Tagged Questions
6
votes
1answer
123 views
Is “cofunctor” an accepted term for contravariant functors?
People are used to the prefix co- flipping arrows in a concept1, and I have seen people using cofunctor to mean a functor that flips arrows, i.e. that takes $A \to B$ to $FB \to FA$. I know this ...
3
votes
1answer
52 views
If monic, then *property*. Does the converse hold?
Theorem. Suppose $f : X \rightarrow Y$ is monic. Then for all $g : \bar{X} \rightarrow Y$ there exists at most one $h : \bar{X} \rightarrow X$ such that $f \circ h = g$.
Question. Does the converse ...
2
votes
1answer
66 views
Direct product of groups in categorical terms
In the category of abelian groups $\mathsf{Ab}$ the coproduct is the "direct" product (only finite support), but the product is the "cartesian" product (may be infinite support).
In the category of ...
6
votes
0answers
69 views
French translation of “well-powered” category
In order to write a report, I'm looking for a French translation of the term "well-powered category".
Does anyone know the canonical term in French?
4
votes
1answer
90 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, ...
3
votes
1answer
77 views
Mono's and Epi's in the category Rel?
Sorry to ask such a trivial question, but I can't find the answer anywhere.
Question. What are the monomorphisms/epimorphisms in Rel?
Furthermore, what's the standard terminology for describing ...
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
49 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 ...
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 ...
0
votes
0answers
70 views
Does this object have a category-theoretic name?
I have morphisms:
$$
f : A \to B \\
g : B \to C
$$
The composition is:
$$
g \circ f : A \to C
$$
In the function $(g \circ f)$ we call $A$ the domain and $C$ the codomain (or range).
I'm ...
3
votes
0answers
150 views
When are two objects essentially the same?
From the comments to this question I have learned, that many (most?) mathematicians are not very interested in the relationship between an object $X$ and its "correspondent" $F(X)$ for an arbitrary ...
1
vote
1answer
70 views
Name of corresponding objects in equivalent categories
This question is only about terminology. Inside a category we have the standard wordings:
An arrow $f: X \rightarrow Y$ is an isomorphism if there is another arrow $g: Y \rightarrow X$ such that $g ...
1
vote
1answer
99 views
Injective Morphisms, Monomorphisms and Left Invertible Morphisms in Abelian Categories
Let $\mathcal{C}$ be an abelian category. A morphism $f:X \rightarrow Y$ is called injective if its kernel is zero. $f$ is called monomorphism if whenever $f \circ g=0$, for $g:Z \rightarrow X$, then ...
2
votes
0answers
40 views
Quivers with a binary operation on the arrows
A set with an arbitrary binary operation is called a magma.
A set of dots with a set of arrows between them is called a quiver.
A category is a quiver with a binary operation on the arrows obeying ...
1
vote
1answer
166 views
Do the terms “quiver” and “meta graph” refer to the same concept?
Do the terms "quiver" and "metagraph" refer to the same concept? Or is there a distinction I am missing.
My sources are
Quiver - http://ncatlab.org/nlab/show/quiver
Metagraph - ...
1
vote
0answers
31 views
Names of certain morphisms in Pos
Pos is the category of small posets and monotone maps.
I call a morphism $f:\mathfrak{A}\rightarrow\mathfrak{B}$ of Pos monovalued iff it maps every atom of $\mathfrak{A}$ either into an atom of ...
1
vote
1answer
166 views
Meaning of commutative diagram
What is the meaning of a commutative diagram in mathematics?
For example, if a map translate an object, then rotate it around the origin and then translate it again, is this a commutative ...
1
vote
0answers
115 views
What would be a more suggestive name for a “Comma Category”?
Comma categories are pretty expressive construction but apparently many mathematicians including their inventor, Dr. Lawvere, dislike the term for its non-informativeness. I was wondering if anyone ...
0
votes
1answer
70 views
Do function with the following property have special name?
I'm writing "a structure preserving surjection" way too much when I need to refer a function of the following property:
$$ Y \subseteq Z, X \subseteq Z. g: Z \to A, g \text{ is some fixed ...
