Tagged Questions
1
vote
1answer
48 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
89 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
1answer
54 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
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 ...
3
votes
1answer
71 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
34 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 ...
-1
votes
5answers
392 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
224 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
64 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
146 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
39 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
292 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 ...
3
votes
1answer
80 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
4answers
1k 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
697 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, ...
