What types of objects can you define in a category?

You can define a group object in a category through some commutative diagrams that loosely correspond to group axioms. For example, the group objects in category of sets are well...groups. Group objects in category of topological spaces are topological groups.

My friend also mentioned monoid objects. I looked online and saw you could define ring objects.

What other types of objects can you define? Can you define a module object? (Maybe through a group object) Can you define field, algebra, etc objects?

A follow up question here.

There are all sorts of things you can do. In any category with finite products you can define models of any Lawvere theory, which roughly speaking lets you define any type of structure given by some operations which satisfy some universal equational axioms. Examples include

• Group objects
• Ring objects
• Module objects over a particular ring $R$ (in $\text{Set}$)
• A pair consisting of a ring object and a module object over it (this requires a multisorted Lawvere theory)

but not fields, because the axiom that every nonzero element is invertible isn't a universal equational axiom, owing to the need to say "nonzero." For more see e.g. this blog post.

In any fixed category $C$ you also can try to write down a monad or comonad on $C$ and talk about algebras or coalgebras over $C$. This is a tremendously general formalism; for example, compact Hausdorff spaces are the algebras of a monad over $\text{Set}$ called the ultrafilter monad. Roughly speaking this defines compact Hausdorff spaces in terms of a bunch of infinitary operations called taking limits with respect to an ultrafilter.

If you require more of $C$ you can define more things in $C$. Generalizing the case of Lawvere theories, if you require that $C$ have all limits you can define models of any limit sketch in $C$, which generalize Lawvere theories. For example, you can define an internal category in $C$ if $C$ has finite pullbacks.

If you require that $C$ have a monoidal structure, the prototypical example being the tensor product of vector spaces, then you can define algebras over operads. These are similar to Lawvere theories but with some constraints on what the axioms can look like; on the other hand monoidal categories are substantially more general than categories with finite products.

Defining fields is complicated because of the need to talk about "nonzero elements." There are various possibilities and in categories other than the usual category of sets they don't agree.

The answer to your first question is that you can define $F$-algebras on a category $\mathcal{C}$ whenever $F$ is an endofunctor of $\mathcal{C}$.

For example, if we take $F$ to be the functor $G \mapsto 1 + G + G \times G$ (in a category with a terminal object $1$, binary products, and coproducts), and insist that some diagrams commute (corresponding to insisting that multiplication is associative and so on), then the $F$-algebras in the category are group objects over that category.

As a rule of thumb, if you can define your algebraic structure by a collection of operations on a set whose codomain is that set, then you can probably form it as an $F$-algebra and hence apply it to many more categories.

A notable example of a structure which is not an $F$-algebra (and which therefore does not obviously generalise to other categories) is the theory of posets, because posets are not "sets with some operations on the sets": there's fundamentally a "is $a$ less than $b$?" operation which maps out of the poset, into $\{\text{less}, \text{greater}, \text{equal}, \text{incomparable}\}$.

Totally ordered sets, however, do form $F$-algebras, because they can be reformulated to omit this "is less than" operation: instead we may consider them to be a set equipped with join and meet operations (which do map into the poset, though the "is less than" operation did not), and then simply define "$a$ is less than or equal to $b$" to mean "the inf is equal to $a$".

• This seems like an awkward formalism to use. The structure of an $F$-algebra alone doesn't encode the various axioms you might want to impose and in this formalism you have to impose them by hand. That's unnecessary: there are a variety of other formalisms that naturally impose them, e.g. Lawvere theories or monads. – Qiaochu Yuan May 16 '18 at 22:54
• Most of the applications I know for $F$-algebras are specifically in defining initial algebras (ncatlab.org/nlab/show/initial+algebra+of+an+endofunctor); arbitrary $F$-algebras are not structures I see showing up in mathematics a lot. – Qiaochu Yuan May 17 '18 at 7:51
• @QiaochuYuan That's probably why I thought of them; I'm a programmer, and initial algebras are quite common in my sphere. – Patrick Stevens May 17 '18 at 21:02