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Spencer Breiner (Jul 19 2020 at 17:54):Hi all!
Does anyone know if the algebras of a (colored, symmetric) operad always form a topos? They look very presheaf-y.
Spencer Breiner (Jul 19 2020 at 17:57):Related: How does Yoneda translate into the operadic context?
Dan Marsden (Jul 19 2020 at 21:25):Wouldn't this imply the category of monoids and the category of commutative monoids are toposes? I don't believe they are. Johnstone has a paper "When is a Variety a Topos" that might prove helpful.
Peter Arndt (Jul 19 2020 at 21:25):There is a variable to fill in here: operad algebras in what kind of symmetric monoidal category? But the general answer will be a resounding "no". Categories of algebraic structures are rarely toposes.
For the case of operad algebras in the category of Sets with the cartesian monoidal structure, the answer has been worked out in: Johnstone, When is a Variety a Topos? , Algebra Universalis 21 (1985) pp.198-212. Roughly the answer is: this can only happen if essentially all your operations are unary. The modifier "essentially" leaves a tiny bit of room: Roughly, you can allow operations of higher arity, if they amount to conditions on your algebras, and not giving extra structure. Better look at Johnstone's article yourself.
Peter Arndt (Jul 19 2020 at 21:27):You were quicker, @Dan Marsden :grinning:
Jens Hemelaer (Jul 20 2020 at 09:38):There is only one topos in which 0=1, i.e. in which the initial object and the terminal object are isomorphic to each other (it is the trivial topos with only one object). In particular, the category of commutative monoids is not a topos. This might be quicker than checking Johnstone's conditions.
The only examples that I know of toposes that are varieties are the ones in Johnstone's paper:
the category of -sets for a monoid, and the category of Jónsson-Tarski algebras.