Various structures are studied in category theory using properties of objects and morphisms between them. Many construction are special cases of categorical limits and colimits (e.g. products in various categories). The notions of functor and natural transformations are very important in category ...
16
votes
0answers
310 views
Abstract nonsense proof of snake lemma
During my studies, I always wanted to see a "purely category-theoretical" proof of the Snake Lemma, i.e. a proof that constructs all morphisms (including the snake) and proves exactness via universal ...
0
votes
1answer
45 views
Relations as arrows and as objects - what are the arrows in the latter?
Since a relation $R$ from $X$ to $Y$ defined as a subset of $X \times Y$, the category of sets and relations is just that: the objects are sets and the arrows are the relations.
Is there a generally ...
1
vote
1answer
103 views
Inverse of power-set functor?
In his answer to this Q: How to interpret $1 \to 0$ in $\mathbf {Set^{op}}$, and $\mathbf {Set^{op}}$ itself? Zhen Lin proposed that $\mathbf {Set^{op}}$ is naturally equivalent to the category of ...
2
votes
2answers
92 views
Final object in fields of characteristic $ 0 $?
In his answer to this question: Category of Field has no initial object, Arturo Madigin indicated that the field of rational numbers is the initial object in the category of fields of characteristic $ ...
1
vote
0answers
111 views
Group of Isomorphisms of a Groupoid
Write a careful proof that every group is the group of isomorphisms of a groupoid. In particular, every group is the group of automorphisms of some object in some category.
First can someone tell me ...
-7
votes
2answers
108 views
Objects in Categories
I need help to structure the following proof. I now know the definitions of each of the terms but I still do not know how to do the proof.
Prove that a final object in a category $C$ is initial in ...
6
votes
2answers
70 views
Internalising the functor action on morphisms (e.g. to exponential objects)
Part of what it means to be a functor between two categories is to have a map of morphisms e.g. $F$ sends $f: A \to B$ to $Ff: FA \to FB$.
Suppose $F$ is a functor from a category to itself, and that ...
4
votes
2answers
69 views
Eilenberg Moore category
I've been trying to code up the Eilenberg-Moore category for a monad in Haskell.
As I understand it, given a category $C$ and a monad $(T,\eta,\mu)$ on $C$ we build the Eilenberg-Moore category $C^T$ ...
9
votes
3answers
293 views
What is category theory useful for?
Okay so I understand what calculus, linear algebra, combinatorics and even topology try to answer, but why invent category theory? In wikipedia it says it is to formalize. As far as I can tell it sort ...
0
votes
0answers
88 views
Long Exact Sequence on Homology in an Abelian Category
Let $\mathcal A$ be an abelian category and let $0 \xrightarrow{} X \xrightarrow f Y \xrightarrow g Z \xrightarrow{} 0$ be an exact sequence of chain complexes in $\mathcal A$. I am using the ...
3
votes
2answers
89 views
Coproducts and products of modules
I've just looked at the book "A Course in Homological Algebra" ( by Hilton and Stammbach) . They show the universal property of the direct sum (coproducts) using injections and the universal property ...
9
votes
1answer
167 views
Completion as a functor between topological rings
In the following all rings are assumed to be commutative and unitary.
Preliminaries:
For any topological ring $R$ we can form its completion $\widehat{R}$ by taking all Cauchy sequences modulo null ...
1
vote
4answers
223 views
Can we modify ETCS to handle structures directly, as objects in their own right?
I have completely rewritten this question; thus, some of the comments/answers may no longer be relevant.
The elementary theory of the category of sets (hereafter, ETCS) is an axiomatic approach to ...
3
votes
1answer
52 views
Are bicategories and lax 2-categories the same?
My question is that whether the definition of bicategories is the same as the definition of lax 2-categories.
I heard that they are both week versions of 2-categories.
Are they the same? If not, how ...
0
votes
1answer
41 views
Is this thing K-finite?
This is related to this question:
Freyd's Geometric Finiteness : An Example Computation
I've essentially reduced the problem to the following question:
Equip $\mathbb{N}$ with the discrete ...
4
votes
2answers
57 views
Is there a “partial function” approach to subobjects in category theory?
Given a relation $f : X \rightarrow Y$, lets define that the source of $f$ is $X$, and that the domain of $f$ is the set of all $x$ such that there exists $y \in Y$ satisfying $(x,y) \in f$. Thus the ...
9
votes
6answers
160 views
Examples of categories where morphisms are not functions
Can someone give examples of categories where objects are some sort of structure based on sets and morphisms are not functions?
