# Tag Info

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
Accepted

• 18.2k
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
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### 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 ...
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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