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Stream: theory: category theory

Topic: Pushouts of Monoids

Jade Master (May 07 2020 at 00:11):

Let $(C,\otimes,I)$ be a monoidal category. Under what conditions does the category $\mathsf{Mon(C)}$ have pushouts? This nlab article says that if $C$ is cocomplete and $\otimes$ is closed then the free monoid construction can be used to construct certain pushouts but not all of them. More specifically, if $T : C \to \mathsf{Mon}(C)$ is the free monoid functor (which exists by a theorem of MacLane section VII), then pushouts over maps
$T(A) \xleftarrow{T(f)} T(B) \xrightarrow{g} X$
exist. A more detailed construction of these pushouts can be found in Lemma 5.2 of this paper. Do pushouts of monoids not exist more generally in a cocomplete monoidal closed category? Does anyone know of a counterexample?

sarahzrf (May 07 2020 at 00:23):

maybe if ⊗ is not symmetric you can have issues?

sarahzrf (May 07 2020 at 00:24):

given that Mon(C) is monadic over C under those conditions if we add that C is locally presentable and ⊗ is symmetric https://ncatlab.org/nlab/show/category+of+monoids#local_presentability

Mike Shulman (May 07 2020 at 00:24):

If $C$ is locally presentable, then so is $\mathsf{Mon}(C)$, hence cocomplete, with colimits constructed by transfinite iteration. Similar arguments should work if $C$ has other good properties like being totally cocomplete or locally bounded. Symmetry plays no role here.

sarahzrf (May 07 2020 at 00:26):

my counterexample-generating trick of suggesting a non-symmetric ⊗ has failed :(

Jade Master (May 07 2020 at 00:26):

sarahzrf said:

my counterexample-generating trick of suggesting a non-symmetric ⊗ has failed :(

oh no

Jade Master (May 07 2020 at 00:27):

Mike Shulman said:

If $C$ is locally presentable, then so is $\mathsf{Mon}(C)$, hence cocomplete, with colimits constructed by transfinite iteration. Similar arguments should work if $C$ has other good properties like being totally cocomplete or locally bounded. Symmetry plays no role here.

What do you mean by transfinite iteration?

Mike Shulman (May 07 2020 at 00:29):

Take the pushout in $C$, generate the free monoid from it, take a coequalizer in $C$ to quotient out the freely added stuff that shouldn't be there, etc.

Mike Shulman (May 07 2020 at 00:29):

https://ncatlab.org/nlab/show/transfinite+construction+of+free+algebras#colimits_of_algebras_for_a_monad

Mike Shulman (May 07 2020 at 00:32):

If I had to bet, I would bet that if $C$ is merely cocomplete and monoidal, $\mathsf{Mon}(C)$ might not have pushouts, but I don't have any counterexamples immediately to hand, since categories that are cocomplete but not locally bounded are hard to come by.

Mike Shulman (May 07 2020 at 00:37):

However, if $C$ is closed monoidal and cocomplete, I'm pretty sure that $\mathsf{Mon}(C)$ is also cocomplete. I think the trick is that under these hypotheses the free-monoid functor preserves reflexive coequalizers, hence monoids are closed under reflexive coequalizers. Now you can build any colimit of monoids out of coproducts of free monoids (which always exist since the free-monoid functor preserves colimits) and reflexive coequalizers.

Jade Master (May 07 2020 at 01:26):

Mike Shulman said:

Take the pushout in $C$, generate the free monoid from it, take a coequalizer in $C$ to quotient out the freely added stuff that shouldn't be there, etc.

I'm sort of confused...maybe you can help. It seems like the proof of cocompleteness you linked relies on the fact that $\mathsf{Mon}(C) \hookrightarrow C$ is a reflective subcategory. I don't see how this is true, if you take the free monoid on a monoid it is in general much larger than the original monoid so the counit of this adjunction isn't an isomorphism?

Mike Shulman (May 07 2020 at 02:08):

$T \mathrm{Alg}$ is a reflective subcategory of $T/C$, not $C$ itself.