All Questions
Tagged with foundations ct.category-theory
59 questions
6
votes
2
answers
319
views
Set theoretical foundations for derived categories
A modern approach to derived functors, that has been shown to be useful in a number of different circunstances is that of a derived category (see the book by Yakutieli, for example, here).
However, it ...
- 3,318
29
votes
3
answers
3k
views
Are there substantive differences between the different approaches to "size issues" in category theory?
In category theory, there are different ways to approach the "size issues" that crop up when we try to formalise the subject in axiomatic set theory. As far as I can tell, there are two main ...
- 961
11
votes
1
answer
1k
views
Existence of skeletons in ZFC
Influenced by this question from a fellow lagomorph, I would like to get to the bottom of existence of a skeleton of a category. I want to stay in ZFC, so I do not assume the global axiom of choice. ...
- 12.3k
3
votes
2
answers
331
views
On the definition of small categories in SGA4
We assume ZFC+U.
A category is an ordered pair $(\operatorname{Ob} \mathcal{C},\operatorname{Mor} \mathcal{C},\operatorname{dom},\operatorname{codom},e,∘)$ of sets (not classes) and maps satifying ...
- 405
16
votes
2
answers
2k
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Why do we care about small sets?
I have been a user of category theory for a long time. I recently started studying a rigorous treatment of categories within ZFC+U. Then I become suspecting the effect of the smallness of sets.
We ...
- 405
1
vote
1
answer
118
views
Is it possible to set up multiple automorphisms over a structureless object inside single-sort defined category?
I was trying to understand the behaviour of the primitive equality (=) in the axiomatization of category, which takes morphisms as primitives and objects as derivatives in bijection to identity ...
- 13
7
votes
3
answers
3k
views
Are the categories of sets, abelian groups, and commutative rings unique?
Are the categories of sets, abelian groups, and commutative rings unique? Independence results like the independence of the generalized continuum hypothesis, the Whitehead problem, and the global ...
6
votes
1
answer
429
views
Joyal arithmetic universes and the Box operator
Last month @godelian, alias Christian Espíndola, in a FOM post has mentioned Joyal's proof of Godel's second incompleteness via the so-called Arithmetic Universes, introduced by Joyal around 1973, ...
- 7,853
7
votes
1
answer
429
views
Does the small object argument need replacement?
Does one need the axiom of replacement in the small object argument and in the transfinite construction of free algebras?
My motivation for the question is that I heard that the axiom of replacement ...
- 395
13
votes
0
answers
362
views
Context of set theory in which one doesn't have to worry about size issues
In this beautiful talk by Colin McLarty, McLarty quotes Grothendieck:
It would be nice to have a context where one doesn't add any real axioms to set theory, and yet one can work with categories ...
- 395
15
votes
2
answers
959
views
Can the opposite of an elementary topos be an elementary topos?
This question is not really about elementary topoi, it is much more about a category $(\mathcal{E}, \Omega)$ admitting a subobject classifier, or about a category with power objects, you can choose ...
- 9,111
10
votes
1
answer
451
views
Is material set theory conservative over structural set theory?
Suppose a statement $\phi$ that doesn't use the global $\in$-relation or the global $=$-relation in an essential way is provable in some material set theory, say bounded Zermelo with choice. (So that ...
- 101
17
votes
2
answers
2k
views
When the definition of a set starts to matter in category theory
In most introductory courses to category theory, the precise definition of a set is more-or-less ignored. The idea being that all basic results in the subject hold for any reasonable definition of a ...
- 141
0
votes
2
answers
1k
views
Has there been any serious attempt at a "circular" foundation of mathematics?
As far as I know, there is no published attempt at a "circular" foundations of mathematics though I'ave seen it noted by many category theorists and logicians without in-depth analysis, e.g ...
- 95
13
votes
3
answers
1k
views
Elementary theory of the category of groupoids?
One axiomatisation of set theory, the Elementary Theory of the Category of Sets, or ETCS for short, comes from category theory and states that sets and functions form a locally cartesian-closed, ...
user173426
63
votes
4
answers
7k
views
When size matters in category theory for the working mathematician
I think a related question might be this (Set-Theoretic Issues/Categories).
There are many ways in which you can avoid set theoretical paradoxes in dealing with category theory (see for instance ...
- 3,318
0
votes
0
answers
183
views
A syntax independent theory of categories
The classic way to encounter the theory of categories is via Set Theory via the typical definition we see for categories. We see all kinds of categories that are equivalent to the category of small ...
- 1,313
11
votes
1
answer
1k
views
Are categories special, foundationally?
