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Stream: learning: reading & references

Topic: Reference for the left adjoint to the inclusion Mon -> Cat

Bryce Clarke (Mar 14 2025 at 15:55):

The fully faithful functor $\mathbf{Mon} \hookrightarrow \mathbf{Cat}$ from the category of monoids to the category of categories admits a left adjoint. Given a category $\mathbf{A}$, we can construct a monoid $L(\mathbf{A})$ which is freely generated by the set of morphisms of $A$, modulo the relations $() \sim 1_{x}$ for each object $x \in \mathbf{A}$ and $(f, g) \sim g \circ f$ for each pair $f$ and $g$ of composable morphisms in $\mathbf{A}$.

Bryce Clarke (Mar 14 2025 at 15:56):

Is there a reference for this fact? I remember learning it from a mathstackexchange/mathoverflow question, but I cannot seem to locate it again. It very possible there is also a standard textbook exercise or lemma which proves it.

Vincent Moreau (Mar 14 2025 at 18:24):

I don't have a reference for you. Instead, let me explain how it comes from very general considerations about essentially algebraic theories. I'm hoping this is interesting to you nevertheless.

Let me write $L_{\mathbf{Mon}}$ and $L_{\mathbf{Cat}}$ for the essentially algebraic theories of monoids and categories respectively. These are the opposite of categories of finitely presented monoids and finitely presented categories. We see a monoid $M$ as a finite limit-preserving functor

$$M \quad:\quad L_{\mathbf{Mon}} \ \longrightarrow\ \mathbf{Set}$$

Vincent Moreau (Mar 14 2025 at 18:31):

I define a category internal to $L_{\mathbf{Mon}}$

• whose finitely presented monoid of objects is the terminal one $0$
• whose finitely presented monoid of morphisms is the free one on one generator $\mathbb{N}$
• whose composition is the morphism $\mathbb{N} \to \mathbb{N} + \mathbb{N} \cong \{a, b\}^*$ which sends 1 on the word $ab$

and the rest of the data is determined. This yields a finite limit-preserving functor

$$P \quad:\quad L_{\mathbf{Cat}} \ \longrightarrow\ L_{\mathbf{Mon}}$$

The precomposition functor $(-) \circ P$ takes a monoid $M$ and produces a category

• whose set of objects is $\mathbf{Mon}(0, M)$, i.e. a singleton
• whose set of morphisms is $\mathbf{Mon}(\mathbb{N}, M)$, i.e. $M$
• whose composition is the one of $M$ given the identification just mentionned

Therefore, $(-) \circ P$ is the functor $\mathbf{Mon} \to \mathbf{Cat}$ that you are interested in. Now, for any such finite limit-preserving $P$, the precomposition functor $(-) \circ P$ has a left adjoint, which is the left Kan extension along $P$ -- indeed, it preserves the preservation of finite limits. This is the left adjoint you mentioned!