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Certain types of categories (like abelian categories) are specified by listing a set of "categorical" properties that the category must have. For example, we might demand the category has finite products, or that all monomorphisms are normal. Sometimes, if you demand enough of these properties, you can uniquely specified a category up to equivalence. (It might be the one-object, one-morphism category, for instance.)

Is there any way to list categorical properties that uniquely specify common categories such as $\mathbf{Grp}$, $\mathbf{Vect}$, etc.?

Although this question is not entirely formal (e.g. what does it mean to list "categorical" properties?), I hope it will be clear what type of answer I'm looking for. I suppose I'd like specifications of categories using "nice" properties, rather than just saying "$\mathbf{Grp}$ is the category whose morphism structure is exactly that given by [groups and group morphisms]." (I'm aware that $\mathbf{Grp}$ may be defined as the full subcategory of Cat whose objects are groups, but this isn't what I'm looking for.)

I would appreciate a formalization of the question as much as an answer.

Thanks!

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    $\begingroup$ See here for a related question, and characterizations of $\mathrm{Set}$ and $\mathrm{Mod}_R$ you might be interested in. $\endgroup$
    – user54748
    Aug 23 '15 at 6:00
  • $\begingroup$ Great suggestion. These characterizations are exactly what I'm after. Though I'm still interested in Grp, for instance, and I'd also like to see comments on what constitutes a categorical characterization. $\endgroup$ Aug 23 '15 at 6:10
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    $\begingroup$ One might even wonder whether it's possible to specify every category in such a way. One fact pointing to "yes" is that Cat is rigid in this sense (well, modulo duality), and objects in rigid categories should be distinguishable. $\endgroup$ Aug 24 '15 at 19:50
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    $\begingroup$ Yes, but that doesn't preclude stupid descriptions, e.g. a generators-and-relations presentation... $\endgroup$
    – Zhen Lin
    Aug 24 '15 at 22:38
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Here is a possible starting point. $\text{Grp}$, as well as other familiar categories of algebraic objects like $\text{Vect}$ or $\text{Ring}$, are distinguished from arbitrary categories by the fact that they are categories of models of Lawvere theories in $\text{Set}$. This is a categorical way of talking about universal algebra.

A categorical characterization of such categories is known: such categories $C$ must

  • be cocomplete
  • admit an object $F$ such that $\text{Hom}(F, -)$ preserves sifted colimits and such that every object in $C$ is a sifted colimit of finite coproducts of copies of $F$.

$F$ ends up being the free object on a one-element set, so in the case of $\text{Grp}$ it's $\mathbb{Z}$. Given a fixed choice of $F$, the corresponding Lawvere theory can be taken to be the opposite of the full subcategory of $C$ on the finite coproducts of copies of $F$.

From here the problem becomes characterizing the Lawvere theory of groups among all Lawvere theories.

A closely related fact is that $\text{Grp}, \text{Vect}, \text{Ring}$ are also all monadic over $\text{Set}$, and a categorical characterization of this condition is also known.

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If you're willing to work with concrete categories, there's a hack that seems to do what you want. Take $\mathbf{Mon}$ for example. We know that:

  • $\mathbf{Mon}$ is a category.
  • There's a forgetful functor $U : \mathbf{Set} \leftarrow \mathbf{Mon}$.
  • There's a natural transformation $\mu:U \Leftarrow U \times U$ that tells us how to multiply the elements of a monoid.

  • There's a natural transformation $\eta:U \Leftarrow 1$ that tells us what the identity element of each monoid is.

  • The following hold:

    1. Associativity. For each $X:\mathbf{Mon}$ and all $x,y,z \in UX$, we have $\mu_X(\mu_X(z,y),x) = \mu_X(z,\mu_X(y,x)).$

    2. Unitality. For each $X:\mathbf{Mon}$ and all $x \in UX$, we have $\mu_X(x,\eta(*)) =x$ and $\mu_X(\eta(*),x)$, where $*$ is the unique element of the set $1$.

So by a candidate for the category of monoids, let us mean a category $\mathbf{C}$ together with a forgetful functor $U : \mathbf{Set} \leftarrow \mathbf{C}$ together with two natural transformations $\mu : U \Leftarrow U \times U$ and $\eta : U \Leftarrow 1$, such that the above axioms hold. There's an obvious notion of morphism between such things, and it seems to be the case that $\mathbf{Mon}$ is the terminal candidate for the category of monoids.

This gives us a way of explaining where certain forgetful functors come from. For example, we can view $\mathbf{Ring}$ as a candidate for the category of monoids, so there's a unique morphism $\mathbf{Mon} \leftarrow \mathbf{Ring},$ which is the relevant forgetful functor.

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  • $\begingroup$ The context for understanding this is the theory of Lawvere theories: see qchu.wordpress.com/2013/06/09/operations-and-lawvere-theories for some details. $\endgroup$ Jan 7 '16 at 17:31
  • $\begingroup$ @QiaochuYuan, that's a very well-written article, Qiaochu. $\endgroup$ Jan 7 '16 at 17:50
  • $\begingroup$ @QiaochuYuan, actually, what I'm describing here seems to be more general than algebraic theories. For instance, we can describe the category of fields in this way. $\endgroup$ Jan 7 '16 at 18:04
  • $\begingroup$ Not as far as I know. As explained in that post, given any concrete category there's a corresponding Lawvere theory given by natural transformations between products of copies of the forgetful functor. Then there's a natural functor from any such concrete category to the category of models of this Lawvere theory, but it need not be an equivalence. When you apply this construction to the concrete category of fields you get the Lawvere theory of commutative rings. $\endgroup$ Jan 7 '16 at 19:22

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