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1. • CommentRowNumber1.
   • CommentAuthorEric
   • CommentTimeAug 6th 2010
   • (edited Aug 6th 2010)
   • PermaLink
   Author: Eric
   Format: MarkdownItexThere have been several occasions where I have found it conceptually useful to want to "extrude" a category. I'm hoping that if I manage to explain what I'm trying to do clearly enough someone can help me legitimize/formalize the process. I think it would make a handy tool once in a while. One of the first times I tried to formalize the idea was in [this comment](http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=908&Focus=6300#Comment_6300): >Is there a name for a process I outlined on the nCafe a while back, where we take a category and "extrude" it into something like "discrete time"? >For example, consider a category with two objects a,b and two non-identity morphisms $$f:a\to b$$ $$g:b\to a$$ >Now I want to "form a cylinder" of this category (please read this losely as I'm most likely not using the terms correctly) or "extrude it in time". To do this, we create a bunch of copies of a and a bunch of copies of b and label then with integers. >We also make a bunch of copies of the morphisms and label them with two integers. >The key is that now the morphisms cause the index to change, i.e. $$f(i,j): a(i)\to b(j).$$ >A consequence of this is that we have non-identity identity-like morphisms $$id(i,j): a(i)\to a(j). >The true identity morphisms are the ones with the same integer in both labels, e.g. $$id(i,i): a(i)\to a(i).$$ >I'm pretty sure this procedure can always be done for groupoids and I can't see anything keeping us from doing it for a general category either. In fact, I sometimes try to use this trick to "unwrap" a groupoid into a category by insisting that each morphism increases the index by 1 so that $$f^{-1}(i+1,i+2)\circ f(i,i+1) = id(i,i+2).$$ Then Mike made a very nice [observation](http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=908&Focus=6333#Comment_6333): >This may not be a problem for what you're looking for (I don't really know what you're looking for), but I observe that the condition $$f^{-1}(i+1,i+2) \circ f(i, i+1) = id (i, i+2)$$ >violates the condition you asked for above, that every morphism be assigned a unique time interval. Namely, here the identity $id$ has both the time interval 0 (giving the actual identity in DC) and the time interval 2 (so that the above composite is possible). Perhaps you only wanted *nonidentity* morphisms to have unique time intervals? But if identities can have a nonzero time interval k, then any morphism with time interval i must also have time interval i+k, i+2k, i+3k, etc., since we can compose it with an identity on either side. Now, I like the idea of extruding a lot. I hope it is not just a bogus pipe dream. So if I am to have any chance of formalizing the idea, I need to find a solution or workaround to Mike's observation. I still don't have an answer, but I had an idea and am hoping if I state it, knowing it isn't correct, someone might see a grain of truth and help make the correct statement. The idea stems from considering idempotents: $e:a\to a$ with $e^2 = e$. How can we extrude this? A first guess might be to write down $$e(i,i+1): a(i)\to a(i+1)$$ and try to make sense of the equation $$e(i+1,i+2)\circ e(i,i+1) = e(i,i+2).$$ As I said, I don't have an answer, but am hoping by trying to write ideas down someone can see the answer I'm looking for. So if I look back at Mike's observation, instead of writing $e(i,i+2)$, which probably shouldn't be allowed to parse, we instead write: $$e(i+1,i+2)\circ e(i,i+1) = id(i+1,i+2)\circ e(i,i+1) = e(i+1,i+2)\circ id(i,i+1).$$ If we take this seriously, it starts raising some conceptual questions about the role of the identity morphism. Especially in a directed category or directed space. For instance, we can take powers of the one-step identity morphism to get multi-step identity morphisms $$id^n(i,i+1) = id(i,i+n)$$ but the only true identity is the 0-step identity morphism $$id^0(i,i+1) = id(i,i).$$ Is there any kind of syntax, or logic, or anything where we examine the role of the identity in a directed space or some computing process that maybe identifies a subtle difference between $1^0$ and $1^1$? Conceptually, I would think of $1^0$ as an "instant" with no duration, but would think of $1^1$ as an interval where "nothing happens". The two concepts are different as processes, but from the traditional viewpoint, we rarely if ever take seriously a distinction between $1^0$ and $1^1$. PS: For some directed spaces or processes, $1^1$ might not even be available as a morphism, i.e. $id(i,i+1)$, might not even exist because if there is some barrier, you do not have an option of just sitting there. You need to move around it. An example would be a directed binary tree, the basis for Brownian motion. There, you have a 50% chance of moving to the right and a 50% chance of moving to the left, but sitting there is not an option. This also appears in QFT in the form of [Zitterbewegung](http://en.wikipedia.org/wiki/Zitterbewegung) (one of my favorite concepts).

