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Plausibly we can construct examples from the power series expansion of formal group laws. We can try to write down such examples by writing down nice functions $q(x)$ and hoping that the formal group …

As far as I can tell, the quantifiers are just hidden in the definition of equality. Your example is too complicated for a humble $1$-category theorist like myself so I am going to replace it with a s …

Not a complete answer. Your definition of an extendible map says that $\beta$ is an endomorphism of the forgetful functor $U$ from the under category $G \downarrow \mathrm{FinGrp}$ to $\mathrm{Set}$ s …

Also I guess it'd also be useful to reason by duality: since $\bf Rel^{\rm op} = Rel$, an endoepimorphism must necessarily also be a monomorphism. This does not follow. Duality only tells us that a …

The Lawvere fixed point theorem asserts that if $X, Y$ are objects in a category with finite products such that the exponential $Y^X$ exists, and if $f : X \to Y^X$ is a morphism which is surjective o …

(Crossposted from math.SE.) Call a category acyclic if only the identity morphisms are invertible and the endomorphism monoid of every object is trivial. Let $C, D$ be two finite acyclic categories. …

You don't give your definition of $\pi_0$ on $\text{Top}$, but since you mention left adjoints I assume it is the left adjoint of the inclusion $\text{Set} \to \text{Top}$ of discrete spaces into $\te …

This is not really in the spirit of the examples you give but it is at least a set of purely categorical properties. Proposition: A category $C$ is the category of models of a Lawvere theory iff it h …

An example similar in spirit to yours is giving explicit examples of affine group schemes. Take, for example, $GL_n$: if we wanted to work solely in $\text{Aff} = \text{CRing}^{op}$ we'd have to write …

There aren't any nontrivial preadditive groupoids; a preadditive category always has zero morphisms, and if zero morphisms are invertible then every object is a zero object. If you think of rings as o …

A magma is a set $M$ equipped with a binary operation $* : M \times M \to M$. In abstract algebra we typically begin by studying a special type of magma: groups. Groups satisfy certain additional ax …

$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Vect{Vect}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Hom{Hom}$The infinitude of the in …

People have done lots of interesting works along these lines. This is a discussion that would be best had at a blackboard to facilitate easy drawing, but here is one version of the story among many. F …

The answer is no already for $Z = \mathbb{R}$. The endomorphism operad of $\mathbb{R}^2 \cong \mathbb{C}$ contains the Lawvere theory of $\mathbb{C}$-algebras, which cannot act on $\mathbb{R}$. This a …

You want to take the category $\text{Ban}_1$ of Banach spaces and short maps (linear maps of operator norm $\le 1$). The unit ball functor $U : \text{Ban}_1 \to \text{Set}$ is represented by $\mathbb{ …