4
votes
0answers
60 views

What does it mean to categorify something? [on hold]

What does it mean to categorify something? Is there any simple illustrative example of this process? How it is done and what is it useful for?
0
votes
0answers
24 views

Definition of quotient category

Is there any reason why only gluing of morphisms sharing domain and codomain is usually allowed in the definition of quotient category?
1
vote
1answer
62 views

What is $1 / \mathbf{Set}$ if $1$ is a one-element set and $\mathbf{Set}$ a category?

What does $1 / \mathbf{Set}$ denote? A pointed set is a set $X$ equipped with an element (a basepoint) $x \in X$. Let $\mathbf{Set_*}$ be the category of pointed sets and basepoint-preserving ...
3
votes
0answers
33 views

What is the desirable function identification when setting up arrows in the category of types?

My question is which functions can not be allowed in a statically typed programming language, so that the "canonical" category is less coarse than what you get if you define it's arrows to be ...
3
votes
3answers
139 views

Has the opposite category exactly the same morphisms as the original?

This is actually a question about categories; not only about the category that I mention here specifically. I only use category $\mathsf{Rel}$ as an example. How to describe a morphism that ...
1
vote
1answer
50 views

Collection vs set in this textbook about category theory, and some related questions.

What is the meaning of collection in this context ? Is it here a synonym of set ? Can someone please explain what the author means by "A moment's though shows that, as sets of functions, these two ...
5
votes
2answers
180 views

Definition of forgetful functor [duplicate]

Is there an actual definition of "forgetful functor?" Wolfram MathWorld defines it in terms of functors from algebraic categories to the category of sets, but then says, "Other forgetful functors..." ...
2
votes
2answers
124 views

Categorial definition of free products?

If $X$ and $Y$ are objects of a concrete category $\mathcal{C}$, is there an accepted definition of "free product of $X$ and $Y$," generalizing the in the special case where $\mathcal{C}$ is the ...
6
votes
2answers
385 views

Difference between being faithful and being injective on arrows

I'm studying the concept of faithful functors, but I cannot grasp the difference between being faithtful and being injective on arrows. Could someone explain the difference and provide some examples? ...
5
votes
1answer
74 views

Is there way to formalize the idea that a category can be “cocomplete from the inside”?

Let $\mathrm{KSet}$ denote the category of all countable sets, including the finite ones. Then $\mathrm{KSet}$ is finitely complete. Furthermore, $\mathrm{KSet}$ admits all countable colimits, or, ...
0
votes
0answers
21 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 ...
10
votes
3answers
481 views

What is category theory?

I have read the page about category theory in wikipedia carefully, but i don't really get what this theory is. Is category theory a content in ZFC-set theory? (Just like measure theory, group theory ...
1
vote
0answers
39 views

Indecomposable groups vs. indecomposable objects

An object $X$ in a category $\cal C$ with an initial object is called indecomposable if $X$ is not the initial object and $X$ is not isomorphic to a coproduct of two noninitial objects. A group $G$ ...
0
votes
2answers
52 views

Double categories

So, I wanted to ask a question about double groupoids until I find myself having the answer, meanwhile I wrote a lot of stuff about double categories, so I decided to create a question and answer my ...
7
votes
2answers
113 views

Definition of truth values in a topos.

I'm trying to understand what is meant exactly by a "truth value" in a topos. Take for example the topos of irreflexive graphs. It is known that the classifying morphism can take nodes to 2 different ...
4
votes
1answer
96 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 ...
3
votes
2answers
124 views

Should axioms be viewed as part of the signature?

I included category theory in the tags in order to get feedback from the categorial logic community. It goes without saying that this isn't really category theory. A semigroup can be defined as a ...
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
73 views

Definitions of adjoints (functional analysis vs category thy)

If I have a linear operator $f$ on a Hilbert space, then I define the adjoint of $f$ to be $f^*$ where, $(fx,y)=(x,f^*y)$ for all $x,y$. I am confused because this definitions is very different to ...
5
votes
1answer
47 views

How to see a 2-group as a 2-category with only one object?

