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I want to convince you that this is the wrong question to ask. First, a little background. There is a relationship between algebras in the mathematical sense and $F$-algebras for $F$ an endofunctor, b …

Take $C = BG$ for some group $G$ and take $D = \text{Set}$. A functor $BG \to \text{Set}$ is a $G$-set. Two $G$-sets are unnaturally isomorphic iff they have the same cardinality, and it's easy to fin …

There is a map $[A, B] \times [B, A] \to [A, A] \times [B, B]$ given by composition in each of the possible directions. There is also a map $1 \to [A, A] \times [B, B]$ which picks out $(\text{id}_A, …

To expand on Mariano's comment, let $A$ be, say, an $\mathbb{R}$-algebra and $\phi : \mathbb{R} \to \text{Aut}(A)$ a one-parameter group of automorphisms of it. Suppose that we can make sense of the d …

Let me elaborate on unknown (google)'s comment. If $\text{Rng}$ denotes the category of unital rings and $\text{Grp}$ denotes the category of groups, then it turns out that "group of units" is a func …

Yes. Instead of working in $C$, you can work in presheaves $[C^{op}, \text{Set}]$ on $C$ using the Yoneda embedding. There is always a terminal presheaf given by sending every object $c \in C$ to $1 \ …

Yes, take $C = A$ with the trivial action of $G$. I will write $BG$ for the one-object category with automorphisms $G$; this is really a different object from $G$ and really should be indicated with d …

As Tim says, for a conceptual ("coordinate-independent") picture we should start with $\text{Bim}(k)$, the 2-category of $k$-algebras, $k$-bimodules between them, and $k$-bimodule homomorphisms, which …

As in the previous question, I continue to suggest that everything is much cleaner if you think in terms of bimodules. So we'll ask the more general question: in the 2-category $\text{Bim}(k)$ of $k$- …

Here's an example I find easier to think about than the examples given so far. Let $G$ be a group and $k$ a field, let $C = BG$ be the category with one object with automorphisms $G$, and let $D = \te …

The nLab page you're looking for is called factorization systems. Here is my favorite one, which I think answers your question. In any category with finite limits and colimits, every morphism $f : X \ …

This statement isn't true in all abelian categories. The morphism $\bigoplus \mathbb{Z} \to \prod \mathbb{Z}$ from an infinite coproduct to an infinite product of copies of $\mathbb{Z}$ is monic but n …

Edit: This answer currently addresses a previous version of the question; here $\text{Vect}$ denotes the category of all vector spaces. We can replace $C$ with its idempotent completion WLOG, so the …

I don't know what counts as an "arithmetical identity" for you, but there's a rich family of interesting examples coming from groupoid cardinality. To say it very tersely, if $X$ is a groupoid with at …

More precisely, is every concrete category C isomorphic to a category C' of small categories such that the morphisms between two elements of C are precisely the functors between their images in C'? A …