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 ...
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 $AB4^*$...
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 ...
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 ...
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

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