipped with the Zariski topology. This topological ...

37

$Spec(\mathbb{Z})$ should only be considered as $S^3$, if you "compactify" that is add the point at the real place. This is demonstrated by taking cohomology with compcat support. The étalé homotopy type of $Spec(\mathbb{Z})$ is however contractible (indeed what do you get by removing a point form a sphere?) to see this (all results apper in Milne's ...

36

One point of synthetic differential geometry is that, indeed, it is "synthetic" in the spirit of traditional synthetic geometry but refined now from incidence geometry to differential geometry. Hence the name is rather appropriate and in particular highlights that SDG is more than any one of its models, such as those based on formal duals of C-infinity rings ...

29

Derived versions of differential topology are becoming prominent tools in symplectic geometry. Whether or not you think of them via topoi is not crucial (I certainly can't), and perhaps the terminology turns off more people than it draws, but these ideas are being put to serious use by very serious no-nonsense mathematicians -- I think an excellent (though ...

27

Short answer: the Kreisel-Putnam axiom $(\lnot p \to (q \lor r)) \to ((\lnot p \to q) \lor (\lnot p \to r))$ is not an intuitionistic tautology but it is valid for any subobjects of an object in the topos of simplicial sets. The longer answer relies on an interesting characterization of the subobject classifier $\Omega$ of the topos of simplicial sets. (...

26

In general the internal language of a topos can only express those statements that make sense in every topos. In essence, this limits you to something like bounded Zermelo set theory, without global membership. The right way to use the internal language of a particular topos, such as your topos of directed graphs, is to enrich the general internal language ...

26

Yes there is: the formal locale of p-adic integer is simply defined as the projective limit of the $\mathbb{Z}/p^k\mathbb{Z}$ (as a pro-finite locale). So internally in any topos a continuous function with values in $\mathbb{Z}_p$ corresponds to an element of the projective limit of the $\mathbb{Z}/p^k\mathbb{Z}$ (as a set this time) You can then define the ...

25

For any space $X$, there's an $\infty$-topos of spaces fibered over $X$. The underlying ordinary topos is the category of representations of the fundamental groupoid of $X$. So if $X$ is simply connected, this is just the category of sets. But the $\infty$-topoi are different for different values of $X$ (two spaces $X$ and $Y$ yield equivalent $\infty$-topoi ...

23

Ok, so, I will try to answer this best I can. first, I'll tell you a skewed-perspective of what an infinity topos is (or ought-to-be). As for how you can "do differential geometry"- this is a bold statement. But, there's certain aspects of differential geometry which extend naturally / have a nice interpretation in (certain) infinity topoi (with extra ...

23

Here are two possible motivating examples. First, for any topos $\mathcal{E}$, if all epimorphisms in $\mathcal{E}$ split, then $\mathcal{E}$ is a boolean topos. In particular, for a topological space $X$, if all epimorphisms in $\mathbf{Sh} (X)$ split, then every open subset of $X$ is also closed (and vice versa), so $X$ is a disjoint union of indiscrete ...

22

The change-of-universe construction is faithful but not full. For example, let X be the topos of sets and let Y be the classifying topos for abelian groups. The category of geometric morphisms from X to Y is equivalent to the category of abelian groups. If you pass to a larger universe, you get more abelian groups.

21

Marc's examples are good ones, but let me add two more (which are closely related to each other): 1) Let $\mathcal{C}$ be an accessible $\infty$-category which admits small filtered colimits, and let $\mathcal{X}$ be the $\infty$-category of functors from $\mathcal{C}$ to $\mathcal{S}$ which preserve small filtered colimits. Then $\mathcal{X}$ is an $\infty$...

20

Here is a concrete example of a consistent geometric theory that has no model in $\mathbf{Set}$ but does have a model in a Boolean Grothendieck topos. Our base theory $T$ is an expansion of the theory of linear orders with a constant $c_q$ for each rational number $q$. In geometric form, the base axioms are: $$x = y \lor x \lt y \lor y \lt x$$ x \lt y \...

19

It would be presumptuous on my part to attempt to answer this question, but I want to share with other MOers this recent paper http://www.ihes.fr/~lafforgue/math/TheorieCaramello.pdf of Laurent Lafforgue and this video https://sites.google.com/site/logiquecategorique/Contenus/20130227_Lafforgue of one of his recent lectures, [dont] le but [] est de ...

