# (Reference) Can every category be turned into a category of categories (monoids)?

Note: The following construction seems very straightforward, so a pointer to a reference discussing it in detail, or identifying a standard term used to describe the construction which would allow one to find multiple references discussing it, would suffice for an accepted answer.

Background: Given an arbitrary category $$C$$, it seems that every object $$O$$ in it can be identified with a category $$\mathfrak{O}$$ consisting of a single object (i.e. a monoid) whose morphisms are the endomorphisms $$O \to O$$ in $$C$$.

So one can construct the category $$\mathfrak{C}$$ whose objects are the categories $$\mathfrak{O}$$ and whose morphisms are functors between the categories $$\mathfrak{O}$$. Question 1: Is there a standard name for this construction?

In particular, given any morphism $$f: O_1 \to O_2$$ in $$C$$, one appears to get an induced functor $$F: \mathfrak{O}_1 \to \mathfrak{O}_2$$ ($$e_1 \mapsto f \circ e_1$$) and thus also a morphism in $$\mathfrak{C}$$. Hence it seems like it is always possible to identify $$C$$ as a subcategory of $$\mathfrak{C}$$.

Question 2: Are there any interesting examples where $$C$$ is a strict subcategory of $$\mathfrak{C}$$? In particular, any interesting examples of functors $$F: \mathfrak{O}_1 \to \mathfrak{O}_2$$ which are not induced by a morphism $$O_1 \to O_2$$ in $$C$$?

Of course, it would be "natural" to consider $$\mathfrak{C}$$ not just as a category, but also as a 2-category, i.e. to also consider the natural transformations between functors $$F: O_1 \to O_2$$, $$G: P_1 \to P_2$$ that constitute the morphisms of $$\mathfrak{C}$$.

Given any commuting square $$h_1: O_1 \to P_1$$, $$h_2: O_2 \to P_2$$ connecting morphisms $$f: O_1 \to O_2$$, $$g: P_1 \to P_2$$ in $$C$$, this obviously induces a commuting square of functors $$H_1: \mathfrak{O}_1 \to \mathfrak{P}_1$$, $$H_2: \mathfrak{O}_2 \to \mathfrak{P}_2$$ that connects the functors $$F:\mathfrak{O}_1 \to \mathfrak{O}_2$$, $$G:\mathfrak{P}_1 \to \mathfrak{P}_2$$. My understanding is that any commuting square of functors implies a natural transformation (but not necessarily vice versa), so it seems that the 2-category of commutative squares in $$C$$ can be identified as a sub-2-category of $$\mathfrak{C}$$.

Question 3: Are there any interesting examples where the commutative squares of $$C$$ are a strict sub-2-category of $$\mathfrak{C}$$? I.e. natural transformations in $$\mathfrak{C}$$ that are not induced by commutative squares of functors?

Or even just natural transformations in $$\mathfrak{C}$$ that are commutative squares of functors, but at least one of the functors in question is not induced (by morphisms in $$C$$) and so the natural transformation is also not induced by a commutative square of $$C$$?

Related questions:

Name for mapping of $(f_1, f_2)$ to $f_2 \circ f_1$?

Commuting square of functors

Why are natural transformations the "right" transformations between functors

Why are natural transformations the morphisms between functors?

What's the name of a morphism the morphism category of the category of categories?

• A morphism $x\to y$ doesn’t induce any function between the endomorphism monoids of $x$ and $y,$ which you seem to be assuming. Aug 5, 2023 at 11:32
• Thank you for pointing this out -- I was making that incorrect assumption. Aug 5, 2023 at 11:53

The induced functor you talk about is not actually a functor except in the trivial cases. Say $$O_1\neq O_2$$, $$f:O_1\to O_2$$. If $$g:O_1\to O_1$$ then $$fg:O_1\to O_2$$ is not an endomorphism of $$O_2$$ so you don't have a valid action on arrows (also, even when $$O_1=O_2$$ your functors are not, in general, preserving the identities).