3
votes
1answer
62 views
Pullbacks and transpose map
Given maifolds $M,N$ and a smooth map $\phi:M \to N$, and a smooth function $f:N \to \mathbb{R}$, we have the pullback of $\phi$ by $f$ to be the function $\phi^* f = f \circ \phi : M \to \mathbb{R}$. ...
1
vote
1answer
39 views
Isomorphism between (source of) kernels of parallel arrow of a pullback square, by adjunction
Let $\mathcal A$ be an abelian category, $\alpha_1 \colon A_1 \to B$, $\alpha_2 \colon A_2 \to B$ two morphisms and $A_1 \leftarrow A_1\times_B A_2 \to A_2$ their pullback (with morphisms, say, $p_1, ...
7
votes
2answers
115 views
Products and pullbacks imply equalizers?
I was reading Herrlich & Strecker's Category Theory, and there is a theorem called The Canonical construction of Pullbacks which states that if a category has products and equalizers, then it has ...
4
votes
0answers
72 views
Freyd's Geometric Finiteness : An Example Computation
In his paper "Numerology in Topoi" available here:
http://www.tac.mta.ca/tac/volumes/16/19/16-19abs.html
Peter Freyd defines an object $A$ in a topos $\mathcal{E}$ to be geometrically finite if ...
0
votes
1answer
31 views
Sections, Transversals and Quotient Maps
Here I read:
Given a quotient space $\bar X$ with quotient map $\pi\colon X \to \bar X$, a section of $\pi $ is called a transversal.
I asked myself how such sections are possible. It must be a ...
4
votes
1answer
94 views
why split epi and mono implies iso?
I was doing some exercises on the definitions of epics, monos, split monos, etc..., and I asked myself that if you could take, for instance an epi which is mono, and then deduce it is an iso, which is ...
2
votes
2answers
80 views
MacLane's Abelian Categories Chapter
I have just started reading MacLane's chapter Abelian Categories in Categories for the Working Mathematician. I am stuck in the second page, where he discusses the equivalence relation on a set of ...
1
vote
2answers
63 views
Is there a category-theoretic perspective on induced functions?
Let $X$ and $Y$ denote sets. Given a function $f : X \rightarrow Y$ and a natural number, there is an induced function $g : X^n \rightarrow Y^n$ defined by $g(x_1,\cdots,x_n) = (f x_1,\cdots,f x_n).$
...
6
votes
3answers
154 views
In what sense is the forgetful functor $Ab \to Grp$ forgetful?
One sometimes hears about "the forgetful functor $Ab \to Grp$." Given that the image of an object under this functor is still abelian, in what sense is this "forgetful"?
3
votes
1answer
51 views
Property kept under base change and composition is preserved by products
The following is true? Why?
Let $P$ be a property of morphisms preserved under base change and composition. Let $X\to Y$ and $X'\to Y'$ be morphisms of $S$-schemes with property $P$.
Then the unique ...
2
votes
1answer
65 views
adjunction relation
Assume I have simplicial sets $X:\Delta^{op}\rightarrow Set$ and $Y:\Delta^{op}\rightarrow Set$, then I can form the simplicial set
$\operatorname{Hom}(X,Y)_n := ...
5
votes
1answer
93 views
How we can understand one category is small
"A category is said to be small if its objects form a set."
Now one question is in my mind and that is although we know lots of sets and always working with them, but how we can show a class of ...
1
vote
1answer
84 views
Is there a categorical construction of the general linear group?
This question is related to the answer of Qiaochu in this one. Since the object $X=\mathbb{F}_2^2$ generates the category of vector spaces of dimension $2^n$ over $\mathbb{F}_2$, and since we know ...
7
votes
1answer
99 views
Projective objects in the category of rings
What are the projective objects in the category of rings with identity ?
Remarks:
The only projectives I could find so far are $\{ 0\}$ and $\mathbb{Z}$.
If $R$ is projective and ...
3
votes
1answer
56 views
What does it mean for a Category to have equalizers or/and pullbacks?
I know the definitions of what pullbacks and equalizars mean, but I don't know what it means that a given category $\mathfrak C$ has pullbacks or equalizers.
Thanks
3
votes
0answers
48 views
Terminology Question: Precompose vs Compose?
I was wondering if there was a standard convention on what 'precompose' means compared to 'compose', as I am often confused between the two when all sorts of text casually use both terminologies. For ...
1
vote
0answers
37 views
Good reference for co-groups, perspective of co-algebra applications
There are lot of applications of state transition systems STS (computer science, planning problems in robotics and so on) and lot of algorithms are devised, but the mathematical background for STS is ...
