Tagged Questions
6
votes
1answer
211 views
Understanding the three isomorphism theorems
I have learnt the following three isomorphisms for a while but without true understanding:
A group homomorphism $\phi:G\to G'$ can be decomposed into ...
3
votes
3answers
98 views
Intuition for Coconstant morphisms
A constant morphism $f \in \mathrm{Hom}(X,Y)$ is a morphism such that for any object $Z$ and any morphisms $g,h \in \mathrm{Hom}(Z,X)$, $f \circ g = f \circ h$. This is very easy to grasp and one can ...
4
votes
1answer
183 views
Canonical example of a cosheaf
Sheaves can, like all modern mathematical constructions and abstractions, be counterintuitive beasts but, like all such constructions, a few examples can allow one to visualise them simply as a ...
6
votes
2answers
206 views
Intuition for limits
My basic intuition for limits/colimits was "limits suck up, colimits suck down". Now, having seen colimits used in presheaf categories, algebraic geometry, and topology, I have much clearer intuition ...
30
votes
3answers
531 views
Why do we look at morphisms?
I am reading some lecture notes and in one paragraph there is the following motivation: "The best way to study spaces with a structure is usually to look at the maps between them preserving structure ...
4
votes
2answers
199 views
Importance of 'smallness' in a category, and functor categories
I feel like, having spent a little time doing category theory now, this is probably a silly question, but I keep coming up to many things (definitions, examples etc.) where smallness is required. I ...
5
votes
1answer
269 views
Mathematical structures
Preamble: My previous education was focused either on classical analysis (which was given in quite old traditions, I guess) or on applied Mathematics. Since I was feeling lack of knowledge in 'modern' ...
67
votes
5answers
3k views
In (relatively) simple words: What is an inverse limit?
I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me.
The inverse limit. I tried to ask one of ...
14
votes
2answers
443 views
Categorical description of algebraic structures
There is a well-known description of a group as "a category with one object in which all morphisms are invertible." As I understand it, the Yoneda Lemma applied to such a category is simply a ...
25
votes
4answers
1k views
Can someone explain the Yoneda Lemma to an applied mathematician?
I have trouble following the category-theoretic statement and proof of the Yoneda Lemma. Indeed, I followed a category theory course for 4-5 lectures (several years ago now) and felt like I understood ...
21
votes
4answers
1k views
Simple explanation of a monad
I have been learning some functional programming recently and I so I have come across monads. I understand what they are in programming terms, but I would like to understand what they are ...
