Various structures are studied in category theory using properties of objects and morphisms between them. Many constructions 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 ...

learn more… | top users | synonyms (2)

6
votes
1answer
385 views

Semidirect Products in category theory?

Here is a quote from the wikipedia entry on semidirect products of groups: There are also far-reaching generalisations in category theory. They show how to construct fibred categories from ...
13
votes
1answer
334 views

Is there a way to make tangent bundle a monad?

The tangent bundle functor $T: \mathbf{Diff} \to \mathbf{Diff}$ together with the bundle projection $\pi: T \Rightarrow 1_\mathbf{Diff}$ basically screams 'monad' at me, especially because both $\pi ...
1
vote
1answer
85 views

On the non-injectivity of forgetful functors on Ob

My book asserts that, for several common categories, such Vec, Grp, Top, etc., the forgetful functor is not injective on the category's class of objects, $\mathcal{Ob}$. I'm looking for examples ...
3
votes
1answer
143 views

Looking for a review of the book Monoids, Acts and Categories

I have been looking for a review of the following book: Monoids, Acts and Categories with Applications to Wreath Products and Graphs by Mati Kilp, Ulrich Knauer, Alexander V. Mikhalev but i was ...
2
votes
0answers
76 views

Making a Family of Problems into a Category and defining the Morphisms

This question relates to my previous question found here: Defining Category of Problems Let $\left\{\Pi_i \right\}_{i \in I}$ be a family of problems. Let the solution $u_i$ of $\Pi_i$ lie in some ...
2
votes
1answer
100 views

Defining Category of Problems

Let $\left\{\Pi_i\right\}_{i \in I}$ be a family of problems. Let problem $\Pi_i$ have solution $u_i$ lying in some solution space $X_i$. I am interested in making this set into a category. Is it ...
11
votes
1answer
479 views

Monic (epi) natural transformations

Let $C$ and $D$ be categories and let $F : C \rightarrow D$, $G : C \rightarrow D$ be two functors such that they are either both covariant or both contravariant. Under what most general hypotheses is ...
4
votes
1answer
91 views

Definition of a groupoid of fractions

The title sums it up : I am looking for a definition of "a groupoid of fractions for a category". I would be interested in any example someone might have as well...
10
votes
0answers
118 views

What is the decategorification of a triangulated category?

The decategorification of an essentially small category $\mathcal C$ is the set $\lvert\mathcal C\rvert$ of isomorphism classes of $\mathcal C$. If $\mathcal C$ carries additional structure, then so ...
7
votes
1answer
415 views

Category of Field has no initial object

Why the category of Field has no initial object? (see page $47$ of Horst Herrlich, George E. Strecker "Category Theory")
2
votes
2answers
511 views

Question on pullbacks and compositions

Wikipedia presents composition as a special case of pullbacks, but I can't quite reconcile that interpretation with the definition of pullback that I know. In its most general form, composition ...
1
vote
0answers
79 views

What is the name of this equivalence relation?

Given any sets $X \subseteq Y$, the relation given by: $$1_Y \; \cup \; (X \times X) \;\; \subseteq Y \times Y$$ (where $1_Y = \{ (y, y): y \in Y \}$ ) is an equivalence relation. Is there a name ...
5
votes
0answers
480 views

When is a pullback also a pushout?

The subject line says it all, but perhaps it would be more reasonable to split the question into two parts: 1) can a pullback diagram also be a pushout diagram?; if so, 2) can necessary and sufficient ...
1
vote
1answer
263 views

Every equalizer is monic

Theorem I of section 3.10 of Goldblatt's Topoi states that every equalizer is monic. I don't understand the proof given. For reference, it is: Suppose $i : e \rightarrow a$ equalizes $f,g : a ...
5
votes
3answers
589 views

Examples of categories where epimorphism does not have a right inverse, not surjective

Epimorphism is defined as following: $f \in \operatorname{Hom}_C(A,B)$ is epimorphism if $\forall Z. \forall h', h'' \in \operatorname{Hom}_C(B, Z)$ the following holds: $h' f = ...
4
votes
2answers
203 views

Can metric properties can be expressed in category theoretical terms?

