3
votes
2answers
207 views

Category with no product?

Is there a family of objects in some category which has no product? If so is there a simple reason for it?
4
votes
1answer
42 views

example diagram of pullbacks and fiber products

I am going through Category Theory for Scientists. I am on section 2.5.1 Pullbacks. I am having trouble visualizing a pullback. Earlier in the book the author gives a nice diagram of an example of ...
1
vote
2answers
122 views

Proof: Categorical Product = Topological Product

Is there a nice way to prove that the categorical product equals the topological product? What I mean is the following: Starting with a given family of topological spaces $X_i$ and any topological ...
3
votes
1answer
56 views

Direct product commutes with direct sum?

Do direct products commute with the direct sums of vector spaces? Basically is $\underset{i \in I}{\prod} \underset{j \in J}{\bigoplus}M_{i,j} \cong \underset{j \in J}{\bigoplus}\underset{i \in ...
0
votes
1answer
92 views

Preserving finite coproducts

i want to prove the following statement: Given a bicartesian closed category $\Bbb{A}$ (thus we have exponentials, finite products and finite coproducts) then the functor $F:\Bbb{A}\rightarrow\Bbb{A}$ ...
0
votes
1answer
63 views

$\textbf{C}$-Monoids and products

i have a question about $\textbf{C}$-Monoids. We can make a new category $\textbf{Mon(C)}$ from the category $\textbf{C}$, namely the category of all $\textbf{C}$-monoids. A $\textbf{C}$-monoid is a ...
0
votes
1answer
36 views

Algebraic formula for co-products in the category of digraphs

I define a digraph as a set $V$ (vertexes) and a relation $E$ (edges) on $V$. Morphisms of digraph are functions which preserve $E$. So we have a category. It is easy to show that products of $n$ ...
0
votes
1answer
61 views

Canonical direct product (in a category)

In some categories there are more than one (isomorphic) direct products: For example in Set there are $A\times B$ and $B\times A$ products (as well as many others). But only one of these products ...
1
vote
1answer
65 views

What are canonical injections for co-products in the category Rel?

What are canonical injections for co-products in the category Rel?
1
vote
1answer
26 views

Products/limits for non-small indexed families of morphisms?

Can the strange requirement that direct products exist only for small indexing families be relaxed, saying that all products (or limits) exists but some are outside of our category (and possibly ...
0
votes
0answers
74 views

Direct products in a partially ordered category

Consider a category, whose set of objects is a poset. Let $f=(f_0,f_1)$ is an indexed family of objects of this category. (Thus $f$ is a 2-indexed family, we can extend it to any index set in an ...
1
vote
1answer
123 views

Whats the diffrence between Products and Coproducts

So I just started in on Category theory (reading the quintessential text, "Categories for the Working Mathematician"), and I am trying to get my head around the difference between Products and ...
0
votes
1answer
55 views

Different direct product in a category and its full subcategory

A question related to Continuing direct product on a subcategory. Let $F$ is a full subcategory of a category $G$. I denote $\operatorname{Ob}X$ the set of objects of a category $X$. Is it possible ...
0
votes
1answer
22 views

Continuing direct product on a subcategory

Let $F$ is a full subcategory of a category $G$, both categories having binary direct product. Is it always true that there is such a binary direct product in $G$ that it is a continuation of a ...
1
vote
3answers
177 views

Why isn't every coproduct a product (and vice-versa)?

So I know that every coproduct is not a product, so I am misunderstanding some part of the definition of (co)products. Saying that U is a coproduct (the disjoint union of X1 and X2 below) of objects ...
4
votes
1answer
106 views

Products in the category of sets and (left-)total relations

By a total (or left-total) relation I mean a binary relation $R \subseteq X \times Y$ where there is, for each $x \in X$, at least one $y \in Y$ with $(x,y) \in R$. Equivalently stated, I mean ...
0
votes
1answer
209 views

Direct products in the category Rel

Please describe direct products in the category Rel.
4
votes
2answers
171 views

When equal products imply equal factors?

Under which additional conditions $a\times b = c\times d \Rightarrow a=c\wedge b=d$ (where $\times$ is a categorical product)? For example, in the case of Cartesian product, for this is enough when ...
2
votes
1answer
80 views

Direct products in subcategories

I have a several categories some of which are subcategories of others. I want to research properties of products in these categories but don't know where to start. How direct products in a category ...
6
votes
1answer
221 views

Uncountable product in the category of metric spaces.

I need to prove that category $\mathrm{Met}$ of metric spaces and continuous maps doesnt possess uncountable product of non-one point spaces. Definition. A pair $(X,\{\pi_\nu:\nu\in\Lambda\})$ where ...