   There have been several occasions where I have found it conceptually useful to want to “extrude” a category. I’m hoping that if I manage to explain what I’m trying to do clearly enough someone can help me legitimize/formalize the process. I think it would make a handy tool once in a while.

   One of the first times I tried to formalize the idea was in this comment:

   Is there a name for a process I outlined on the nCafe a while back, where we take a category and “extrude” it into something like “discrete time”?

   For example, consider a category with two objects a,b and two non-identity morphisms

   $$f:a\to b$$ $$g:b\to a$$

   Now I want to “form a cylinder” of this category (please read this losely as I’m most likely not using the terms correctly) or “extrude it in time”. To do this, we create a bunch of copies of a and a bunch of copies of b and label then with integers.

   We also make a bunch of copies of the morphisms and label them with two integers.

   The key is that now the morphisms cause the index to change, i.e.

   $$f(i,j): a(i)\to b(j).$$

   A consequence of this is that we have non-identity identity-like morphisms

   $$id(i,j): a(i)\to a(j). The true identity morphisms are the ones with the same integer in both labels, e.g.$$

   I’m pretty sure this procedure can always be done for groupoids and I can’t see anything keeping us from doing it for a general category either. In fact, I sometimes try to use this trick to “unwrap” a groupoid into a category by insisting that each morphism increases the index by 1 so that

   $$f^{-1}(i+1,i+2)\circ f(i,i+1) = id(i,i+2).$$

   Then Mike made a very nice observation:

   This may not be a problem for what you’re looking for (I don’t really know what you’re looking for), but I observe that the condition

   $$f^{-1}(i+1,i+2) \circ f(i, i+1) = id (i, i+2)$$

   violates the condition you asked for above, that every morphism be assigned a unique time interval. Namely, here the identity $id$ has both the time interval 0 (giving the actual identity in DC) and the time interval 2 (so that the above composite is possible). Perhaps you only wanted nonidentity morphisms to have unique time intervals? But if identities can have a nonzero time interval k, then any morphism with time interval i must also have time interval i+k, i+2k, i+3k, etc., since we can compose it with an identity on either side.

   Now, I like the idea of extruding a lot. I hope it is not just a bogus pipe dream. So if I am to have any chance of formalizing the idea, I need to find a solution or workaround to Mike’s observation.

   I still don’t have an answer, but I had an idea and am hoping if I state it, knowing it isn’t correct, someone might see a grain of truth and help make the correct statement.

   The idea stems from considering idempotents: $e:a\to a$ with $e^2 = e$. How can we extrude this? A first guess might be to write down

   $$e(i,i+1): a(i)\to a(i+1)$$

   and try to make sense of the equation

   $$e(i+1,i+2)\circ e(i,i+1) = e(i,i+2).$$

   As I said, I don’t have an answer, but am hoping by trying to write ideas down someone can see the answer I’m looking for. So if I look back at Mike’s observation, instead of writing $e(i,i+2)$, which probably shouldn’t be allowed to parse, we instead write:

   $$e(i+1,i+2)\circ e(i,i+1) = id(i+1,i+2)\circ e(i,i+1) = e(i+1,i+2)\circ id(i,i+1).$$

   If we take this seriously, it starts raising some conceptual questions about the role of the identity morphism. Especially in a directed category or directed space.

