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Results tagged with big-picture
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Questions designed to get an overview of a specific subject or body of results or to understand the relations among similar definitions, techniques or concepts appearing in different sub-fields of mathematics. While such questions by their very nature sometimes cannot be made very narrow and focused, it can be helpful to keep in mind that the design of MathOverflow does not make it a good fit for questions that are too broad.
10
votes
Equality vs. isomorphism vs. specific isomorphism
I think a great example of this is groupoid cardinality. The cardinality of a groupoid $G$ is defined to be
$$ \sum_{[x] \in \pi_0(G)} \frac{1}{|Aut(x)|} $$
If the groupoid is a discrete set, this re …
- 66.8k
34
votes
Equality vs. isomorphism vs. specific isomorphism
Suppose you have two categories C and D, and functors $F:C\to D$ and $G:D\to C$ such that for all $x\in C$, $G(F(x))$ is isomorphic to $x$, and for all $y\in D$, $F(G(y))$ is isomorphic to $y$. If yo …
- 66.8k
10
votes
What do you use categorical glueing/sconing/Freyd covers for?
I think the terminology "gluing" comes from the following example. Let X be a topological space, let U be an open subset of X, and let $K=X\setminus U$ be its complementary closed subset. Then the i …
- 66.8k
45
votes
Accepted
Why is Set, and not Rel, so ubiquitous in mathematics?
Regarding question 3, one can make an argument that actually the fundamental object is "Set together with Rel". The bijective-on-objects inclusion of Set into Rel is a categorical structure that can …
4
votes
Sober spaces vs. spatial frames-a big picture
Yes, it is always true that $Pt(F) \cong Pt(\mathcal{O}(Pt(F)))$ (i.e. $Pt(F)$ is sober), whether or not $F$ is spatial. Indeed, this follows formally from the fact that $\mathcal{O}(Pt(\mathcal{O}(X …
- 66.8k
41
votes
Why did Voevodsky consider categories "posets in the next dimension", and groupoids the corr...
The other answers are quite good and nothing is wrong with them, but lest a wrong impression be given (e.g. by the implicit suggestion that the idea of "categories as sets in the next dimension" in ea …
- 66.8k
29
votes
Why is the definition of the higher homotopy groups the "right one"?
I think there's something fundamental missing from all the other answers so far: the modern realization that topological spaces are distinct from $\infty$-groupoids.
Suppose you didn't know about hig …
12
votes
Locales as geometric objects
While Simon's answer is very good, I think one can also say something a little more along the lines of what you may be thinking. I haven't seen this written out in this way before, so I may have made …
- 66.8k