Can we modify ETCS to handle structures directly, as objects in their own right?

Question

I have completely rewritten this question; thus, some of the comments/answers may no longer be relevant.

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The elementary theory of the category of sets (hereafter, ETCS) is an axiomatic approach to the category of sets and functions. As such, it can provide a foundations for mathematics, in much the same way that ZFC can.

Now personally, I don't find sets with no additional structure to be very interesting. I'm interested in groups, posets, topological spaces etc. So I want to know if there's a way to modify ETCS such that the objects are no longer just "sets", they're "sets+additional structure."

I envision something like the following. If $Y$ is an object and $f : X \rightarrow Y$ is a function, then we can form a new object $(Y,f)$ which is essentially "$Y$ + the additional structure provided by $f$." For instance, if $Y$ is an object and $f : 1 \rightarrow Y$, then $(Y,f)$ is just like $Y$, except with a distinguished element. More generally, if $X$ is a set and $Y$ is an object, and if $f : X \rightarrow Y$ is an injection, then $(Y,f)$ is just like $Y$, except with a distinguished subset. Similarly, if $W$ is a well-ordered set and $Y$ is a structure, and if $f : W \rightarrow Y$ is a function, then $(Y,f)$ is just like $Y$, except with a distinguished sequence (possibly transfinite).

Note that $(Y,f)$ is not an ordered pair, but a new object. We could have instead denoted it $Y + f$, to emphasize that its $Y$ plus some additional structure.

So anyway, I'm after ideas for axioms and also general comments. Also, if this sort of thing has been done before, TELL ME!!! A reference would be nice.

Now, a technical question. How should we represent binary relations on an object? Observation: Given an object $Y$ and a family of functions $\{f_i : 2 \rightarrow Y\}_i$, where $2$ is a well-ordered set of two elements and $i$ is understood to range over an unordered set, the family $\{f_i\}_i$ essentially encodes a binary relation on $Y$. So $(Y,\{f_i\}_i)$ might be viewed as the structure $Y$ equipped with the binary relation $\{f_i\}_i$. But should we view this as being a valid object? Perhaps a better approach would be to consolidate the family $\{f_i\}_i$ into a single morphism $g$, and then regard $(Y,g)$ as the new object. But what is the appropriate notion of consolidation?

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EDIT: It's been suggested that the question is too unfocused, so let me try to clarify. I'm looking for an axiom system for the category of structures and functions. A "structure" being a set together with some generalized elements. So I guess the novel aspect of this idea is that given a structure $Y$ and a generalized element $f : X \rightarrow Y$, we want to define a new structure $Y+f$, or $(Y,f)$ if you prefer, which is just the old structure equipped with an additional generalized element.

For instance, supposed $Y$ is a set (the simplest kind of structure), and let $f : 1 \rightarrow Y$ denote a function. Then $Y+f$ is a new kind of structure; it's a set with a distinguished element called $f$. Now suppose additionally that $g : X \rightarrow Y$ is an injection, and suppose $X$ is a set. Then $(Y+f)+g$ is again a new structure; it's $Y+f$ together with a distinguished subset called $g$.

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EDIT 2: I think I finally understand what the confusion is about. The word "morphism" is typically understood to refer to a structure-preserving map. This is NOT how I'm using the word. Recall that I'm interested in a category of structures and functions. So the morphisms between two structures are precisely the functions between their underlying sets. That is, the morphisms in the category of structures and functions aren't necessarily structure-preserving. They're arbitrary functions. So if $G$ and $H$ are groups, then a morphism $f : G \rightarrow H$ is ANY FUNCTION between the underlying sets of $G$ and $H$. Of course, we can define a subcategory wherein the objects are all groups and the morphisms are all group homomorphisms, and in this subcategory the meaning of "morphism" is more restrictive.

Hopefully, that clears things up.

Answer 1

Working over ETCS is, if I recall correctly equivalent to working over a well pointed topos with choice. A good reference for topos theory is Sheaves in Geometry and Logic by Moerdijk and MacLane.

In topos theory, a binary relation on an object $X$ is just a subobject of $X \times X$, ie an object $R$ together with a monomorphism $r:R \rightarrowtail X \times X$. Based on this idea, in ETCS you can construct a set whose elements correspond to binary relations on $X$ by simply constructing the product $X \times X$ by the axiom of products and then applying the axiom of power sets to get the set of binary relations $P(X \times X)$. Since ETCS is already able to handle binary relations and in fact $n$-ary relations there isn't any need to extend it at all.

