All Questions
Tagged with topos-theory reference-request
54 questions
11
votes
3
answers
671
views
Merging single-sorted and multi-sorted theories
The general theory of single-sorted (say, algebraic) theories is very similar to the general theory of multi-sorted (algebraic) theories. Each variable gets a sort, but apart from that nothing really ...
- 63.1k
7
votes
2
answers
284
views
Bounded geometric morphisms, origin and motivation for the terminology
Bounded geometric morphisms serve as a generalization of Grothendieck topoi; $T$ being a Grothendieck topos iff global section $T \rightarrow Set$ is bounded. I managed to only track this down to the ...
- 1,347
7
votes
2
answers
292
views
Quotient topoi as quotient objects
In Lawvere's Open problems in topos theory; quotient topoi are treated as connected geometric morphisms of Grothendieck topoi.
Is there a good reference for where these come from? Is there any sense ...
- 1,347
6
votes
1
answer
388
views
Decimal expansion definition of real numbers, constructively
The two most common definitions of $\mathbb{R}$ are as Dedekind cuts or Cauchy sequences of rational numbers.
A real analysis student of mine is working out of the book Real Analysis and Applications ...
- 10.1k
7
votes
1
answer
432
views
Strict toposes as a finite limit theory
For some motivation I have been wondering about generalizing the topos of coalgebras theorem to relative monads in my previous question. This brought me to wonder about topos objects.
NLab on ...
- 1,347
2
votes
1
answer
98
views
What can be said about a semigroup $M$ from its topos $\mathit{Sets}^M$?
Please recommend some review "what can be said about a semigroup $M$ from its topos $\mathit{Sets}^M$". For example, for which semigroups are de Morgan's laws true?
4
votes
0
answers
102
views
Topos as a totally cocomplete object in a 2-category CART
In the preface to Sketches of an elephant, Peter Johnstone gives a list of characterizations of topos, some applicable to elementary (ii- finite limits and power objects) other to Grothendieck (i- ...
- 1,347
7
votes
0
answers
192
views
Constructive theory of Lie algebras
I'm looking for references on constructive Lie algebra theory, e.g. the sort of theory you could develop in Martin-Löf type theory or internal to some topos with a NNO. Obviously excluded middle is ...
- 6,025
8
votes
0
answers
190
views
A reference on a result by Steve Schanuel
In the Author Commentary section of the TAC reprint of the paper of 1968 Diagonal arguments and cartesian closed categories., Bill Lawvere wrote:
‘Nilpotent infinitesimals fall far short of even one-...
- 774
7
votes
1
answer
242
views
Dissolution of a topos
The dissolution of the locale associated with a frame $F$ is the locale associated with the frame $N(F)$ of nuclei of $F$ (see, e.g., Johnstone, “Stone Spaces” (1982), §2.5). Note that there is a ...
- 32.5k
7
votes
0
answers
173
views
Beck-Chevalley conditions for the local geometric morphisms $\pi:\mathrm{Zar}/X\to \mathrm{Sh}(X)$
$\newcommand{\Zar}{\mathrm{Zar}}\newcommand{\Sh}{\mathrm{Sh}}$The category of schemes is a full subcategory of the big Zariski topos $\Zar$. For each scheme $X$, there is a local geometric morphism $\...
- 775
5
votes
0
answers
139
views
When is the topos of algebras well-pointed?
The monad version of the theorem at Topos of coalgebras over a comonad is as follows:
Let $\mathcal{E}$ be an (elementary) topos. Then if a monad $T : \mathcal{E} \rightarrow \mathcal{E}$ has a right ...
- 1,347
17
votes
0
answers
368
views
Joyal's topos in which $[0,1]$ fails to be compact
Some time around 1977, André Joyal constructed a topos (actually a locale, i.e., a localic topos, necessarily non-spatial) in which the closed unit interval $[0,1]$ fails to be compact. There are ...
- 32.5k
2
votes
0
answers
101
views
Concrete topological objects and notions in the category of locales
I have read Peter Johnstone's “The Point of Pointless Topology” and the idea that topological spaces are not quite the right abstraction for topology seems, at least philosophically, rather appealing. ...
- 121
10
votes
1
answer
368
views
Deductive system involving only geometric sequents
A geometric theory is made up of sequents of restricted form: It may only be of the form $$\phi \vdash \psi$$
possibly with free variables (which are implicitly taken universal closure). $\phi, \psi$ ...
- 1,263
11
votes
2
answers
405
views
Equivalence between geometric theories and frames internal to the free topos
What is a reference for "the equivalence between geometric theories and frames internal to the free topos"? [1] This seems to be an extremely interesting theorem.
[1] André Joyal, “A crash ...
- 714
8
votes
1
answer
910
views
Questions about SGA 4
What is Deligne's motivation in Appendice 9 of Exposé VI to prove that every coherent topos has enough points? For instance, does that have applications in étale cohomology (or other parts of ...