2
votes
1answer
138 views
A pedantic question about defining new structures in a path-independent way.
Sometimes there are multiple equivalent ways of defining the same structure; for example, topological spaces are determined by their open sets, but also by their closed sets. I'm looking for a way of ...
0
votes
1answer
62 views
Name/notation for the subgroup generated by all stabilizers
Say we have a group $G$ acting on a set $X$.
I'm interested in the subgroup generated by all isotropy groups $G_x$, and looking for a designation for it.
Thanks in advance!
PS1: I thought about the ...
1
vote
1answer
49 views
Definition of monoids in slice category
Can someone please tell me what would be an appropriate definition of an internal monoid in the slice category?
Or better yet, suppose you have an object $p : X \rightarrow A \in \mathcal{C}/{A}$ ...
2
votes
2answers
76 views
Does an adjoint pair fix a unit/counit pair?
From Ravi Vakil, Fundations of Algebraic Geometry.
I want to ask if anyone can give a hint in how to prove Execrise 1.5.B(page 43). I tried to draw the diagram for half an hour but the resulting ...
4
votes
2answers
144 views
If $gf$ is an equalizer , is $f$ an equalizer?
Suppose $gf$ is an equalizer in a category $\mathfrak C$, I think that $f$ not necessarly is an equalizer, but I don't know how to come up with a counterexample; i've really tried it so hard. Thanks ...
5
votes
1answer
79 views
What is the minimum required background to understand articles in the nLab?
I am interested in learning more about the nLab categorical perspective on several mathematical subjects such as topology and logic, but found that my understanding of category theory was not ...
1
vote
0answers
53 views
Are finitely additive measures 'topological'?
The category of measurable spaces are
topological over $Set$ in that they support initial & final structures similarly to that topological spaces.
A measurable space is a set supporting ...
2
votes
1answer
58 views
Why $Kom(\mathcal{A})$ may not be triangulated, while $D(\mathcal{A})$ may not be abelian?
Let $\mathcal{A}$ be an abelian category, let $Kom(\mathcal{A})$ be the category of complex with a shift functor $T$, and Let $D(\mathcal{A})$ be the derived category of $\mathcal{A}$.
Why:
(1) ...
3
votes
2answers
100 views
Category-Theory and the modelling of $n$-ary functions (especially the $0$-ary functions)
Hi have a questioning regarding the modelling of $n$-ary functions and constants.
First in the category of sets I know that the empty set $\emptyset$ is initial because for every other set $X$ there ...
4
votes
2answers
88 views
Is duality an exact functor on Banach spaces or Hilbert spaces?
Let $V,V',V''$ and $W$ be vector spaces over $k$. Then, it is known that $\operatorname{Hom}(\cdot,V)$ is a contravariant exact functor, i.e. for each exact sequence
$0\to V'\to V\to V'' \to 0$,
and ...
5
votes
0answers
103 views
Closed model categories in the sense of Quillen [1969] vs the modern sense
The modern definition of (closed) model category differs in two ways from Quillen's 1969 definition:
Model categories are now required to be complete and cocomplete, whereas Quillen only asked for ...
0
votes
1answer
38 views
Mono/Epi of sections of presheaves
Let $\mathsf{C}$ be a category with initial and terminal objects, and $\phi:\mathscr{F}\to\mathscr{G}$ a morphism of presheaves on $X$ taking values in $\mathsf{C}$. I have a rather messy proof that ...
0
votes
3answers
147 views
Reference request - being rigorous about a common abuse of notation.
I've completely rewritten this question, in accordance with this advice.
As a motivating example, suppose we're working in ETCS. Let $\bar{1}$ denote the canonical singleton set, and assert that by ...
5
votes
4answers
209 views
Real world applications of category theory
I was reading some basic information from Wiki about category theory and honestly speaking I have a very weak knowledge about it. As it sounds interesting, I will go into the theory to learn more if ...
4
votes
3answers
107 views
Saying $a \in b$ in category theory
Suppose I have a category $C$ of sets, and $a,b \in C$. How can I express, in the language of category theory, that $a \in b$? (To clarify: the objects of $C$ are actually sets, and I want to express ...
2
votes
1answer
47 views
Is there any non-trivial relationship between kernels & kernel pairs?
Kernels are inspired by group theory, and kernel pairs by a similar concept in monoids where kernels aren't sufficient to capture the information necessary for the first isomorphism theorem.
When a ...
