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Stream: learning: questions

Topic: Monoids and Groups in a CCC

Martti Karvonen (Sep 09 2020 at 20:27):

Most textbook level treatments of algebraic theories mostly discuss models in Set in detail. Is anyone aware of a book/article that discusses properties of $Mod_T(C)$ for some other $C$? I don't mean in full generality, but when $C$ is e.g. cartesian closed and (co)complete? In particular, I'd like to understand conditions under which the inclusion $Grp(C)\to Mon(C)$ has a right adjoint, which in the case of $C=Set$ is given by taking the group of invertible elements of the monoid. Explicit descriptions of free monoids/groups and coproducts of them would also be neat - at the moment I'm just aware of the free monoid on $X$ being given by $\sum_i X^i$ under suitable assumptions.

fosco (Sep 10 2020 at 08:58):

If $C$ is itself the category of models of an algebraic theory (which is a reasonably general, yet well-behaved head-start) you can say many things; but few are cartesian closed (take, e.g., $R$-modules)

if your $C$'s aren't algebraic varieties, what's your main example instead?

Morgan Rogers (he/him) (Sep 10 2020 at 10:08):

This works in any Grothendieck topos; I would recommend thinking about a category of presheaves over a nice small category (eg the walking arrow, or over a monoid), or if you're familiar with them, over a category of sheaves. You'll find that once you express all of the free constructions categorically, they work in any such setting; in particular, the adjoint to the forgetful functor you mention does exist.

André Beuckelmann (Sep 10 2020 at 10:19):

Johnstone's book on topos theory talks about algebraic theories in arbitrary topoi for a bit. For example, it shows how free structures exist for nice algebraic theories in any topos with a natural numbers object, but it doesn't really get into too much detail (and is a bit too concise, in general).

Reid Barton (Sep 10 2020 at 10:24):

It works in any locally presentable category, in fact.

Martti Karvonen (Sep 10 2020 at 21:26):

Besides Set I'd actually like to understand the case of Cat. Of course, the correct approach there is 2-categorical (so instead of the adjunction between strict monoidal cats and strict 2-groups, one gets the one between monoidal cats and (weak) 2-groups), but I'd like to see how far I can push the enriched/strict theory before seeing how to relax it when one goes properly 2-categorical. Of course, I know by constructing the adjunction by hand that it exists, but I'd like to understand what makes it work structurally. In any case, this means that requiring one is in a topos is a bit more than I'm ok with assuming.

Nathanael Arkor (Sep 10 2020 at 22:00):

Cat is locally finitely presentable, so this is a well-behaved setting.

fosco (Sep 11 2020 at 09:26):

Martti Karvonen said:

Besides Set I'd actually like to understand the case of Cat. Of course, the correct approach there is 2-categorical (so instead of the adjunction between strict monoidal cats and strict 2-groups, one gets the one between monoidal cats and (weak) 2-groups), but I'd like to see how far I can push the enriched/strict theory before seeing how to relax it when one goes properly 2-categorical. Of course, I know by constructing the adjunction by hand that it exists, but I'd like to understand what makes it work structurally. In any case, this means that requiring one is in a topos is a bit more than I'm ok with assuming.

Sometimes what makes a construction work structurally is that a monad is analytic: for example, the free commutative monoid over a set is the exponential power series $e^X = \sum_{k=0}^\infty \frac{X^k}{k!}$ where the set $X^k = X\times\dots \times X$ is quotiented out by the action of the symmetric group on $k$ elements. A monad is analytic if and only if it commutes with $\omega$-filtered colimits, and with weak pullbacks. This is true over bases of enrichment other than $\sf Set$: http://www.tac.mta.ca/tac/volumes/21/11/21-11abs.html