Question. Is there a category $\mathcal{C}$ that is equivalent (or anti-equivalent) to the category of smooth manifolds $\mathbf{Man}$ such that the definition of the objects of $\mathcal{C}$ does not involve a set of local models?
This is an attempt to frame the question Is there an intrinsic approach to defining manifolds? in the language of category theory, whose accepted answer I find rather unsatisfying. I am asking this question just out of curiosity. I am not saying that there is anything wrong with the usual definition (this is another difference to the linked question).
If it helps to answer the question, feel free to drop properties such as smooth, paracompact, Hausdorff from the manifolds, but of course not the property of being locally Euclidean. Feel also free to answer the question to manifold-like objects, but whose definition still a priori involves local models. But I will require that we should not limit the dimension, since otherwise potential classification results are available, which goes in a different direction. I am not looking for generalizations such as diffeological spaces which are defined precisely by using local models. I am looking for something more global (as the OP did in the linked question, I assume).
I am aware that such a definition might not exist.
Something rather silly but which goes in that direction is the observation that $\{\mathbf{R}\}$ is dense in $\mathbf{Man}$. It follows formally that $\mathbf{Man}$ is equivalent to a full subcategory of $M{-}\mathsf{Set}$, where $M$ is the monoid of all continuous maps $\mathbf{R} \to \mathbf{R}$. But (a) $M$ is vast, (b) it is not clear which $M$-sets lie in the image (without essentially repeating the definition of a manifold), (c) it still involves the local model $\mathbf{R}$ in some sense.
I understand that this is a rather soft question since I didn't specify exactly when a definition does not involve local models. But I claim that every mathematician will immediately recognize a suitable definition as such. Let me therefore give you an example how a global definition might look like. Assume we have constructed a functor $T : \mathbf{Top} \to \mathbf{Top}$ just by working abstractly with topological spaces. Then the category $T{-}\mathbf{Alg}$ of $T$-algebras (objects are morphisms $T(X) \to X$) has a global definition. Of course, $\mathbf{Man}$ will not be of this form since it lacks fiber products. The name $T$ here intentionally looks like the tangent bundle, which does satisfy a universal property for manifolds.
The category $\mathsf{LRS}$ of locally ringed spaces also counts as global (even though it involves some gluing of local data, we don't have a set of local models), and $\mathsf{Man}$ is a full subcategory of $\mathsf{LRS}$, so it will be sufficient to describe the image without referring to local models, but probably that is not possible, and we should rather go in a different direction.
By Gelfand duality, the category of compact topological manifolds is anti-equivalent to a full subcategory of the category of $C^*$-algebras. But I am not sure if the image can be described without local models, i.e., how to detect the $C^*$-algebra $A := C(M)$ as coming from a manifold without just saying that $\mathrm{Spec}(A)$ is a manifold.