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Question: Internal structure vs. External relationships (the terminal objects of Cat and Set)

I'm noticing the following issue in defining Cat, Set and other categories.

In defining Cat, the category of small categories, Wikipedia states that the terminal object (the terminal category) is the trivial category 1 with a single object and morphism.

Is that trivial category necessarily unique?

In defining Set, the category of sets, Wikipedia states that every singleton is a terminal object.

This seems to be an essential distinction that I wish to voice and discuss. In the case of Cat, we would say that there is only one trivial category because any other such trivial category has the exact same structure and so they are the same. Whereas in the case of Set, we would say that a singleton is defined by the letter, the element that it consists of, such as {a} or {1} or {0}.

But this, I am thinking, is the essential contribution of category theory. Category theory is studying the duality between internal structure and external relationships. And, in principle, the idea would be to restrict our attention to external relationships and to show where and how that is sufficient to deduce internal structure. It's a very typical situation in the real world, where we observe affairs in the real world and try to deduce models of what people are thinking or why they are behaving as they do or what are the physical laws dictating the course of the universe.

However, most of the verbage about category theory ends up being centered on the internal structure, and so it's very hard for a beginner like me to learn to focus just on the external relationships.

From the external point of view, a category is a diagram (of nodes and arrows - they can be labelled for convenience) plus (and this is very important and I am only slowing realizing) a commutative diagram showing which compositions are deemed equivalent (of course, the commutative diagram can typically be described more succinctly than by drawing them all out). The diagram can have unfathomable cardinality but still it's a diagram.

Then a category Set and some other category are the same category if they have the same external relationships, which is to say, the same diagram and the same commutative diagram, which can be labelled the same.

If that's basically correct, then I would find it helpful to have it stated up front in the beginning. It's interesting to think more about what can be garnered from a "mystery" category, as such, just knowing facts about its diagram and commutative diagram. Because it's confusing and misleading to always be told, look at the category of sets, of vector spaces, of abelian groups, of Z, because those are all cases of internal structure. It would seem more relevant, at least pedagogically, to say, here's some facts about a category. What can you say, if anything, about its internal structure? I have an example in mind that I hope to discuss.

Aside from that, I really appreciate starting out with focus on preorders. John, I am very grateful for your explanation of adjoints. It's the first time I have felt that I am understanding it. Thank you!

Comments

  • 1.
    edited April 21

    Hi,

    Is that trivial category necessarily unique?

    I don't think so. A fast argument would say terminal objects are a kind of limit (somewhat degenerate) and hence they are unique only up to isomorphism. So whatever thing with whatever loop arrow from and to it, and suitable composition, is a terminal object in \(Cat\). Two of them are isomorphic because you can have back and forth functors between them, their compositions giving identity functors. One of those compound identity functors sends the unique object of the category to itself and the loop arrow to itself.

    This seems to be an essential distinction that I wish to voice and discuss. In the case of Cat, we would say that there is only one trivial category because any other such trivial category has the exact same structure and so they are the same.

    Isomorphism of structures is a notion of such an importance, and it is so frequent that we need to speak of equivalence classes of isomorphic structures, that it sometimes eclipses the fact that the concrete objects showing the sturcture are different. One speaks of, say, "the" free group on two generators, but the quotes are mandatory, and I think in the wikpedia article they were neglected or shrugged out. They should have said "the" terminal object.

    From other side, one can think of the process of passing from concrete structures to their isomorphism types (i. e., the quotient of the isomorphic-to equivalence) as passing from the category of structures under consideration to its skeleton. Saying that there is only one trivial category would be true in the skeleton of \(Cat\), but not in \(Cat\) itself.

    From the external point of view, a category is a diagram (of nodes and arrows - they can be labelled for convenience) plus (and this is very important and I am only slowing realizing) a commutative diagram showing which compositions are deemed equivalent.

