1
vote
1answer
87 views

Arrows-only implication & disjunction in $\mathbf{Set}.$

Just before the truth-arrows in a topos subsection of Goldblatt's "Topoi: A Categorial Analysis of logic," descriptions of the truth functions $\Rightarrow$ and $\smallsmile$ are given in ...
0
votes
1answer
99 views

Understand the $\operatorname{Hom}$ Functor.

Via Wikipedia I see that $\operatorname{Hom}_C(A,-): C \rightarrow \textbf{Set}$ a covariant functor which maps each object $X$ in $C$ to the set of morphisms $\operatorname{Hom}_C(A,X)$. I am trying ...
1
vote
1answer
99 views

What is the relationship between the second isomorphism theorem and the third one in group theory?

The second isomorphism theorem [wiki] in group theory is as follows: Let $G$ be a group. $H \triangleleft G, K \le G$. Then: $HK \le G$, $(H \cap K) \triangleleft K$, and $K/(H \cap K) ...
0
votes
2answers
51 views

What is a zero morphism in an abelian category

I am trying to familiarize myself with some basic category theory and I am getting confused with what a $0$-morphism is. If we are in category of say $k$-vector spaces then I am guessing ...
5
votes
2answers
112 views

What does the Yoneda lemma say for the identity functor and finite sets?

So I try to plug in the simplest arguments into the Yoneda lemma and see how to interpret it. I'll try it for the identity functor and the category of finite sets, in particular, I use an three ...
3
votes
1answer
52 views

Difference between the simplicial nerve and the nerve of a simplicial category

In Jacob Lurie's Higher Topos Theory book, he defines the following notion of a simplicial nerve: Definition 1.1.5.5. Let $\mathcal{C}$ be a simplicial category. The simplicial nerve ...
8
votes
3answers
137 views

Intuition or Motivation behind definition of Homomorphism - Fraleigh p. 29

p.29: A binary algebraic structure is a set $S$ together with a binary operation $*$ on $S$ and is denoted $<S, *>$ p.29: Let $<S,*>$ and $<S',*'>$ be binary algebraic ...
4
votes
1answer
97 views

How does this definition capture the intuitive notion of an algebra?

On page 15 of this document, the author writes: Definition 1.1.1. Let $\mathcal{E}$ be any category. Given an endofunctor $\Gamma : \mathcal{E} \rightarrow \mathcal{E}$, a $\Gamma$-algebra ...
0
votes
1answer
47 views

Definition for the action of a category on a set.

I'm trying to understand the definition of the action of a category on a set which is given in nLab, more particularly the first one. If one has a functor $\rho: C \to Set$, one takes the set S as the ...
4
votes
1answer
44 views

Do there exist faithful functors $\mathrm{Grp} \rightarrow \mathrm{Set}$ that aren't naturally isomorphic to the underlying set functor?

I'm trying to get some intuition for the notion of natural isomorphism. To that end, my question is: do there exist faithful functors $\mathrm{Grp} \rightarrow \mathrm{Set}$ that aren't naturally ...
1
vote
1answer
225 views

Adjunctions via Universal Arrows: Understanding a Proof.

I've having trouble understanding something in Turi's Category Theory Lecture Notes from The University of Edinburgh, which can be found here. It's the proof of Theorem 7.1, part (3). Here's the ...
5
votes
1answer
211 views

Foundation of category theory

In the first pages of "category theory for the working mathematician" Saunders claims that category can be introduced, without set theory, as objects and arrows without some "operations" satisfying ...
2
votes
1answer
234 views

Intuition behind the definition of Adjoint functors

I think of adjoint functors as some sort of inverses. So, the first part of the definition looks reasonable that there exists natural transformations $$\epsilon : FG \rightarrow 1_C$$ $$\eta : 1_D ...
7
votes
1answer
446 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
118 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 ...
6
votes
1answer
348 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 ...
7
votes
2answers
301 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
4answers
763 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 ...
9
votes
2answers
362 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 ...
12
votes
3answers
446 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' ...
89
votes
5answers
5k 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 ...
18
votes
2answers
576 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 ...
28
votes
4answers
2k 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 ...
24
votes
4answers
2k 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 ...