A simple example: If you are given the category of Hilbert spaces with the bounded linear mappings as morphism sets, then dualization is a contravariant endofunctor. So we can talk about ...
0
votes
1answer
312 views

Meaning of 'pullback of a pullback square'

I'm trying to solve a problem which asks me to deduce that "the pullback of a pullback square is a pullback", using the result of http://ncatlab.org/nlab/show/pullback (under 'Pasting of pullbacks') ...
8
votes
2answers
440 views

What's the dual of “inverse image” in Set?

My book defines the "inverse image" as the pullback of a morphism $f:A\to B$ and a monomorphism $i:C \rightarrowtail B$. If the category under consideration is Set, and we interpret $i$ as the ...
0
votes
2answers
148 views

Canonical functor associated to a ring homomorphism

I've encountered this problem on my Non commutative algebra handouts wich says: given $R,S$ rings and $f:R\to S\:$ a ring homomorphism, define a canonical functor $$F:\textbf{Mod-S}\to ...
3
votes
1answer
153 views

Do all functors preserve split coequalisers?

I have a homework problem asking me "which kind of functors preserve split coequalisers?" - I have seen multiple online sources (such as the comments in ...
10
votes
2answers
797 views

Is the axiom of universes 'harmless'?

Usually when you start studying category theory you see the usual definition: a category consists of a class $Ob(\mathcal{C})$ of objects, etc. If you take ZFC to be your system of axioms, then a ...
4
votes
1answer
475 views

Category theory, a branch of abstract algebra?

In Steve Awodey's book on category theory, he claims the latter is a branch of abstract algebra. I've never seen such a classification before. Is this really correct?
2
votes
1answer
211 views

Category theory for describing systems?

In Rosetta Stone quantum description is interpreted from the category theory point of view. Systems (Hilbert spaces of wave functions) are objects and processes (linear operators) are arrows. But the ...
1
vote
2answers
514 views

Group as a category

Is it possible to define a group as a category? What exactly will be objects of this category and how will we say that every element should have an inverse?
9
votes
3answers
647 views

A Concrete Approach to Category Theory

Is there a way to learn Category Theory without learning so many concepts of which you have never seen examples?
12
votes
3answers
464 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' ...
-1
votes
2answers
219 views

When does $|\mathrm{Hom}(A,B)| = |\mathrm{Hom}(B,A)|$ imply $A \simeq B$?

In which categories does $|\text{Hom}(A,B)| = |\text{Hom}(B,A)| \neq 0$ and $|\text{Hom}(A,A)| = |\text{Hom}(B,B)|$ imply that $A$ and $B$ are isomorphic?
6
votes
1answer
291 views

Concrete Categories Where Epis are Just Surjections

Before I begin let me provide some background to fix notation/make the post more readable to interested outsiders. In a category $\mathscr{C}$ we say that a morphism $X\xrightarrow{f}Y$ is an ...
2
votes
2answers
642 views

Tensor product as a colimit

I've been dealing with category theory for three weeks now and we just started covering limits and colimits, meanwhile in my geometry class we defined the tensor product of vector spaces. Then I ...
3
votes
0answers
130 views

Simple Category Theory/Grps Problem

I'm doing a little homework but I think my brain has ceased to function late at night, was hoping you could help me out with what I am certain is a very simple group/cat theory problem. Similarly to ...
0
votes
1answer
73 views

Natural isomorphism of endofunctors

Let $A,B:C\to C$ be two endofunctors. I wonder if it follows from a natural isomorphism $A\circ A\xrightarrow{\cong}B\circ B$, that $A$ and $B$ are naturally isomorphic.
20
votes
4answers
1k views

Proving the snake lemma without a diagram chase

Suppose we have two short exact sequences in an abelian category $$0 \to A \mathrel{\overset{f}{\to}} B \mathrel{\overset{g}{\to}} C \to 0 $$ $$0 \to A' \mathrel{\overset{f'}{\to}} B' ...
1
vote
1answer
76 views

Decomposition of simplicial G-sets as a colimits of its simplicial G-subsets

Every simplicial set is the colimit of its finite simplicial subsets. Suppose $G$ be a finite discrete set. Is every simplicial $G$-set a colimit of its finite simplicial $G$-subsets? I'm ...
2
votes
1answer
121 views

Adjointness of the corresponding simplicial functors associated to an adjoint pair between categories