Some folks over at nLab want to use categories as a foundation for all of mathematics, I'm guessing as an alternative to sets. Sets work fine, and so do categories, so I have started wondering what ...
- 237
7
votes
2
answers
2k
views
What is a good definition of a mathematical structure?
At the moment I am writing a textbook in Foundations of Mathematics for students and trying to give a precise definition of a mathematical structure, which is the principal notion of structuralist ...
- 41.8k
74
votes
8
answers
14k
views
Category theory and set theory: just a different language, or different foundation of mathematics?
This is a question to research mathematicians, as well as to those concerned with the history and philosophy of mathematics.
I am asking for a reference. In order to make the reference request as ...
- 6,917
38
votes
4
answers
6k
views
Could groups be used instead of sets as a foundation of mathematics?
Sets are the only fundamental objects in the theory $\sf ZFC$. But we can use $\sf ZFC$ as a foundation for all of mathematics by encoding the various other objects we care about in terms of sets. The ...
- 3,137
11
votes
0
answers
342
views
Categorial foundations via "categories of algebras"
There are categorical foundations for mathematics axiomatizing the category of sets (Lawvere's ETCS), cartesian closed categories (type theory), and the category of spaces (homotopy type theory). ...
user30211
8
votes
3
answers
2k
views
How much of concrete mathematics can be expressed in the language of category theory?
Question 1
How much of group/ring/lattice/... theory can be expressed in purely categorical terms (only using the notions object, morphism, identity morphism, and composition), that is, as properties ...
4
votes
0
answers
140
views
Is well-pointedness the reason that the internal/external distinction seems not to apply to $\mathbf{Set}$?
When reasoning about the category of sets, we usually don't have to worry about the internal/external distinction. For example, if $f : X \rightarrow Y$ is a morphism of sets, then $f$ is either ...
- 3,793
6
votes
1
answer
993
views
Which branches of mathematics can be done just in terms of morphisms and composition?
Consider the first-order language $L_{\omega\omega}$ of the signature $L:=\{\mathrm{dom}, \mathrm{cod}, \mathrm{comp}\}$, where $\mathrm{dom}$ and $\mathrm{cod}$ are unary function symbols and $\...
user103598
47
votes
7
answers
7k
views
What is an explicit bijection in combinatorics?
A standard way of demonstrating that two collections of combinatorial objects have the same cardinality is to exhibit a bijection between them. Browsing through some examples (here, there, yonder) ...
- 48.8k
12
votes
1
answer
1k
views
ZF(C) and category theory
Is there an axiomatisation of some kind of category theory and a definition of sets in this framework such that the axioms of ZF resp. ZFC are theorems?
- 1,648
2
votes
0
answers
305
views
Does this axiomatic system satisfy requirements for founding mathematics?
In this article, the author, F.A.Muller, suggests criteria for a founding theory of mathematics (pp:14-16). The author proposes $ARC$ Class Theory to embody these requirements. The motivation is ...
- 11.3k
4
votes
0
answers
215
views
Formalization and set-theoretic issues in the definition a functor category
Universes are used in a category theory to handle size-issues. Assuming Grothendieck's axiom UA that
every sets is contained in some universe
there are two approaches to $U$-smallness given a ...
- 1,115
6
votes
3
answers
445
views
How do we formally construct the successor universe $\mathscr{U}^+$ of a universe $\mathscr{U}$ in $\mathsf{ZFC}$?
A set $\mathscr{U}$ is a universe if the following conditions are met:
For any $x \in \mathscr{U}$ we have $x \subseteq \mathscr{U}$
For any $x,y \in \mathscr{U}$ we have $\{x,y\} \in \mathscr{U}$,
...
- 1,115
12
votes
1
answer
601
views
Translating Grothendieck axiom UB into ZFC
In SGA 4, Grothendieck introduced a set-theoretic device called a Grothendieck universe. He and his collaborators worked in Bourbaki set theory, which is practically similar to $\mathsf{ZFC}$ but ...
- 1,115
5
votes
2
answers
474
views
Limits, colimits and universes
For many purposes in category theory, we consider limit and colimits of diagrams $F\colon\mathsf{J\to C}$ where $\mathsf{J}$ is small category, that is, a category where the classes of objects and ...
- 1,115
5
votes
1
answer
309
views
Does the notion of a compactly generated space (or $k$-space) depend on the choice of universe?
We recall the notion of a $k$-space (or compactly generated space) to fix our notations. For every topological space $X$, we can define a category $\mathfrak{M}_X$. The class of objects of $\mathfrak{...
