0
votes
0answers
19 views

Is there a term for an endomorphism defined up to conjugation by an automorphism?

Is there a standard term to designate the equivalence class of endomorphisms where two endomorphisms $\phi$ and $\psi$ are considered equivalent if there exists an automorphism $\alpha$ such that ...
1
vote
1answer
45 views

Confusion about “horizontal composition” of natural transformations

I'm having trouble with an exercise from Rotman's Homological Algebra. It has to do with what Wikipedia calls "horizontal composition" of natural transformations. Namely, given $F, ...
0
votes
0answers
142 views

Collections of Homomorphic (defined) structures via $f$

Long ago I read a text about a collection of algebraic sturctures all homomorphic (or isomorphic) via a unique homomorphism An Example similar to the construction I found was this: Lets take define ...
-1
votes
1answer
76 views

What are morphisms in the category of sets $\mathbf{Set}$?

Do i understand correctly that morphisms in the category of sets $\mathbf{Set}$ are ordered triples $(f, A, B)$ where $f$ is a function $A\to B$? It seems that it is often claimed, even in the ...
3
votes
1answer
43 views

How do morphisms in a comma category single out commuting squares?

I'm trying to teach myself the rudiments of Category Theory. I have a doubt about the definition of comma categories, more precisely about the morphisms. Suppose have two functors ...
2
votes
1answer
47 views

Can objects repeat in commutative diagrams?

Are objects allowed to repeat in commutative diagrams? This seems to be necessary when representing endomorphisms such as the morphism $f : X \to X$ in the category $\mathbf{Set}$, such as when $f$ is ...
2
votes
1answer
65 views

Exponentials “commutes” explicitly

i have a question about exponentials in a category $\Bbb{A}$. I have to prove that the following holds: $C^{A\times B}\cong(C^A)^B$. Therefore i have to give two arrows. This can be done on an ...
1
vote
0answers
59 views

Functor whose values on morphisms are monomorphisms

Is there a name for a functor whose values on morphisms are monomorphisms?
2
votes
3answers
54 views

A notation for a morphism in a thin category

Consider a thin category with objects $A\leq B$. There exists a unique morphism $A\rightarrow B$. Is there a standard notation for this morphism (given $A$ and $B$)?
0
votes
0answers
33 views

Finding coproduct of category(specified in the question!) [duplicate]

I asked a question few minutes ago, and when I saw the answer to my question, I found that I had explained my question wrongly (so the answer was not what I wanted to know). So I decided to write new ...
1
vote
2answers
105 views

What is the product and coproduct of Morphism category(Arrow category)?

Given category C, Its morphism category D means a category that has 1) "morphisms of C" as its objects 2) "pair (f,g) s.t. the diagram(square) commutes" as its morphisms The above definition is ...
3
votes
2answers
89 views

How can I quantify over the class of all cardinalities?

I'd like to quantify over all cardinalities of sets. My end goal is to make a category-theoretic arguement: For all cardinalities of sets, in the category of sets with maps as morphisms: the ...
0
votes
3answers
83 views

From where comes the horizonal composition in a 2-category?

Viewing a 2-category as a category enriched in $\mathsf{Cat}$, I can see from where comes the vertical composition: morphisms of a 2-category are objects of $\mathsf{Cat}$ and 2-morphisms of this ...
1
vote
1answer
49 views

Replacing a morphism with composition with this morphism

I have a certain category. I feel it is better to study the functor $x\mapsto f\circ x$ (where $\circ$ is the composition in my category) than the morphism $f$ itself. How is it called when a ...
0
votes
0answers
82 views

Morphisms in Bourbaki “Theory of Sets”

In Bourbaki "Theory of Sets" there is notion of "morphisms" and different kind of morphisms such as "initial morphisms". These are defined in terms of order theory. It seems that Bourbaki treatment ...
1
vote
1answer
58 views

Systematizing graph morphisms

Trying to systematize possible notions of graph morphisms I came about the following classification: A morphism $f$ which sends a graph $G$ to another graph $G'$ is – first of all – ...
4
votes
2answers
141 views

An example of a monomorphism that is not an equalizer in “Abstract and Concrete Categories — The Joy of Cats”

I am sorry, maybe i am just confused or have not understood some definition, but i do not understand the following remark in Abstract and Concrete Categories -- The Joy of Cats on page 117: ...
4
votes
1answer
37 views

How to denote an 'atomic' morphism in category?