    I blinked at that and made my gears spin, and started to assimilate it. A quiver is graph with loops and parallel arrows allowed. If you extend the diagram you speak about, with the implicit compound arrows (i. e., you pass to the set of paths), those paths form the arrows of the free category corresponding to the quiver (with path concatenation as composition). Now and resembling when normal objects are quotients of free objects, you can paint a set of conmutative diagrams showing when two paths are to be made equal. What seems troublesome is to condense all those diagrams into a single big one. You can find the non sloppy version of the argument here in the nLab.

    I concur that it's intriguing to put in the candle light your external aspects. Since we are speaking of paths, some in all the overwhelming talk about homotopy can apply here. I started to look at how homotopic concepts apply in the context of graphs just with this in mind.

    Comment Source:Hi, > Is that trivial category necessarily unique? I don't think so. A fast argument would say terminal objects are a kind of limit (somewhat degenerate) and hence they are unique only up to isomorphism. So whatever thing with whatever loop arrow from and to it, and suitable composition, is a terminal object in \\(Cat\\). Two of them are isomorphic because you can have back and forth functors between them, their compositions giving identity functors. One of those compound identity functors sends the unique object of the category to itself and the loop arrow to itself. > This seems to be an essential distinction that I wish to voice and discuss. In the case of Cat, we would say that there is only one trivial category because any other such trivial category has the exact same structure and so they are the same. Isomorphism of structures is a notion of such an importance, and it is so frequent that we need to speak of equivalence classes of isomorphic structures, that it sometimes eclipses the fact that the concrete objects showing the sturcture are different. One speaks of, say, "the" free group on two generators, but the quotes are mandatory, and I think in the wikpedia article they were neglected or shrugged out. They should have said "the" terminal object. From other side, one can think of the process of passing from concrete structures to their isomorphism types (i. e., the quotient of the isomorphic-to equivalence) as passing from the category of structures under consideration to its skeleton. Saying that there is only one trivial category would be true in the skeleton of \\(Cat\\), but not in \\(Cat\\) itself. > From the external point of view, a category is a diagram (of nodes and arrows - they can be labelled for convenience) plus (and this is very important and I am only slowing realizing) a commutative diagram showing which compositions are deemed equivalent. I blinked at that and made my gears spin, and started to assimilate it. A quiver is graph with loops and parallel arrows allowed. If you extend the diagram you speak about, with the implicit compound arrows (i. e., you pass to the set of paths), those paths form the arrows of the free category corresponding to the quiver (with path concatenation as composition). Now and resembling when normal objects are quotients of free objects, you can paint a set of conmutative diagrams showing when two paths are to be made equal. What seems troublesome is to condense all those diagrams into a single big one. You can find the non sloppy version of the argument [here](https://ncatlab.org/nlab/show/quiver) in the nLab. I concur that it's intriguing to put in the candle light your external aspects. Since we are speaking of paths, some in all the overwhelming talk about homotopy can apply here. I started to look at how homotopic concepts apply in the context of graphs just with this in mind.
  • 2.

    terminal objects are a kind of limit (somewhat degenerate) and hence they are unique only up to isomorphism

    I’m still pretty new to category theory but what I have seen previously is unique up to unique isomorphism. If there exists more than one isomorphism then you do not have a terminal object. In the present context, there is only one isomorphism between any two single element sets.

    Comment Source:> terminal objects are a kind of limit (somewhat degenerate) and hence they are unique only up to isomorphism I’m still pretty new to category theory but what I have seen previously is unique up to *unique* isomorphism. If there exists more than one isomorphism then you do not have a terminal object. In the present context, there is only one isomorphism between any two single element sets.
  • 3.
    edited April 21

    Nice reminder. Here we would still have several terminal objects in \(Cat\), all of them related pairwise with just one isomorphism (and its inverse in the other direction), just as in \(Set\).

    Comment Source:Nice reminder. Here we would still have several terminal objects in \\(Cat\\), all of them related pairwise with just one isomorphism (and its inverse in the other direction), just as in \\(Set\\).
  • 4.

    Jesús and Matthew, thank you very much for your comments, which I need more time to absorb.

