# Tag Info

Accepted

### Nontrivial advantages of thinking of groups as groupoids with one object?

One statement you can prove by viewing groups as one-object groupoids is that, for any group $G$, the product functor $G\times-\colon\mathrm{Grp}\to\mathrm{Grp}$ preserves connected colimits, so in ...
• 5,617

### Nontrivial advantages of thinking of groups as groupoids with one object?

$\newcommand{\B}{\mathbf{B}}$Another fascinating thing is Mackey's formula. In fact this groupoidal view is the only way I have to this day of remembering the details of the formula at all! As a ...
• 4,707
Accepted

### Commutativity of second square when first square and outer rectangle commute

No. If $C'$ is the terminal object, then the outer rectangle and right square always commute, so it's easy to cook up a counterexample in which the left square fails to commute. For an explicit ...
• 77.7k
Accepted

### Natural transformation picking out the map from the initial object

Let $\mathscr{C}^\triangleleft:\simeq[0]\star\mathscr{C}$ be the $\infty$-category obtained from $\mathscr{C}$ by adjoining an initial object, and write $-\infty$ for this initial object. It suffices ...
• 5,617
Accepted

### Forgetful functor $V: \underline{\mathbf{PSet}} \rightarrow \underline{\mathbf{Set}}$ is not full

By definition, a functor $F\colon\mathcal{C}\to\mathcal{D}$ is full if, for any $x,y\in\mathcal{C}$ the induced map $\mathcal{C}(x,y)\to\mathcal{D}(Fx,Fy), f\mapsto Ff$ is surjective. Your ...
• 5,617

### Can individual topological space be considered as category?

As people in the comments have noted, there are many ways to view a topological space $X$ as a category. One obvious way is to look at its poset of open sets $\mathcal{O} X$, viewed as a category. ...
• 38.4k

• 11.1k
1 vote
Accepted

• 1,226
1 vote

### Is there a connection between the topology of $\mathbb{R}/\mathbb{Q}$ and the logic fundamental to Smooth Infinitesimal Analysis?

You may be confusing $\mathbb R/\mathbb Z$ and $\mathbb R/\mathbb Q$. The former can indeed be viewed as the interval $[0.1]$ with endpoints identified, but the latter is much more complicated. ...
• 43.3k
1 vote

### Concrete examples of internal categories (other than small categories)

I would say Hopf-Algeboids, which are categories/groupoids internal to schemes are good examples. One very important example in Algebraic Topology is the one represented by something called \$(MU_*,MU_*...
• 141

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