We'll take the following definition of a 2-group: A 2-group $\mathsf{G}$ is a category internal to $\mathsf{Grp}$ Namely, it is a group $\mathsf{G_0}$ of objects, a group $\mathsf{G_1}$ of ...
0
votes
1answer
60 views

$K$-Category $M(0, 0) = M(A, 0) = M(0, A)$ using definition from Swan's 'Sheaf Theory'

I'm using the following definition: A category $\mathcal C$ is given by the following: A collection of objects $A$ A set $M(A, B)$ for any two objects $A, B \in \mathcal C$. A function $M(B, C) ...
3
votes
1answer
128 views

Conglomerate in mathematical literature.

I rememeber I downloaded a pdf in category theory and after talking about classes it defined something called conglomerates, but I can't remember what that is and wikipedia has no article on it. ...
10
votes
6answers
734 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 ...
8
votes
2answers
582 views

Definition of a monoid: clarification needed

I'm only in high school, so excuse my lack of familiarity with most of these terms! A monoid is defined as "an algebraic structure with a single associative binary operation and identity element." ...
2
votes
1answer
98 views

Defining Test-Objects

In various categories one encounters what are referred to as 'test-objects'. For example, singleton set serves as test-object (in testing the equality of functions) in the category of sets. In the ...
4
votes
1answer
125 views

Allegories in easy words?

1) What is, in easy words, the definiton of an allegory? 2) And when are allegories useful? What does it have to do with the category theory and categories? With the definiton of category, ...
2
votes
2answers
115 views

In a groupoid, do any two objects have a morphism between them?

This is a question about the definition of a groupoid (in category theory). I've read the Wikipedia article, and it says that a groupoid is a small category in which every morphism is an isomorphism. ...
3
votes
1answer
80 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 ...
3
votes
1answer
85 views

Are distinctions in definitions of “finite” material in, eg, topology or measure theory?

There are several definitions of "finite", like Dedekind's and Tarski's (Thanks to A.K. for point out the latter - first time I've heard of it): From the Wikipedia entry on Finite Set: (Richard ...
2
votes
0answers
42 views

Name for the initial object of a connected component of a category

I was just wondering whether there is already a name for an object of a category as described below. Recall that an initial object in a category $\mathcal C$ is an object $I\in\mathcal C$ such that ...
0
votes
5answers
554 views

Question about the definition of a category

I am confused about the definition of a category given in the Wikipedia article on Category theory: It seems to me that the structure being described (the "arrows" between objects in some class) is ...
2
votes
1answer
262 views

Definition of a universal example

I'm not sure how the term is being used here: Let $R$ be a commutative ring and $X_1,\ldots, X_n$ indeterminates over $R$. Set $P = R[X_1, \ldots, X_n]$. Given a ring homomorphism $\phi: R ...
1
vote
1answer
78 views

In category theory, is there any such thing as “compatibility” for arrow composition?

Is this a properly defined category? Objects $\{P, R, S\}$ Arrows $f_{1} : P \rightarrow R$ $f_{2} : P \rightarrow R$ $g : R \rightarrow S$ $h_{1} : P \rightarrow S$ $h_{2} : P \rightarrow S$ ...
-1
votes
2answers
234 views

Definition of a point and object

Is there any theory in which a point has a definition? What is the definition of "object" as seen in category theory?
2
votes
1answer
45 views

Reducts of categories

There are several ways to reduce a category. The skeleton of a category is the category with isomorphic objects collapsed into one i.e. the only isomorphisms that remain are the identities. What's ...
8
votes
2answers
375 views

Exact sequence in a nonabelian category [previously: “Exact sequence for topological groups?”]

If $A$, $B$, and $C$ are topological groups, and $f: A \to B$ and $g: B \to C$ are two continuous group homomorphisms, what does it usually mean for $$1 \to A \stackrel{f}{\to} B \stackrel{g}{\to} C ...
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...
18
votes
5answers
2k views

What is a universal property?

Sorry, but I do not understand the formal definition of "universal property" as given at Wikipedia. To make the following summary more readable I do equate "universal" with "initial" and omit the ...
4
votes
3answers
1k views

Definitions of direct product and of direct sum

I was wondering if there are some general definitions for direct product and for direct sum, for example in category theory or in set theory, so that the concepts for vector spaces, Abelian groups, ...