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Given a category $\mathcal{C}$ we have a class $\mathrm{Obj}(\mathcal{C})$ of objects of $\mathcal{C}$. Depending on the category these objects can be things like groups, vector spaces (over a given field), topological spaces, members of some poset, etc.

Are there any objects that are defined as an object in a particular category but which have no definition apart from this?

EDIT:

I'm sorry it took me so terribly long to get back to this.

Let me ask this:

Is there any way to define, say, a group as a member of $\bf{\text{Grp}}$? This would require a way to define the category $\bf{\text{Grp}}$ without reference to the definition of a group as a pair $(G,\cdot)$ that satisfies the group axioms.

I know that when defining a category we have a class of objects and morphisms between them. Is there any way to construct a category with a particular structure built in terms of generic objects and morphisms, such that there is only one (up to equivalence of categories?) class of objects that is compatible with the structure?

Here's a rough idea on how one might go about formalizing this:

Let $\mathcal{C}$ be a non-concrete category, just thought of as a multidigraph (quiver) that might be infinite in size. Let $F$ be a map $\mathcal{C} \to X$, where $X$ is some class of objects (that is the class of objects of some category). If we look at $\mathcal{C}$ 'locally,' i.e. at just a finite subset of objects and morphisms or maybe certain infinite subsets, then we can probably realize $F$ as a diagram $\mathcal{C} \to X$ for different $X$.

Considering $\mathcal{C}$ 'globally,' i.e. with all (generic, non-concrete) objects and morphisms considered at once, can there be only a single (non-trivial, whatever that means in this context) $X$ such that $F$ is an equivalence of categories?

After all that... Maybe what I'm asking is just if a category is uniquely determined by its nerve, $\mathcal{N(C)}$.

Related, and maybe (also) what I'm actually trying to ask: can every non-concrete category be 'concretized' by a functor to some concrete category?

I realize this is very sketchy but I wanted to try somehow to express the nebulous thoughts I have been having about this.

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    $\begingroup$ I honestly do not understand your question. $\endgroup$ – Mariano Suárez-Álvarez Feb 1 '15 at 4:39
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    $\begingroup$ Are you maybe looking for non concrete categories? Take e.g. the category with two objects $A,B$ and two arrows $A\to B$ besides the identities.. $\endgroup$ – Berci Feb 2 '15 at 23:15
  • $\begingroup$ The question is really a bit vague. Does the definition of a simplicial set as an object of the cocompletion of the simplex category count? $\endgroup$ – Martin Brandenburg Apr 8 '15 at 1:16
  • $\begingroup$ What about the following? If A,B are rings, then an (A,B)-bimodule is a cocontinuous functor Mod(A) ---> Mod(B). Of course, there is also a usual definition of a bimodule. And I think that this will be true for every other categorical definition of objects. Otherwise we would not really work with these objects? The only thing which would perhaps count is the following: A set is an object of a well-pointed elementary topos with natural numbers object and axiom of choice (ETCS). $\endgroup$ – Martin Brandenburg May 8 '15 at 7:30
  • $\begingroup$ @MartinBrandenburg Your third sentence gets at my question: is there ever NOT a 'usual definition'? I need a lot more background to fully grasp your examples, but your ETCS example made me do a lot of reading in the nLab, in particular the pages on classifying topoi and Lawvere theory. I will do some more thinking about this and (eventually) formulate a more specific question to ask. Thanks. $\endgroup$ – Alex Petzke May 13 '15 at 3:12

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