- 714
6
votes
1
answer
243
views
Stability properties of essential geometric morphisms
Notation.
$\mathsf{Topoi}$ is the bicategory of topoi, geometric morphisms and natural transformations between left adjoints.
$\mathsf{Topoi}_{\text{ess}}$ is the bicategory of topoi, essential ...
- 9,111
7
votes
0
answers
234
views
How much is known about the consistency strength of toposes and topos-like categories?
It's a well-known fact that the theory of a well-pointed topos with a natural numbers object (NNO) has the same consistency strength as MacLane set theory (also known as bounded Zermelo). There are ...
- 12.5k
10
votes
0
answers
391
views
How do properties of a partial order $\mathbb{P}$ affect the logic of the functor category $\mathsf{Set}^\mathbb{P}$?
$\DeclareMathOperator\true{\mathsf{true}}$I am very suspicious the answer to this (family of) question(s) is well-known, but I couldn't find anything after a bit of searching so I'll ask anyway.
I am ...
- 613
9
votes
1
answer
478
views
From Topoi to Grothendieck categories
This question is mostly about a reference request. Let $\mathcal{E}$ be a Grothendieck topos. I am looking for a reference of the following two facts. I am aware that $(2) \Rightarrow (1)$ by Gabriel-...
- 9,111
9
votes
1
answer
265
views
Object classifiers in 1-toposes
In a Grothendieck $\infty$-topos, it is known that, for arbitrarily large regular cardinals $\kappa$, there is a classifier for the class of relatively $\kappa$-compact morphisms. It is also easy to ...
- 1,071
16
votes
2
answers
2k
views
Categorification of probability theory: what does a "probability sheaf" tell us (if anything) about probability theory?
Disclaimer: I only have a superficial knowledge of what category theory and related subjects are concerned with.
So, my understanding is that category theory and related fields of higher mathematics ...
- 6,853
17
votes
1
answer
411
views
Topos extensions
In set theory, starting from a model $V$ of $ZFC$, a forcing notion $\mathbb{P}$, and a generic filter $G \subset \mathbb{P}$ over $V$, we can find a generic extension which is a model of $ZFC$ and is ...
- 32.1k
3
votes
1
answer
373
views
Universal property of the category of $\mathcal{S}$-sheaves and the definition of Topos
Let $\mathcal{C}$ be a small category; let $\mathcal{S}$ be any family of maps in $\text{Psh}\left(\mathcal{C}\right)$.
Call $X\in \text{Psh}\left(\mathcal{C}\right) $ an $\mathcal{S}$-sheaf when $\...
- 1,514
9
votes
1
answer
511
views
Free models of finitely presented essentially algebraic theories in elementary toposes?
The following result is well-known in folklore (I think), but I’ve been unable to find a reference in the literature:
Let $T$ be a finitely presented essentially algebraic theory, and $\newcommand{\...
- 21.3k
11
votes
2
answers
664
views
'Continuity' of the étale topos
In certain concrete situations, I can show that the small étale topos of an inverse limit of schemes is the inverse limit of the associated toposes, for example, if $X$ is a (qcqs) scheme relative ...
- 19.6k
4
votes
1
answer
241
views
Geometric morphism whose counit is epic
Is there a name for the class of geometric morphisms whose counits are epic (equivalently, whose direct images are faithful)?
This notably includes the class of inclusions/embeddings of toposes. By ...
- 519
7
votes
1
answer
405
views
A list of proofs of "Coherent topoi have enough points"
For my research I would like to read all the known proofs of the very classical result "Coherent topoi have enough points", by Deligne.
Ref 1: D3.3.13 in Sketches of an Elephant
provides ...
- 9,111
10
votes
1
answer
723
views
Reference request about “internal language of categories”
I've tried to become familiar with the so-called "internal language of a category" for the last months. However, I'm still not confident enough when, for instance, I find a subobject (of a given ...
- 203
5
votes
1
answer
251
views
Smash product and the integers in a Grothendieck $(\infty, 1)$-topos
Let $\mathcal{H}$ be a Grothendieck $(\infty,1)$-topos. According to this page in nlab, for any $X \in \mathcal{H}$, the suspension object $\Sigma X$ is homotopy equivalent to the smash product $B \...
- 3,019
5
votes
0
answers
156
views
Reference Request: Cohomology and limits of coherent topoi. (Non-abelian case)
SGA 4 VI Discusses finiteness conditions one can impose on topoi to make limits behave correctly. I am not that familiar with SGA but it is my impression that this expose only discusses abelian ...
- 435
9
votes
0
answers
545
views
Topos with enough projectives
It is often observed that every presheaf topos has enough projectives, as a corollary of the result that representables are projective and every presheaf is a colimit of representables. We also have ...