5
votes
1answer
148 views
What are “Lazard” sheaves?
Early in Categories, Allegories, by Freyd and Scedrov (p.12, in the section on basic examples) there appears the following example:
Let $\mathcal{LH}$ be the category whose objects are topological ...
1
vote
1answer
71 views
Graph of a Rel-morphism
Let $F=(f;A;B)$ is a morphism of the category $\mathbf{Rel}$ (the category whose objects are sets and morphisms are defined as binary relations).
How to name and how to denote $f$ when we know $F$?
...
1
vote
1answer
229 views
what is the difference between functor and function?
As it is, what is the difference between functor and function? As far as I know, they look really similar.
And is functor used in set theory? I know that function is used in set theory.
Thanks.
3
votes
1answer
161 views
Why bifunctors?
Why are bifunctors called "bifunctors"? They are just functors, are they not?
After all, we don't call functions of two arguments "bifunctions", or natural transformations with two parameters ...
12
votes
3answers
265 views
Is the thingie/cothingie distinction absolute?
Is there some inherent quality of a mathematical object that marks it as being "naturally" a thingie or a cothingie?
Suppose, for example, that two mathematical concepts, say, doodad and ...
1
vote
0answers
72 views
Need terminology for components of pullback/pushout squares (or of limits/colimits)
Is there a name for the two morphisms $X\to Z$ and $Y\to Z$ that determine a pullback? Likewise, is there a name for the two morphisms $Z\to X$ and $Z\to Y$ that define a pushout? Also, is there a ...
2
votes
2answers
141 views
Is there a name for a non-iso monomorphism?
I am really bummed out to find that the term "strict monomorphism" is already used to mean something else.
Can anybody console me with the knowledge that there is another name I can use for a ...
4
votes
2answers
217 views
Is there a meaningful distinction between “inclusion” and “monomorphism”?
The title pretty much says it all. As far as I can tell, the terms "inclusion" and "monomorphism" are equivalent. (Ditto for $\hookrightarrow$ and $\rightarrowtail$.) Is this the case?
Edit: ...
6
votes
2answers
163 views
Meaning of “a mapping factors over another”?
I was wondering what "a mapping factors over another mapping" generally means? Does it have something to do with commutative diagram in category theory?
I have seen this usage in different ...
1
vote
1answer
92 views
Not a functor not prefunctor
Are there any special term for the following?
A function from the set of morphisms of a category to the set of morphisms of an other category preserving source and destination of every morphism.
I ...
5
votes
1answer
108 views
Is there a term for a morphism which is surjective on generalized points?
Let $C$ be a category. Recall that a morphism $f : a \to b$ in $C$ is said to be a monomorphism if, for any morphisms $g_1, g_2 : c \to a$, it is true that $f g_1 = f g_2$ implies $g_1 = g_2$. ...
1
vote
1answer
99 views
Instantiate spaces in commutative diagram by “appropriate” elements - name of this idea?
I wonder whether the following concept has a name.
Suppose you are given a commutative diagram $\mathcal C$, that we think of a small category where each hom-class (i.e. hom-set) consists of at most ...
11
votes
3answers
998 views
What is a natural isomorphism?
I came across the adjective "natural" many times in my reading and I think this has something to do with category theory.
Could someone please illustrate the idea behind this adjective to me?
Many ...
4
votes
1answer
181 views
Alternative name for “closed set”
It is usually argued (and also joked about) that classifying sets into open and closed is a bit paradoxical, since sets can be open and closed at the same time, or neither. This can be analyzed very ...
2
votes
1answer
101 views
The name for a subobject(subgroup) which is annihilated by action
I know this question is easy, but for the life of me, I cannot remember what we call this thing. Googling for this has offered no help.
Consider an object $A$ and a second object $B$(let them be ...
4
votes
2answers
319 views
Is this the right categorical generalisation of dual space
Motivatation
I am presently looking at a structure that I am trying to pin down- my strategy being to pull the thing up into the greatest possible generality (based on the bits I'm sure about) and ...
