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1. • CommentRowNumber1.
   • CommentAuthorTomas Chamberlain
   • CommentTime7 days ago
   • (edited 7 days ago)
   • PermaLink
   Author: Tomas Chamberlain
   Format: MarkdownItexI have an issue with the current definitions of filtered and directed categories and filtered/directed limits/colimits on the nLab wiki. TL;DR: 1. Filtered categories are currently defined as a generalisation of upward-directed sets. I think they should be defined to have the "direction" going the other way. This is consistent with the use of the term "filter" in order theory, and allows for a solution to the next point. 1. Using the current definitions, the phrase "filtered limit" could mean more than one thing, and the phrases "directed limit" and "codirected limit" mean the same thing. If "filtered" instead meant a generalisation of "downward-directed", then the ambiguity and overloading problems are solved. In set theory/order theory, the term "filter" means, in particular, something which is _downward_-directed (and upward-closed, which is an independent condition), while the term "directed" usually means "_upward_-directed" by default. The standard (?) interpretation of a poset as a category is to say that there is an arrow $x \to y$ if $x \leq y$. In other words, the upper bounds of an element $x$ correspond to the objects which admit an arrow _from_ the object $x$, while the lower bounds of an element $x$ correspond to the objects which admit an arrow _to_ the object $x$. And, of course, the upper bounds of a subset $S$ correspond to objects which admit an arrow _from_ the diagram $S$ (i.e. co-cones under $S$), while the lower bounds correspond to objects which admit an arrow _to_ the diagram $S$ (i.e. cones over $S$). Thus, it seems to me that the "correct" choice is to say that a _filtered category_ is one in which every finite diagram has a _cone_ (cf. lower bound), and that a _directed category_ is one in which every finite diagram has a _co-cone_ (cf. upper bound). The nLab wiki currently uses a conflicting convention. The page on [[filtered categories]] says that, in a filtered category, every finite diagram has a co-cone (i.e. upper bound). There is no page on directed _categories_, but the page on directed _sets_ ([[direction]]) says that "directed" means "upward-directed" by default, so, according to this wiki, every directed set is a filtered category. I see a number of problems with this. One of them is obviously inconsistency with the use of "filter" to refer to downward-directed and upward-closed things. Another problem is that the phrase "filtered limit" is ambiguous. Using the ncatlab's definitions, a [[directed limit]] is the limit of a functor whose domain is a **downward-directed** set, while a [[directed colimit]] is the colimit of a functor whose domain is an **upward-directed** set. In other words, the words "limit" and "colimit" tell you which way the direction goes. For "directed limit", we observe that limits have arrows _to_ the diagram, so the direction on the domain should give you arrows _to_ finite sub-diagrams (and hence a reverse/downward direction). For "directed colimit", analogous reasoning tells you that the direction is upward. (This is another conflict: the page on directed sets says that "directed" is understood as "upward-directed" by default, while the pages on directed limits/colimits suggest that "directed" means "either upward or downward directed".) I don't think this way of defining "directed limit" and "directed colimit" is very kind to the reader. It requires them to do extra (in my opinion, unnecessary) work in order to figure out what is meant. If they know what the words "directed", "limit", and "colimit" mean, then the meaning of "directed colimit" is easy, but they need to remember that "directed limit" is an exception. Even worse, this implies that "directed limit" and "codirected limit" mean the same thing! (And surely one would want "X" and "co-X" to have different definitions, as is the case literally everywhere else!) Using this idea, the phrase "filtered colimit" makes sense if you use the current nLab wiki definition of "filtered". However, the phrase "filtered limit" is ambiguous, because "filtered" suggests an upward direction while "limit" suggests a downward direction. (The page on [[filtered colimits]] suggests "cofiltered limit" for this notion.) Furthermore, a poset is directed (as a poset) if and only if it is filtered (as a category), but a filter on a poset is **not necessarily** a filtered category! Filters are intimately connected to notions of limits and convergence (cf. [[convergence spaces]] and the characterisation of topology in terms of filters), so I think we'd better have good terminology for working with limits and colimits of filtered things. I hope you will agree with me that this is not an ideal situation. Definitions have two opposing requirements, namely that they should be as clear and consistent as possible while also not being so detailed as to be cumbersome. My proposed definitions, as I see it, are more clear and consistent (both internally and with the usage outside category theory), and they are no more cumbersome than the existing ones. In summary, my suggested definitions are: **Filtered category**: a category in which every finite diagram has a cone **Directed category**: a category in which every finite diagram has a co-cone (equivalently, a category whose opposite is filtered) **Filtered limit**: limit of a functor whose domain is a filtered category **Filtered colimit**: colimit of a functor whose domain is a filtered category **Directed limit**: limit of a functor whose domain is a directed category **Directed colimit**: colimit of a functor whose domain is a directed category Notice also that this does not require us to define the phrases "cofiltered limit", "cofiltered colimit", "codirected limit", and "codirected colimit". Since directedness and filteredness are now dual notions, "cofiltered" just means "directed" and "codirected" means "filtered". (PS: A remark about the definition of a filter in a poset. A filter in a poset $P$ is defined to be a subset $F$ which is downward-directed, upward-closed, non-empty, and proper (i.e. is not equal to the whole power set $\mathcal{P}(P)$). Thus, using my definition, a filtered subcategory of $P$ is not necessarily a filter. A filter is a filtered subcategory which is additionally upward-closed, non-empty, and proper. It seems to me that these three extra conditions are not really "fundamental" to the intuitive notion of limits and convergence, while downward-directedness is really what we are interested in when, for instance, we visualise a principal neighbourhood filter "filtering down" to a point. Requiring non-emptiness and properness is merely to exclude degenerate cases (the improper filter $\mathcal{P}(X)$ of subsets of a topological space $X$ converges to every point), and a prefilter (a filter without the condition of upward-closure) converges to a point if and only if the filter it generates (i.e., its upward closure) does. Also, a function between topological spaces is continuous if and only if, whenever a prefilter in the domain converges to a point, the image of that prefilter converges to the image of the limit point. The image of a filter under a function is always a prefilter, but never a filter unless the function is surjective, while the image of a prefilter is always a prefilter. This is pretty convincing evidence for the argument that it is really downward-closure that is important for understanding limits and convergence in this context, so requiring extra conditions to get a filter on a poset from the definition of a filtered category is not a major loss.)

