4
votes
Accepted
Two simple questions about representing and generating (symmetric) groups
Cool ideas! It sounds like you're looking for the concept of an action groupoid: https://ncatlab.org/nlab/show/action+groupoid. This is a category you get from letting a group act on a set. ...
- 6,630
4
votes
Accepted
How can we quickly verify naturality?
As with so many things, you proceed in small steps.
I will not go into too many details because basic things like this depend on definitions and you have not provided any.
First:
$$\textrm{Hom} (X, (A ...
- 85.9k
4
votes
Accepted
Question on why the isomorphism $A \cong \mathbf{Z}^n \oplus \text{Tor} A$ is not natural -- A clarification of Riehl's choice of group?
The argument isn't quite how you've presented it. Let's write $TA$ for the torsion part of $A$, as Riehl does. (I'm working from p. 26 of https://math.jhu.edu/~eriehl/context.pdf.)
First suppose that ...
- 5,175
3
votes
How to internalize the extension operator of a monad in a Cartesian closed category?
It's not always possible to define such a morphism.
Regard the 4-element Boolean algebra $\{ \emptyset, \{0\}, \{1\}, \{0,1\} \}$, ordered by subset inclusion, as a poset category $\mathbb{B}_4$. This ...
- 7,233
3
votes
Accepted
Does the direct sum have a universal property in the category of groups?
The (improperly named) "direct sum" of a family $(G_i)_{i\in I}$ of (non necessarily abelian) groups (I prefer to call it "restricted sum") is the subgroup
$$\sum_{i\in I}G_i:=\...
- 18.2k
3
votes
Accepted
The functor category $[\mathbf{G},\textbf{Set}]$ is Cartesian closed. What is the explict description of the closed monoidal structure of it?
$\newcommand{\Set}{\mathsf{Set}}\newcommand{\GSet}{\mathsf{GSet}}$The symmetric closed monoidal structure has nothing to do with function composition. You can not compose two presheaves $X,Y:G\to\Set$,...
- 3,066
2
votes
Accepted
Examples of abelian categories satisfying AB3 (but not AB4) and AB4 (but not AB5)
There are well-known and naturally occurring examples for the dual questions, so you can just take the opposite categories.
Categories of sheaves of abelian groups are $AB3^*$ but typically not $AB4^*$...
- 26.6k
1
vote
Accepted
Net convergence wrt. intersection of decreasing family of topologies
No. If $\tau$ and $\tau'$ are two topologies on a set, then $(x_j)\to_\tau x$ implies $(x_j)\to_{\tau'}x$ for all nets $(x_j)$ and all points $x$ in $X$ iff $\tau'\subseteq\tau$. So, the reverse ...
- 312k
1
vote
Accepted
Generalizing Adamek's categories of $T$-spaces
Rephrasing Def. 5.40, pg. 73:
Let $\boldsymbol{X}$ and $\boldsymbol{Set}$ be categories (the latter is the usual suspect). Then take
$$T:\boldsymbol{X}\to \boldsymbol{Set}$$
to be a [covariant] ...
- 677
1
vote
Accepted
Isomorphism between distinguished triangles in a triangulated category
Answering my own question so it can be closed.
What I was missing to complete the question was the following observation :
One of the axioms of triangulated categories is the following statement :
A ...
- 657
1
vote
Accepted
What are the free categories generated from a graph with a single object and no edge and a single object with one directed edge?
The free category generated from a (directed) graph is defined as follows: objects are vertices of a graph, while morphisms are paths in that graph. Not arrows. To complete the definition we need to ...
- 36.5k
1
vote
Why is $G \lim F \to \lim (G \circ F)$ natural in $F$?
I believe this is an answer in line with Alessandro's hint. Fix $j \in \text{ob}(\mathsf{J})$ and append to the square the projections to $j$:
The two sectors commute by definition of $\tau$; the ...
- 495
Only top scored, non community-wiki answers of a minimum length are eligible