- 519
13
votes
1
answer
570
views
Configuration spaces, Ran spaces, free semilattices, Vietoris spaces and power objects
These are five important constructions and I would like to know how they are related.
The $n$th unordered configuration space of a space $X$ is
$$
\operatorname{UConf}_n(X):=\{\text{embeddings of $\{...
- 18.4k
12
votes
1
answer
878
views
Higgs paper ``A category approach to Boolean valued set theory''
As Philip Scott says
about Denis Higgs:
In category theory, he wrote an influential and beautiful long paper, "A
category approach to Boolean valued set theory", which initiated many
early students ...
- 32.1k
2
votes
1
answer
481
views
Open and Dense Substack
I am looking for a definition of open and dense substack of a Deligne-Mumford stack $\mathcal X$. I have encountered this notion many times, but I am not able to find any references in which dense ...
- 123
4
votes
0
answers
392
views
On nearby cycle sheaves and a 2-fibered product of topoi
In SGA7 Exposé XIII, Deligne introduces an algebraic theory of nearby cycle sheaves and vanishing cycle sheaves on schemes over the $S$, where $S$ is the spectrum of a Henselian discrete valuation ...
- 181
6
votes
0
answers
477
views
The topos of a graph
If $G$ is, for example, a finite directed graph, one can attach to it a topos $T_G$ whose objects are "$G$-sheaves". A $G$-sheave $F$ is the data of:
For each verticies $x$ a set $F(x)$, for each ...
- 42.4k
12
votes
2
answers
3k
views
Alternatives to "Sketches of an Elephant" Volume 3
The third volume of Peter Johnstone's massive compendium of topos theory, "Sketches of an Elephant", is yet to be published. The volume is supposed to discuss cohomology and mathematical universes in ...
- 3,309
9
votes
1
answer
402
views
Cocontinuous product-preserving functor between Grothendieck toposes
What is an example of a functor $$F : \mathcal{C} \to \mathcal{D}$$ between two Grothendieck toposes which preserves colimits and finite products, but is not left exact (i.e., does not preserve ...
- 5,482
3
votes
0
answers
170
views
Internal language type of power objects
It is a basic fact that in a category with finite limits the following are equivalent
Each object $X$ has a (membership relation to a) power object $PX\times X\hookleftarrow\ni_X$ subject to the ...
- 572
17
votes
2
answers
2k
views
Foundations of topology
I recently went to a talk of Oleg Viro where he expressed his dissatisfaction with current foundations of differential topology parallel to what has been discussed here.
Also some time ago I read ...
- 1,190
36
votes
0
answers
1k
views
Grothendieck's "List of classes of structures"
In Lawvere's article Comments on the Development of Topos Theory, the author writes:
Similarly, Grothendieck and others unerringly recognized which kinds of mathematical structures are 'preserved ...
- 10.5k
15
votes
3
answers
2k
views
Ordinals in constructive mathematics ? (references)
I'm looking for references presenting a constructive treatment of the theory of ordinals. By constructive I mean valid in the internal logic of a topos (so no axiom of choice and no law of excluded ...
- 42.4k
12
votes
2
answers
702
views
Definition of internal field objects
Let $\mathcal{C}$ be a category with finite limits and a strictly initial object $\mathbf{0}$. The final object is denoted by $\mathbf{1}$. I propose the following definition of a field object ...
- 63.1k
1
vote
0
answers
116
views
Properties of the internal language of the category of sheaves
Consider a simple case of set-valued sheaves on some topological space $X$, $\operatorname{Sh}(X)$. All of these are Grothendieck toposes but clearly not all of them are equivalent.
Is there an ...
- 408
4
votes
0
answers
113
views
When is the localic reflection of a topos discrete?
Recall that the inclusion of locales into topoi has a left adjoint, called the localic reflection. It sends a topos $\mathcal{E}$ to the poset of subobjects of the terminal object, which is a locale. ...
- 15.5k
7
votes
0
answers
315
views
Generalizing prime numbers to product-indecomposable objects in toposes
Any object $A$ in any topos decomposes as $A\times1$ and $1\times A$, and in $FinSet$ objects with no other product decompositions are "prime numbers". Is there any extension of the theory of product-...
- 457
0
votes
0
answers
230
views
Is there a translation invariant measure on an infinite dimensional space 'without points'?
This is just a reference request. I thought I'd come across a paper demonstrating that there is a translation invariant measure on an infinite-dimensional space without 'points' whilst browsing the ...
- 2,369
7
votes
0
answers
518
views
Where else has Proposition B1.3.17 in the Elephant been proved?
(I asked the same question here and got some helpful comments, but thought I'd re-ask in case I get a more direct response.)
This is a sort of reference request. Proposition B1.3.17 in Johnstone's ...
- 497