    But for now I would add that it seems that we can at least argue for defining a category like \(Cat\) in two very different ways. That's because it depends on what we mean by Object. In the case of a set, then we typically suppose that the objects exist independently of, and prior to, the set. Sets are defined bottom up, and there can be different sets based on different objects. {1,2,3} is different from {a,b,c}.

    But the whole point of Category theory, as much as I imagine, is that you can define things top down. They exist only to the extent that YOU define them. And you define them by structuring them. So they must be defined by their relationships with themselves. And so from the Category theory point of view, (or let us say, the Structure point of view), you don't define a single set, but rather, you define ALL sets at once. You define "Set-ness". And "Set-ness" is defined by set-functions. But my understanding is that it's not how set-functions works (as maps between sets) that's relevant. Instead, it is the structure of all of the maps that is relevant. If you had another category which had the same relationships (though with nominally different objects), then it wouldn't just be isomorphic, it would be EQUAL to the category of sets. Because it's an equality of CATEGORIES that matters, not an equality of objects. Similarly, 3 apples is different from 3 oranges, but the two 3s are equal! 3 apples is a different set than 3 oranges, but the category 3-ness is the same.

    Maybe I'm wrong. But at least you see how I'm thinking.

    Comment Source:Jesús and Matthew, thank you very much for your comments, which I need more time to absorb. But for now I would add that it seems that we can at least argue for defining a category like \\(Cat\\) in two very different ways. That's because it depends on what we mean by Object. In the case of a set, then we typically suppose that the objects exist independently of, and prior to, the set. Sets are defined bottom up, and there can be different sets based on different objects. {1,2,3} is different from {a,b,c}. But the whole point of Category theory, as much as I imagine, is that you can define things top down. They exist only to the extent that YOU define them. And you define them by structuring them. So they must be defined by their relationships with themselves. And so from the Category theory point of view, (or let us say, the Structure point of view), you don't define a single set, but rather, you define ALL sets at once. You define "Set-ness". And "Set-ness" is defined by set-functions. But my understanding is that it's not how set-functions works (as maps between sets) that's relevant. Instead, it is the structure of all of the maps that is relevant. If you had another category which had the same relationships (though with nominally different objects), then it wouldn't just be isomorphic, it would be EQUAL to the category of sets. Because it's an equality of CATEGORIES that matters, not an equality of objects. Similarly, 3 apples is different from 3 oranges, but the two 3s are equal! 3 apples is a different set than 3 oranges, but the category 3-ness is the same. Maybe I'm wrong. But at least you see how I'm thinking.
  • 5.
    edited April 24

    Andrius wrote:

    In defining Cat, the category of small categories, Wikipedia states that the terminal object (the terminal category) is the trivial category 1 with a single object and morphism.

    Is that trivial category necessarily unique?

    No: there are infinitely many different categories with one object and one morphism. These terminal categories are not equal. The reason is that object could be anything you want, e.g. \(\sqrt{2}\) or \(\{\emptyset\}\), and so could the morphism. Since \(\sqrt{2} \ne \{\emptyset\}\), the terminal category with \(\sqrt{2}\) as its own object is not equal to the terminal category with \(\{\emptyset\}\) as its only object.

    However, by virtue of the definition of "terminal", any two terminal categories are isomorphic. Even better, as Matthew pointed out, they are isomorphic in a unique way.

    This enable category theorists to talk about "the" terminal category. In category theory, unlike set theory, we are happy to talk about "the" thing with a property if any two things with that property are isomorphic in a canonical way. "In a canonical way" requires some explanation, but here we are in a simple case: any two terminal categories are isomorphic in a unique way, and "unique" implies "canonical".

    A category Set and some other category are the same category if they have the same external relationships, which is to say, the same diagram and the same commutative diagram, which can be labelled the same.

    Saying "the same" is sloppy, but okay if one is talking to experts who know exactly what that means in this context. It does not mean "equal". It means "isomorphic".

    If you had another category which had the same relationships (though with nominally different objects), then it wouldn't just be isomorphic, it would be EQUAL to the category of sets.