Suppose you have an adjoint pair of functors $F:C\to D$ and $G:D\to C$. Let ${\mathrm{Simp}}C$ and ${\mathrm{Simp}}D$ be the category of simplicial objects in the categories $C$ and $D$ respectively. ...
0
votes
1answer
167 views

Factorization of functors into full+faithful and object-bijective factors

I'm doing a problem to show that any functor $F: \mathcal{A} \to \mathcal{C}$ between categories can be factorized as $F_L: \mathcal{A} \to \mathcal{B},\,F_R: \mathcal{B} \to \mathcal{C}$, where ...
5
votes
1answer
89 views

Automorphisms and bicategories

I'm reading Lang's section on field theory and he stresses that, unlike typical "universal" constructions which are determined up to unique isomorphism, algebraic closures (and by extension, their ...
1
vote
1answer
51 views

Question on “distinguished morphisms”

For this question I will use $[A, B]$ to represent the class of all morphisms $A\to B$. Also, if $f \in [A, B]$ and $g \in [B, C]$, I will use $f;g$, instead of $g \circ f$, to represent the ...
5
votes
3answers
291 views

An arrow is monic in the category of G-Sets if and only if its monic the category of sets

Let $G$ be a group, regarded as category with one object $*$ in which each arrow is invertible. Then the category of $G$-Sets is just the category of functors from $G$ to $\mathbf{Set}$. Now I've read ...
4
votes
0answers
121 views

Is there a name for a section and a retraction together?

I call $(s, r)$ such that $r\circ s =\operatorname{id}$ a split morphism. (Other names are “section-retraction pair”, “split pair.”) Notice that this is not a section, because any section is one ...
6
votes
1answer
193 views

Why does Frank Adams demand a finite CW-complex?

On page 145 of J.F. Adams' "Stable Homotopy and Generalised Homology", there is a proposition: Let $E$ be the suspension spectrum of a finite CW-complex $K$, and $F$ and spectrum (of topological ...
1
vote
1answer
61 views

Do elements in a filtered colimit of compact objects factor through a finite stage?

Suppose in some nice enough category, say abelian groups, we have a filtered colimit of compact objects, might as well say a colimit indexed by $\mathbb{N}$. If we are given an element $x$ in the ...
1
vote
0answers
72 views

Do comonads always induce cosimplicial objects? Vice-Versa?

So Charles Weibel, in his book "Introduction to Homological Algebra" discusses the idea of a cotriple or a comonad. I believe he say that, given a comonad $\bot$ and an object $X$ which is ...
5
votes
0answers
136 views

Is $\mathrm{Mod}(-)$ a functor?

Note that if we have a ring $R$, we can talk about $R$-modules, and if we have a ring homomorphism $R\to S$, there is a map from $R$-modules to $S$-modules given by $-\otimes_RS$ (just assume ...
22
votes
3answers
1k views

Natural and coordinate free definition for the Riemannian volume form?

In linear algebra and differential geometry, there are various structures which we calculate with in a basis or local coordinates, but which we would like to have a meaning which is basis independent ...
3
votes
3answers
194 views

Model of a group in category of rings

I'm curious, is a model for a theory of groups (in the sense of Lawvere's algebraic theories) in the category of rings a group ring? Similarly, is a model for a theory of rings in the category of ...
15
votes
3answers
469 views

*writing* proofs involving commutative diagrams

This question is a little fuzzy so might be closed, but I'll give it a shot. I'm sorry this question has quite a long introduction, I don't see how to formulate it more concisely. In modern algebraic ...
10
votes
2answers
463 views

Homotopy pushouts and induced maps

Suppose we are in a proper closed model category and consider a commutative square $$ \begin{array}{rcl} A&\to& B\\ \downarrow&&\downarrow\\ C&\to&D \end{array} $$ in its ...
2
votes
1answer
165 views

What type of limit is used when?

I have some trouble understanding what type of limit is sometimes used. The definition I have is the one when a limit is defined as a universal cone over a functor (say from a small category). So my ...
2
votes
1answer
149 views

Representability of nilradical

Another question on representability: let $F$ be the covariant functor from the category CommutativeRing to Set which sends a ring to its nilradical. Is it representable? What if you restrict to the ...
3
votes
1answer
98 views

Representability of strange functor

Consider the category Top of topological spaces. Consider the contravariant functor from Top to Set sending a topological space X to the set of all opens of X. Is this functor representable? What if ...