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As a first step, it may help to choose some relatively simple examples of $X$ and $Y$. If you can find a characterization of objects in your category as representable presheaves among all presheaves, …

An absolute Galois group is an inverse limit of finite Galois groups over a system of finite Galois extensions of fields, so it represents a functor on groups defined by a compatible system of homomor …

If your category has a monoidal structure, you can ask for an object to be dualizable. This is a generalization of finite-dimensionality for vector spaces.

The answer is yes, provided your action has the following data that you didn't specify: For any g in G and any f: a -> b, an assignment gf: ga -> gb, in a way that strictly respects multiplication i …

As you mentioned, the commutor doesn't generally make sense if your 2-category has morphisms between distinct objects. Therefore, here's a first try at a definition: a collection of braided monoidal …

Yes. It seems you're only looking for an object up to isomorphism, so all you really need is divisibility and addition to yield objects that are well-defined up to isomorphism. Assuming your notion …

Assuming you have such an equivalence of categories, the object $1$ is sent to the trivial representation of your affine $k$-group $G$. This is the representation that factors through the canonical h …

If you have any category that admits finite products, you have a notion of $G$-pseudotorsor, namely an object $X$ equipped with a map $act: G \times X \to X$, such that $act \circ (id_G, act) = act …

The notes you are reading seem to disagree with more commonly accepted language (cf. SGA1 Exp 13, Vistoli's notes, or the Stacks project). Some of this seems to be an attempt at expository ease, e.g. …

You might not find this satisfying, but I think the standard description $\displaystyle{\bigoplus_{s \in S} \mathbb{Z} e_s}$ not only makes the adjunction clear on the level of objects, but also makes …

The group ring of any group $G$ yields a special case of Noah's answer, where $C$ is the monoidal category of $G$-graded vector spaces. I wrote this up in a blog post a few years ago.

The answer to your question is "yes". We need the following properties of $X_1^{iso}$: The map $X_1^{iso} \to X_1$ given by $(f,g) \mapsto f$ on scheme-valued points is a monomorphism. This is str …

If you want to capture the structure of the category together with its monoidal structure, you may need a $k$-fold simplicial set for $k>1$, i.e., a functor from $(\Delta^{op})^k$ to sets. One of the …

If you want a categorical proof that encompasses nonabelian groups and rings-without-identity, then you need to work with concepts such as "normal subobject" that are (I think) not really part of the …

Symmetric algebras (aka free commutative associative unital algebras) are given by a functor, and they satisfy a universal property: If M is a module over a commutative ring k and R is a commutative k …