    No.

    Comment Source:Andrius wrote: > In defining Cat, the category of small categories, Wikipedia states that the terminal object (the terminal category) is the trivial category 1 with a single object and morphism. > Is that trivial category necessarily unique? No: there are infinitely many different categories with one object and one morphism. These terminal categories are not equal. The reason is that object could be anything you want, e.g. \\(\sqrt{2}\\) or \\(\\{\emptyset\\}\\), and so could the morphism. Since \\(\sqrt{2} \ne \\{\emptyset\\}\\), the terminal category with \\(\sqrt{2}\\) as its own object is not equal to the terminal category with \\(\\{\emptyset\\}\\) as its only object. However, by virtue of the definition of "terminal", any two terminal categories are isomorphic. Even better, as Matthew pointed out, they are isomorphic in a _unique way_. This enable category theorists to talk about "the" terminal category. In category theory, unlike set theory, we are happy to talk about "the" thing with a property if any two things with that property are isomorphic in a canonical way. "In a canonical way" requires some explanation, but here we are in a simple case: any two terminal categories are isomorphic in a _unique_ way, and "unique" implies "canonical". > A category Set and some other category are the same category if they have the same external relationships, which is to say, the same diagram and the same commutative diagram, which can be labelled the same. Saying "the same" is sloppy, but okay if one is talking to experts who know exactly what that means in this context. It does not mean "equal". It means "isomorphic". > If you had another category which had the same relationships (though with nominally different objects), then it wouldn't just be isomorphic, it would be EQUAL to the category of sets. No.
  • 6.

    John, Thank you for correcting me.

    But then what terms could we or should we use to talk about the structures in the way I would like to?

    Suppose that we are attending the first Interplanetary Math conference, and you are an Earthling mathematician and I am a Martian mathematician. Here is how I would try to explain how we do it on Mars.

    "Dear Earthling, I am trying to understand you, and maybe I do. But what you are saying about sets and categories is so strange that it sounds like fairyland. You talk about objects but you don't say what you could mean by that. It seems like there could be sets of clouds or moments or dreams or ghosts or movie stars or fruits or nonentities and all manner of ill defined things. And it seems that if you use different names, like {uno, dos, tres} or {1, 2, 3} or {A, B, C} or {do, re, mi}, then you mean different objects or sets. And there can be infinitely many kinds of such different sets which, on the surface, seems to be just the same set but denoted with different mathematical symbols. And then you use this word "infinitely" but you have no explanation that we can fathom on what that could possibly mean. But then in other situations you insist that the particular labels do not matter, and we shouldn't worry about what labels we use. I know that you do so much impressive math, but for the sake of eliminating human bias in math, please hear us out on how we do it on Mars, and please explain how we should talk about that with you."

    "Glorious Earthling, please let me tell you about M-categories. An M-category is an M-graph and a path equivalency. An M-graph is a directed graph where there may be multiple edges between two nodes. We also insist that each node has an identity edge from itself to itself. For the sake of convenience, we can label all of the edges and all of the nodes. Then that makes it straightforward to think about finite paths on those edges. And finally, we have a path equivalency that tells us if two paths are to be considered equal. The path equivalency has to honor composition of paths, namely P1 P3 = P2 P3 if and only if P1 = P2, and similarly P3 P1 = P3 P2 if and only if P1 = P2.

    "I think we use M-categories for many of the same reasons you use categories. But we have defined them this way to assist in our study of how internal structure relates to external relationships. So, for example, we like to study what can be done with n letters, let us call them 1,2,...,N. We define M[1,2,...,N] to be the M-category which has a node for each subset of {1,2,...,N} and has an edge from subset S1 to subset S2 for every set function from S1 to S2. Indeed, that set function can be thought of as the name for that edge. And then we define the path equivalency to be such that two paths are the same if and only if the composition of set functions along the one path yields the same set function as the composition of set functions along the other path. But really the point of M[1,2,...,N] is that it is just the directed graph and the path equivalency, and indeed, the labels do not matter, as they were just for convenience. If there is some other M-category with the same unlabelled directed graph and the same path equivalency, then we say that they are the same M-category. We find that very exciting because it means that an M-category which arose from some internal structure may be equal to another M-category which arose from some entirely different definition, whether internal or external. We then study the duality between internal commitments and external relationships."