- 1,586
5
votes
1
answer
470
views
What is the definition of a $\mathcal{U}$-category?
Let $\mathcal{U}$ be a universe. The adaptation of the concept of a locally small category to universes is a $\mathcal{U}$-category.
There are two definitions of $\mathcal{U}$ category I've met.
$(1)$...
- 1,115
7
votes
1
answer
665
views
Enhancing Grothendieck's universes and Grothendieck's axiom: Feferman's universe
A Grothendieck's universe is such a set $U$ so that
$\forall x \in U, x \subseteq U$,
$\forall x,y \in U, \{x,y\} \in U$,
$\forall x \in U, \mathcal{P}(x) \in U$,
given a family $(X_i)_{i \in I}$ ...
- 1,115
28
votes
0
answers
2k
views
Is Feferman's unlimited category theory dead?
In 2013 Solomon Feferman in Foundations of unlimited category theory: what remains to be done (The Review of Symbolic Logic, 6 (2013) pp 6-15, link) laid out three desirable axioms for "...
- 1,446
4
votes
0
answers
424
views
What are the requirements of a foundational theory?
There are multiple languages to describe all of mathematics, and there are some equivalences between them, some more successful then others.
My question is can we describe some requirements (in some ...
- 773
0
votes
1
answer
236
views
The Abstraction of Equality [closed]
In finitely presented groups, we can define equivalence classes simply by writing equations in the generators : $abc=d$. In this equivalence class we find elements like this $a(aa^{-1})bc$. We can ...
- 1,313
8
votes
1
answer
998
views
Category theory without axiom of choice
I'm looking for references on the development of (some of) Category theory without the axiom of choice. One possible axiom system (that, to me, seems the natural setting) is ZF + there are arbitrarily ...
- 773
6
votes
1
answer
375
views
What drawbacks are there to using NF(U) for category theory?
In category theory, you often run into what is known as "size" issues. That is, you run into the issue that the categories you try to define are too "big" to be sets, and so you need to use classes or ...
- 6,353
19
votes
3
answers
1k
views
Large categories vs. $\mathrm{U}$-categories: why is the loss of category-theoretic information inessential?
I've asked a related question about nine months ago here, however, apparently, I lacked expertise to ask the precise question I want to ask here, as I wish to revisit the matter of universes. I hope ...
- 1,115
8
votes
1
answer
403
views
Can ETCC/ETCS talk about 'size issues'?
In material set theories (like ZFC), one can prove that there is no set of all sets. Can one prove a similar statement in ETCS? This exact statement "there is no set x such that y in x for every set y"...
- 81
9
votes
1
answer
1k
views
How are material set theory and structural set theory related from the point of view of category theory?
In his comments to both cody and Nik Weaver regarding his answer to user7280899's mathoverflow question "What kind of foundation are mathematicians using when proving metatheorems?", Mike Shulman ...
- 6,132
1
vote
0
answers
223
views
What should one know about abstract sets and structural foundations?
Recently I came by accident across the book sets for mathematics by Lawvere. It says:
First we deplete the object of nearly all content. We could think of an idealized
computer memory bank that ...
55
votes
10
answers
11k
views
How should a "working mathematician" think about sets? (ZFC, category theory, urelements)
Note that "a working mathematician" is probably not the best choice of words, it's supposed to mean "someone who needs the theory for applications rather than for its own sake". Think about it as a ...
- 1,115
2
votes
0
answers
264
views
About the limitation by size
This could be a big post, so I'll try to summarize my thoughts and divide them into several questions.
When working in category theory, I used to choose the following definition. A category $C$ is ...
- 1,017
4
votes
1
answer
301
views
internalization of the concept of large and small category
I have been poking around the internet and nlab looking at the concept of large and small categories. My original focus was locally presentable categories of categories and I was thinking of finding ...
- 1,313
0
votes
0
answers
410
views
Set as a (strict) infinite-category?
First, let me say that I have no idea if such a post has its place here. However, I believe that the ideas I'm going to present are important. The goal of this thread is three fold:
1) trying to ...
- 438
36
votes
6
answers
6k
views
Who needs Replacement anyway?
The set theory ETCS famously comes without the Replacement axiom schema (or an equivalent) that is part of ZFC. One (to me, not apparently useful) set that one cannot build in ETCS is $\coprod_{n\in \...
- 35.4k
3
votes
0
answers
142
views
Hilb as a Colimit in the Category of Scott Complete Categories (foundations)
Here is a paper I found by Adamek that generalizes Domain theory into categories of categories called Scott Complete Categories. The category of Scott Complete categories is denoted SCC. For years, ...
- 1,313