19

Jacob's answer to your "for example" question is a good one, but let me be so bold as to try to address the general question. I think there are two different issues in play: the fact that the site of a topos may not be truncated, and the fact that an $(\infty,1)$-topos may not be hypercomplete. The first is just as much the case for 1-toposes, while the ...

19

There are known statements that are true in any Grothendieck topos, but not in every elementary topos with NNO. For instance: Freyd's theorem that a complete small category is a preorder is not constructively provable and can fail in elementary toposes with NNO, such as the effective topos; but can be shown to hold in any Grothendieck topos by an "external"...

18

As you pointed out, your question (for general sheaves of abelian groups) follows from the vanishing of the derived functors $R^if_*{\mathcal F}$ for $i>0$, and that's all one can say. A good example is the Beilinson-Lichtenbaum conjectures, which states that for a certain complex of sheaves $\mathbb Z(n)$, the etale and Zariski cohomology groups agree ...

18

If the question is not really about principal bundle theory but just about: why do we need higher differential geometry at all, then of course there are plenty of further answers: Classical differential geometry includes orbifolds http://ncatlab.org/nlab/show/orbifold as objects that handle non-free quotients of smooth manifolds. These are really the ...

18

The undesirable properties of higher-order logic are created by an insufficient notion of model. That is, we cannot have all three, soundness, completeness and effectivness (decidability of proof checking), if we insist that formulas be interpreted in the "standard" set-theoretic way. Henkin semantics does not suffer from this defficiency. What this says is ...

18

I would like to explain why I think the answer is no, but of course there is no way to prove this, and probably some way to use some geometric insight when talking about elementary toposes. My main point is that the geometrical aspect of Grothendieck toposes is not related to the fact that they are elementary toposes, but rather to the fact that they are ...

18

I'll give a counter-example to the claim that having a subobject classifier and being cartesian closed implies the existence of all finite limits. However, this is based on the definition of sub-object classifier given on wikipedia (linked in the comment above) that I would consider as incorrect: The wikipedia definition (at the time this is written) only ...

17

Here is one way to say it, which makes the relation to principal bundle theory most manifest ( http://ncatlab.org/schreiber/show/Principal+%E2%88%9E-bundles+--+theory,+presentations+and+applications ): An $\infty$-topos is a context for homotopy theory that satisfies three extra axioms, the "Giraud-Rezk-Lurie"-axioms (for all keywords see the pointers ...

17

There is a Grothendieck topos $\textbf{Set}[\mathbb{O}]$ with the following universal property: for all Grothendieck toposes $\mathcal{E}$, the category $\textbf{Geom}(\mathcal{E}, \textbf{Set}[\mathbb{O}])$ of geometric morphisms $\mathcal{E} \to \textbf{Set}[\mathbb{O}]$ and "geometric transformations" (a misnomer; they actually code algebraic data!) is ...

17

I'm not sure if this constitutes a full answer, and a lot of it has already been said in some form by HeinrichD in the comments. Because your question is ultimately one of philosophy, I will focus mostly on history and philosophy (and a few applications); not so much the categorical or logical interpretations. History. In functional analysis, the Gelfand ...

17

A classical example is $G$-$Set$ for a large group $G$. That this is a cocomplete elementary topos is not hard to see. Limits and colimits are formed at the underlying set level, and exponentials $Y^X$ are formed as usual as the set of functions $f: X \to Y$ with the $G$-action $(g, f) \mapsto g f$ defined by $g f: x \mapsto g f(g^{-1} x)$. The subobject ...

16

This is not exactly what you asked for but I think it's reasonably close to what you want... The idea of recasting Gödel's results in the context of category theory has led André Joyal to develop arithmetic universes, a minimalistic category tailored for that purpose. Unfortunately, Joyal never published this as explained by Paul Taylor in this recent ...

16

The consistency of ZFC + IC is perhaps a little bit too much to ask, but I believe the next best thing is true: Conjecture. Every boolean topos1 with dependent choice in which every set of reals is Lebesgue measurable contains a well-founded model of ZFC. In fact, every real is contained in a well-founded model of ZFC. The proof that Con(ZF + DC + LM) ...

15

Here are examples appearing in algebraic topology. The category of $\mathrm{Ext}$-$p$-complete abelian groups, as discussed in Homotopy limits, completions and localizations (Chapter VI, Sections 2-4) is locally $\aleph_1$-presentable but not locally $\aleph_0$-presentable. This follows from the fact that the $p$-adic integers $\mathbb{Z}_p$ are an object ...

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