Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.


Stream: learning: reading & references

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


view this post on Zulip Bryce Clarke (Mar 14 2025 at 15:55):

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

view this post on Zulip 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.

view this post on Zulip 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 LMonL_{\mathbf{Mon}} and LCatL_{\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 MM as a finite limit-preserving functor

M:LMon  SetM \quad:\quad L_{\mathbf{Mon}} \ \longrightarrow\ \mathbf{Set}

view this post on Zulip Vincent Moreau (Mar 14 2025 at 18:31):

I define a category internal to LMonL_{\mathbf{Mon}}

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

P:LCat  LMonP \quad:\quad L_{\mathbf{Cat}} \ \longrightarrow\ L_{\mathbf{Mon}}

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

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