Edit: Based on the comments I'm going to extend this a bit.

In my opinion it isn't necessary to combine the two objects $X$ and $R$ into one. However it is possible to do this. In topos theory you could do this by taking the coproduct $X + R$ of $X$ and $R$.

Edit 2: On further thought, coproduct probably doesn't have all the properties you want it to. Maybe it's better to just use pairing in the "meta universe." Given any language $\mathcal{L}$ and any topos you can produce a new category of $\mathcal{L}$-models whose objects are n-tuples $\langle X, R_1,\ldots,R_n\rangle$ of objects (edit 3: as Hurkyl pointed out $R_i$ is actually an object and a monomorphism) in the topos and whose morphisms are morphisms $X \rightarrow X'$ that are homomorphisms in the internal language of the topos.

Edit 4: I think maybe the crux of the matter is that you want to take all the information from $X$ and $r : R \rightarrowtail X \times X$ and package it into one object, but this goes against the spirit of category theory because you shouldn't be thinking in terms of objects but in terms of the relationships between them.

Hope this helps.

Answer 2

$\newcommand{\map}[1]{\stackrel{#1}{\to}}$I wanted to expand briefly on mjqxxxx's comment "Arguably, category theory".

Consider a category with a terminal object $1$.

For any other object $X$, call any morphism $1\map{x}X$ an element of $X$.

If we have another morphism $X\map{f}Y$ then we can form the composition $1\map{x}X\map{f}Y$, and the resulting morphism $f\circ x$ is an element of $Y$ (that you might more often see written as $f(x)$).

So you can define the relation "is an element of" in terms of morphisms, and function evaluation in terms of composition of morphisms.

The empty set is a set with no elements - equivalently, an object which has no morphisms to it.

Similarly you can build up notions of cartesian product, images, inverse images, natural numbers, subsets etc out of the more primitive concepts of 'objects' and 'morphisms'. Tom Leinster talks more about this in this paper.

Answer 3

The question is not really focussed and not clear enough. There are various precise interpretations of the question, but in each case I can only offer the trivial answer: Well that's what universal algebra and categorical algebra are all about! I will not even try to summarize what has been done in these large areas of mathematics; this is not possible. The basic principle is, that for many general mathematical ideas the category of sets can be replaced by an arbitrary category with sufficiently nice properties. If these properties are ETCS, we get nothing new, since every category satisfying ETCS is equivalent to the category of sets. Typical properties are the existence of certain limits or colimits and some compatibilities between them. In order to get a fealing for this, try to read the following nlab articles.

• generalized element

• internal relation

• congruence

• algebra and the references, for example group object

• regular category

Of course this is just a selection and hopefully answers some of the questions. If not, please clarify the question, then I will try to improve my answer.

Answer 4

So, it's seems like you want a sort of category in which you can encode structures with one sort. This kind of structures usually are well encoded as models of a Lawvere's algebraic theory.

So it seems to me that what your looking for could be this sort of category, having:

• objects: models of a Lawvere theory, i.e. some special kind of set valued functors (note that for every such model $\mathcal F \colon \mathbf T \to \mathbf {Set}$ the underlying set is the object $\mathcal F(1)$, where $1$ is the generator of the objects in the Lawvere theory;

• morphisms: should just simply be functions (i.e. $\mathbf{Set}$ morphisms) between the underliyng sets.

Edit: I've just noted that we can do even another kind of category: the categories of algebras for monads.

You could consider a category having as objects pairs of the kind $(T(X),h)$ where $T \colon \mathbf{Set} \to \mathbf {Set}$ is a monad, $X$ is a set and $h \colon T(X) \to X$ is a function which commute with multiplication and unit of the given monad $T$. For any give pair of this sort the underlying set is the codomain of the structure map $h$. As morphisms of this category you could take the functions between the underlying sets.

I know that you can build a lot of algebraic structures, but also other ones like topological spaces, as algebras for monads. In particular there's the free group monads whose algebras are groups and (if I'm not mistaken) the ultrafilter monad having topological spaces as algebras.

You can find a lot more Googleing around.

Is this what you wanted?