I want to distinguish between two disjoint classes of morphisms in a category: (1) those morphisms that are composed of other morphisms (other than identities) and could conceivably be factored into a ...
10
votes
6answers
684 views

What does “homomorphism” require that “morphism” doesn't?

I'm starting to learn category theory, but there's one thing I don't get: all morphisms seem to be homomorphisms; the definition seems to be the same. What's the difference between these two? Can you ...
1
vote
4answers
146 views

To what extent are morphisms required to be functions?

Just beginning category theory, and I am looking for clarification on the precise nature of morphisms. The most familiar categories, e.g. $\mathbf{Top}$, have morphisms that are functions in the ...
7
votes
1answer
143 views

Category with endomorphisms only

How is called a category with endomorphisms only? How is called a subcategory got from an other category by removing all morphisms except of endomorphisms?
4
votes
3answers
168 views

Does every category have a functor?

Is there any one (or more) categories that doesn't have a functor? Functors go between categories, so is there any category that only has an identity functor but no other functor that maps it to ...
3
votes
3answers
94 views

Given a functor between categories, how to denote a morphism between particular objects of that category

I have a very common situation, for which I need both: (1) notation; and, if available, (2) a general relative term. Let's say that: there is a functor between categories, $f:C_1\to C_2$, $c_1$ is ...
2
votes
1answer
43 views

What is the name for the intermediary object(s) of functional composition?

Consider two morphisms: $f : X \to Y$ and $g : Y \to Z$ , and their composition: $g \circ f : X \to Z$. What is the name given to the role of $Y$ with respect to $g \circ f$? Is there a naming ...
1
vote
2answers
69 views

Morphisms generated by functions

Given a function $f: A \to B$, I can construct a morphism $g : A^* \to B^*$ where $X^*$ denotes some free structure generated by $X$ (Could be monoid, group, module, etc.). I'd like to study ...
0
votes
0answers
82 views

Verifying that a function is a morphism by checking a generating set

In some categories, to verify that a map $f : X \to Y$ is a morphism, it suffices to check only a generating set for $Y$ (or, rather, a generating set for some structure on $Y$ such as a topology or a ...
7
votes
3answers
245 views

Category theory without codomains?

A surjection is a function whose range equals its codomain. Thus, the distinction between functions and surjections requires the notion of a codomain. Similarly, a bijection is an injection whose ...
1
vote
3answers
180 views

Are monomorphisms of rings injective?

Let $R$ and $S$ be rings and $f:R\to S$ a monomorphism. Is $f$ injective?
5
votes
2answers
187 views

Where is the symmetric group hidden in the Yoneda lemma?

In extension to the question Yoneda-Lemma as generalization of Cayley`s theorem?, can someone point out to me where, in the categorical notation and analyzation of the Cayley's theorem, the symmetrc ...
-2
votes
2answers
213 views

Morphims with unique domain

A morphism $m$ of a category has the following property: No morphism (except of the identity morphism) of the category has codomain equal to the domain of $m$. In other words, $m$ cannot be composed ...
3
votes
2answers
183 views

Basic questions about definitions in category theory

I'm just getting started in category theory, and I'm not understanding the basic definitions. For example, a common example of a category is a poset. So suppose I have a trivial poset $P$ of 10 ...
3
votes
3answers
520 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 = ...
9
votes
2answers
426 views

Morphisms in the category of natural transformations?

I am learning the basics of category theory, so this question is probably obvious to anyone who knows the subject. The resources I've seen all take the following approach: 0) A category is a ...