   For instance, we can take powers of the one-step identity morphism to get multi-step identity morphisms

   $$id^n(i,i+1) = id(i,i+n)$$

   but the only true identity is the 0-step identity morphism

   $$id^0(i,i+1) = id(i,i).$$

   Is there any kind of syntax, or logic, or anything where we examine the role of the identity in a directed space or some computing process that maybe identifies a subtle difference between $1^0$ and $1^1$? Conceptually, I would think of $1^0$ as an “instant” with no duration, but would think of $1^1$ as an interval where “nothing happens”. The two concepts are different as processes, but from the traditional viewpoint, we rarely if ever take seriously a distinction between $1^0$ and $1^1$.

   PS: For some directed spaces or processes, $1^1$ might not even be available as a morphism, i.e. $id(i,i+1)$, might not even exist because if there is some barrier, you do not have an option of just sitting there. You need to move around it. An example would be a directed binary tree, the basis for Brownian motion. There, you have a 50% chance of moving to the right and a 50% chance of moving to the left, but sitting there is not an option. This also appears in QFT in the form of Zitterbewegung (one of my favorite concepts).

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeAug 6th 2010
   • (edited Aug 6th 2010)
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI would think that you need some kind of graded category, where each morphism is assigned a degree, and the composite of degree-$p$ morphism and a degree-$q$ morphism has degree $p + q$. The identities would still have to have degree $0$. Edit: In light of David's comment below, note that a grading is precisely a functor to $\mathbf{B}(\mathbb{Z},+)$.

   I would think that you need some kind of graded category, where each morphism is assigned a degree, and the composite of degree-$p$ morphism and a degree-$q$ morphism has degree $p + q$. The identities would still have to have degree $0$.

   Edit: In light of David’s comment below, note that a grading is precisely a functor to $\mathbf{B}(\mathbb{Z},+)$.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeAug 6th 2010
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexOne easy way to do this is (and not the real solution) to form $C \times \{ \ldots \lt -2 \lt -1 \lt 0 \lt 1 \lt 2 \lt \ldots\}$ where the second factor is the usual category associated to a poset. If you don't want to be able to 'do nothing' and stay at the same object, then you could replace the second factor with the same thing with no identity arrows. Then an identity arrow in $C$ is 'forced to move in time'. In particular, endomorphisms don't end up at the same object they left. This is not the solution to your problem, because arrows in $C$ aren't given a specified 'cost' or 'time'. But this is perhaps to be expected, because one can always move to a spot, then stay there and wait. The trick, perhaps, is to specify some minimum cost or time for an arrow, by grading (this is done by a functor to a monoid, for example) and then seing what can be done...

   One easy way to do this is (and not the real solution) to form $C \times \{ \ldots \lt -2 \lt -1 \lt 0 \lt 1 \lt 2 \lt \ldots\}$ where the second factor is the usual category associated to a poset. If you don’t want to be able to ’do nothing’ and stay at the same object, then you could replace the second factor with the same thing with no identity arrows. Then an identity arrow in $C$ is ’forced to move in time’. In particular, endomorphisms don’t end up at the same object they left.

   This is not the solution to your problem, because arrows in $C$ aren’t given a specified ’cost’ or ’time’. But this is perhaps to be expected, because one can always move to a spot, then stay there and wait. The trick, perhaps, is to specify some minimum cost or time for an arrow, by grading (this is done by a functor to a monoid, for example) and then seing what can be done…