    "That is why we speak of isomorphisms when we are comparing two different internal structures. The sets {a, b, c} and {1, 2, 3} are isomorphic as sets. But there is only one M-category which consists of only 3 nodes and their identity edges. Similarly, we typically say that there is only one number 3. Thus we never speak of M-categories as being isomorphic. They are either equal or they are not. Whereas you never speak of categories as being equal. They are either isomorphic or they are not. But you can see why we work with M-categories. There is only one of each kind. And that brings home the point that defining "top-down" in terms of external relationships is so very different than defining "bottom-up" in terms of distinct "things". Apparently, you believe in the reality of the world and so that is where your mathematical objects are coming from. Whereas we believe in the reality of the mind and so we are interested in how we carve that up. Thus our M-categories have no internal structure. But apparently your categories are always categories "of" something and so they always have the internal baggage of that something. We would prefer to liberate you of that but I suppose that would be Martian bias."

    "Mathematically, one consequence is that we are very interested in certain numbers that you don't seem to care about. For example, the most important sequence on Mars is the number of edges in M[1,2,...,N]. This is the formula

    $$ a(n) = \sum_{i=0}^{n}\sum_{j=0}^{n}\binom{n}{i}\binom{n}{j}i^j $$ We call it Sequence A0, namely, 1, 3, 18, 170, 2200... You call it Sequence A074932 in Sloane's Online Encyclopedia of Integer Sequences. Honestly, that's the whole reason we contacted you. Apparently, the way that you think about it is the sum of the absolute values of the coefficients of the Sidi polynomial in this table. We have no inkling what you mean by that. We are, of course, intrigued.

    Anyways, for us it is the most beautiful sequence. For example, if n=4, then we can construct the following table illustrating the number of functions possible for various domains and codomains:

    $$ \begin{matrix} f() & \varnothing & {x} & {x,y} & {x,y,z}\end{matrix} $$ $$ \begin{matrix} \varnothing & 0^0 & 1^0 & 2^0 & 3^0 \end{matrix} $$ $$ \begin{matrix} {a} & 0^1 & 1^1 & 2^1 & 3^1 \end{matrix} $$ $$ \begin{matrix} {a,b} & 0^2 & 1^2 & 2^2 & 3^2\end{matrix} $$ $$ \begin{matrix} {a,b,c} & 0^3 & 1^3 & 2^3 & 3^3\end{matrix} $$ And by the binomial theorem we count multiplicities (pairs of subsets) as follows:

    $$ \begin{matrix} 1 & 3 & 3 & 1\end{matrix} $$ $$ \begin{matrix} 3 & 9 & 9 & 3\end{matrix} $$ $$ \begin{matrix} 3 & 9 & 9 & 3\end{matrix} $$ $$ \begin{matrix} 1 & 3 & 3 & 1\end{matrix} $$ Multiplying the entries:

    $$ \begin{matrix}1 & 3\cdot1^0 & 3\cdot2^0 & 1\cdot3^0 \end{matrix} $$ $$ \begin{matrix}0 & 9\cdot1^1 & 9\cdot2^1 & 3\cdot3^1 \end{matrix} $$ $$ \begin{matrix}0 & 9\cdot1^2 & 9\cdot2^2 & 3\cdot3^2 \end{matrix} $$ $$ \begin{matrix}0 & 3\cdot1^3 & 3\cdot2^3 & 1\cdot3^3 \end{matrix} $$ "Adding them all up we get the most auspicious A0(4)=170 in your stubby but cute ten-finger notation."