4. • CommentRowNumber4.
   • CommentAuthorEric
   • CommentTimeAug 6th 2010
   • (edited Aug 6th 2010)
   • PermaLink
   Author: Eric
   Format: MarkdownItexSuper super busy right now, but I should have included [Mike's suggestion](http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=908&Focus=6310#Comment_6310): >Yes, in the construction I described, each noninvertible f goes from a(i) to b(j) whenever $i\le j$. Perhaps what you want is an "N-graded category"? In general, if M is any monoid, then an **M-graded category** is a category in which every morphism is assigned an element of M, called its "degree," such that identities have degree 0 and $deg(f\circ g) = deg(f) + deg(g)$. If C is an M-graded category, then you could define a category with object set $ob(C)\times M$ and with a morphism from (a,m) to (b,n) being a morphism f from a to b in C such that m+deg(f)=n. If M is the natural numbers, then this category will be direct. But in general, for an arbitrary category C, there needn't be any nontrivial way to make it N-graded. For instance, in an M-graded category, any isomorphism must have a degree which is invertible in M, which when M=N means that it must be 0; thus no groupoid admits any nontrivial N-grading. Likewise, any idempotent in an N-graded category must have degree 0, and hence so must any split monic or split epic, so you can construct lots more categories admitting no nontrivial N-grading.

   Super super busy right now, but I should have included Mike’s suggestion:

   Yes, in the construction I described, each noninvertible f goes from a(i) to b(j) whenever $i\le j$. Perhaps what you want is an “N-graded category”? In general, if M is any monoid, then an M-graded category is a category in which every morphism is assigned an element of M, called its “degree,” such that identities have degree 0 and $deg(f\circ g) = deg(f) + deg(g)$. If C is an M-graded category, then you could define a category with object set $ob(C)\times M$ and with a morphism from (a,m) to (b,n) being a morphism f from a to b in C such that m+deg(f)=n. If M is the natural numbers, then this category will be direct. But in general, for an arbitrary category C, there needn’t be any nontrivial way to make it N-graded. For instance, in an M-graded category, any isomorphism must have a degree which is invertible in M, which when M=N means that it must be 0; thus no groupoid admits any nontrivial N-grading. Likewise, any idempotent in an N-graded category must have degree 0, and hence so must any split monic or split epic, so you can construct lots more categories admitting no nontrivial N-grading.

5. • CommentRowNumber5.
   • CommentAuthorTim_Porter
   • CommentTimeAug 6th 2010
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexThree separate thoughts: (i) The simplicial resolution used to give simplicial descriptions of homotopy coherence has some of the features that you want, (but not all). (ii) Some of the ideas discussed by Marco Grandis in his new book (Directed Algebraic Topology) may be relevant. (His section on the fundamental category of a directed space and `minimal' models for that has some nice bits that have some of the same flavour). (iii) Another reverberation is with rewrite theory, where an expression may be given but needs some number of rewrites to get it to a normal form (if one exists). So you form up $e^2$ but this is not equal to $e$ it can be reduced to $e$ by application of the rule / relation $e^2\to e$ and requires one application of the rule. If you are given $e^3$ then you need to apply the rule twice, of course, and the two ways are linked by a 2-cell so are equivalent. You could try to add a weight to applications of rewrites and then take a shortest path through the complex made up of the rewrites and higher identities. There is stuff on that sort of thing in combinatorial group theory. This is still wide of the mark but gives a way to measure the time or effort needed to return an expression to the normal form.

   Three separate thoughts:

   (i) The simplicial resolution used to give simplicial descriptions of homotopy coherence has some of the features that you want, (but not all).

   (ii) Some of the ideas discussed by Marco Grandis in his new book (Directed Algebraic Topology) may be relevant. (His section on the fundamental category of a directed space and ‘minimal’ models for that has some nice bits that have some of the same flavour).

   (iii) Another reverberation is with rewrite theory, where an expression may be given but needs some number of rewrites to get it to a normal form (if one exists). So you form up $e^2$ but this is not equal to $e$ it can be reduced to $e$ by application of the rule / relation $e^2\to e$ and requires one application of the rule. If you are given $e^3$ then you need to apply the rule twice, of course, and the two ways are linked by a 2-cell so are equivalent. You could try to add a weight to applications of rewrites and then take a shortest path through the complex made up of the rewrites and higher identities. There is stuff on that sort of thing in combinatorial group theory. This is still wide of the mark but gives a way to measure the time or effort needed to return an expression to the normal form.