    "We find rather curious your fascination with counting the nodes, which is, of course, in this case, simply

    $$2^{4-1}=8$$ "In particular, you miss out on the two ways of summing the array: summing first over outgoing edges, or summing first over incoming edges. Isn't that choice what intelligence is all about? And isn't it all about the edges, the relationships? That's how we model how our attention moves. Your attention seems fixated on the individual objects and not the entire web of relationships. So it's interesting how category theory has taken you almost there. Or are we missing something?"

    "Thus we fly around the universe looking for this sequence as a sign of intelligent life. We were gone and when we came back we finally spotted it on your internet."

    "We understand that this is not how you typically do things and is perhaps very disagreeable. But you may understand this is the tradition on Mars."

    "We thus humbly ask you, what terms would you use to talk about M-categories and M[1,2,...N]?"

    Comment Source:John, Thank you for correcting me. But then what terms could we or should we use to talk about the structures in the way I would like to? Suppose that we are attending the first Interplanetary Math conference, and you are an Earthling mathematician and I am a Martian mathematician. Here is how I would try to explain how we do it on Mars. "Dear Earthling, I am trying to understand you, and maybe I do. But what you are saying about sets and categories is so strange that it sounds like fairyland. You talk about objects but you don't say what you could mean by that. It seems like there could be sets of clouds or moments or dreams or ghosts or movie stars or fruits or nonentities and all manner of ill defined things. And it seems that if you use different names, like {uno, dos, tres} or {1, 2, 3} or {A, B, C} or {do, re, mi}, then you mean different objects or sets. And there can be infinitely many kinds of such different sets which, on the surface, seems to be just the same set but denoted with different mathematical symbols. And then you use this word "infinitely" but you have no explanation that we can fathom on what that could possibly mean. But then in other situations you insist that the particular labels do not matter, and we shouldn't worry about what labels we use. I know that you do so much impressive math, but for the sake of eliminating human bias in math, please hear us out on how we do it on Mars, and please explain how we should talk about that with you." "Glorious Earthling, please let me tell you about M-categories. An M-category is an M-graph and a path equivalency. An M-graph is a directed graph where there may be multiple edges between two nodes. We also insist that each node has an identity edge from itself to itself. For the sake of convenience, we can label all of the edges and all of the nodes. Then that makes it straightforward to think about finite paths on those edges. And finally, we have a path equivalency that tells us if two paths are to be considered equal. The path equivalency has to honor composition of paths, namely P1 P3 = P2 P3 if and only if P1 = P2, and similarly P3 P1 = P3 P2 if and only if P1 = P2. "I think we use M-categories for many of the same reasons you use categories. But we have defined them this way to assist in our study of how internal structure relates to external relationships. So, for example, we like to study what can be done with n letters, let us call them 1,2,...,N. We define M[1,2,...,N] to be the M-category which has a node for each subset of {1,2,...,N} and has an edge from subset S1 to subset S2 for every set function from S1 to S2. Indeed, that set function can be thought of as the name for that edge. And then we define the path equivalency to be such that two paths are the same if and only if the composition of set functions along the one path yields the same set function as the composition of set functions along the other path. But really the point of M[1,2,...,N] is that it is just the directed graph and the path equivalency, and indeed, the labels do not matter, as they were just for convenience. If there is some other M-category with the same unlabelled directed graph and the same path equivalency, then we say that they are the same M-category. We find that very exciting because it means that an M-category which arose from some internal structure may be equal to another M-category which arose from some entirely different definition, whether internal or external. We then study the duality between internal commitments and external relationships." "That is why we speak of isomorphisms when we are comparing two different internal structures. The sets {a, b, c} and {1, 2, 3} are isomorphic as sets. But there is only one M-category which consists of only 3 nodes and their identity edges. Similarly, we typically say that there is only one number 3. Thus we never speak of M-categories as being isomorphic. They are either equal or they are not. Whereas you never speak of categories as being equal. They are either isomorphic or they are not. But you can see why we work with M-categories. There is only one of each kind. And that brings home the point that defining "top-down" in terms of external relationships is so very different than defining "bottom-up" in terms of distinct "things". Apparently, you believe in the reality of the world and so that is where your mathematical objects are coming from. Whereas we believe in the reality of the mind and so we are interested in how we carve that up. Thus our M-categories have no internal structure. But apparently your categories are always categories "of" something and so they always have the internal baggage of that something. We would prefer to liberate you of that but I suppose that would be Martian bias." "Mathematically, one consequence is that we are very interested in certain numbers that you don't seem to care about. For example, the most important sequence on Mars is the number of edges in M[1,2,...,N]. This is the formula $$ a(n) = \sum_{i=0}^{n}\sum_{j=0}^{n}\binom{n}{i}\binom{n}{j}i^j $$ We call it Sequence A0, namely, 1, 3, 18, 170, 2200... You call it Sequence [A074932](https://oeis.org/A074932) in Sloane's Online Encyclopedia of Integer Sequences. Honestly, that's the whole reason we contacted you. Apparently, the way that you think about it is the sum of the absolute values of the coefficients of the Sidi polynomial in [this table](https://oeis.org/A075513/table). We have no inkling what you mean by that. We are, of course, intrigued. Anyways, for us it is the most beautiful sequence. For example, if n=4, then we can construct the following table illustrating the number of functions possible for various domains and codomains: $$ \begin{matrix} f() & \varnothing & \{x\} & \{x,y\} & \{x,y,z\}\end{matrix} $$ $$ \begin{matrix} \varnothing & 0^0 & 1^0 & 2^0 & 3^0 \end{matrix} $$ $$ \begin{matrix} \{a\} & 0^1 & 1^1 & 2^1 & 3^1 \end{matrix} $$ $$ \begin{matrix} \{a,b\} & 0^2 & 1^2 & 2^2 & 3^2\end{matrix} $$ $$ \begin{matrix} \{a,b,c\} & 0^3 & 1^3 & 2^3 & 3^3\end{matrix} $$ And by the binomial theorem we count multiplicities (pairs of subsets) as follows: $$ \begin{matrix} 1 & 3 & 3 & 1\end{matrix} $$ $$ \begin{matrix} 3 & 9 & 9 & 3\end{matrix} $$ $$ \begin{matrix} 3 & 9 & 9 & 3\end{matrix} $$ $$ \begin{matrix} 1 & 3 & 3 & 1\end{matrix} $$ Multiplying the entries: $$ \begin{matrix}1 & 3\cdot1^0 & 3\cdot2^0 & 1\cdot3^0 \end{matrix} $$ $$ \begin{matrix}0 & 9\cdot1^1 & 9\cdot2^1 & 3\cdot3^1 \end{matrix} $$ $$ \begin{matrix}0 & 9\cdot1^2 & 9\cdot2^2 & 3\cdot3^2 \end{matrix} $$ $$ \begin{matrix}0 & 3\cdot1^3 & 3\cdot2^3 & 1\cdot3^3 \end{matrix} $$ "Adding them all up we get the most auspicious A0(4)=170 in your stubby but cute ten-finger notation." "We find rather curious your fascination with counting the nodes, which is, of course, in this case, simply $$2^{4-1}=8$$ "In particular, you miss out on the two ways of summing the array: summing first over outgoing edges, or summing first over incoming edges. Isn't that choice what intelligence is all about? And isn't it all about the edges, the relationships? That's how we model how our attention moves. Your attention seems fixated on the individual objects and not the entire web of relationships. So it's interesting how category theory has taken you almost there. Or are we missing something?" "Thus we fly around the universe looking for this sequence as a sign of intelligent life. We were gone and when we came back we finally spotted it on your internet." "We understand that this is not how you typically do things and is perhaps very disagreeable. But you may understand this is the tradition on Mars." "We thus humbly ask you, what terms would you use to talk about M-categories and M[1,2,...N]?"
  • 7.
    edited May 12

    Would what I'm calling an M-category be the skeleton of a category?

    Comment Source:Would what I'm calling an M-category be the [skeleton](https://en.wikipedia.org/wiki/Skeleton_(category_theory)) of a category?
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