   I have an issue with the current definitions of filtered and directed categories and filtered/directed limits/colimits on the nLab wiki.

   TL;DR:

   1. Filtered categories are currently defined as a generalisation of upward-directed sets. I think they should be defined to have the “direction” going the other way. This is consistent with the use of the term “filter” in order theory, and allows for a solution to the next point.
   2. Using the current definitions, the phrase “filtered limit” could mean more than one thing, and the phrases “directed limit” and “codirected limit” mean the same thing. If “filtered” instead meant a generalisation of “downward-directed”, then the ambiguity and overloading problems are solved.

   In set theory/order theory, the term “filter” means, in particular, something which is downward-directed (and upward-closed, which is an independent condition), while the term “directed” usually means “upward-directed” by default. The standard (?) interpretation of a poset as a category is to say that there is an arrow $x \to y$ if $x \leq y$. In other words, the upper bounds of an element $x$ correspond to the objects which admit an arrow from the object $x$, while the lower bounds of an element $x$ correspond to the objects which admit an arrow to the object $x$. And, of course, the upper bounds of a subset $S$ correspond to objects which admit an arrow from the diagram $S$ (i.e. co-cones under $S$), while the lower bounds correspond to objects which admit an arrow to the diagram $S$ (i.e. cones over $S$). Thus, it seems to me that the “correct” choice is to say that a filtered category is one in which every finite diagram has a cone (cf. lower bound), and that a directed category is one in which every finite diagram has a co-cone (cf. upper bound).

   The nLab wiki currently uses a conflicting convention. The page on filtered categories says that, in a filtered category, every finite diagram has a co-cone (i.e. upper bound). There is no page on directed categories, but the page on directed sets (direction) says that “directed” means “upward-directed” by default, so, according to this wiki, every directed set is a filtered category.

   I see a number of problems with this. One of them is obviously inconsistency with the use of “filter” to refer to downward-directed and upward-closed things. Another problem is that the phrase “filtered limit” is ambiguous. Using the ncatlab’s definitions, a directed limit is the limit of a functor whose domain is a downward-directed set, while a directed colimit is the colimit of a functor whose domain is an upward-directed set. In other words, the words “limit” and “colimit” tell you which way the direction goes. For “directed limit”, we observe that limits have arrows to the diagram, so the direction on the domain should give you arrows to finite sub-diagrams (and hence a reverse/downward direction). For “directed colimit”, analogous reasoning tells you that the direction is upward. (This is another conflict: the page on directed sets says that “directed” is understood as “upward-directed” by default, while the pages on directed limits/colimits suggest that “directed” means “either upward or downward directed”.)

   I don’t think this way of defining “directed limit” and “directed colimit” is very kind to the reader. It requires them to do extra (in my opinion, unnecessary) work in order to figure out what is meant. If they know what the words “directed”, “limit”, and “colimit” mean, then the meaning of “directed colimit” is easy, but they need to remember that “directed limit” is an exception. Even worse, this implies that “directed limit” and “codirected limit” mean the same thing! (And surely one would want “X” and “co-X” to have different definitions, as is the case literally everywhere else!)

   Using this idea, the phrase “filtered colimit” makes sense if you use the current nLab wiki definition of “filtered”. However, the phrase “filtered limit” is ambiguous, because “filtered” suggests an upward direction while “limit” suggests a downward direction. (The page on filtered colimits suggests “cofiltered limit” for this notion.) Furthermore, a poset is directed (as a poset) if and only if it is filtered (as a category), but a filter on a poset is not necessarily a filtered category! Filters are intimately connected to notions of limits and convergence (cf. convergence spaces and the characterisation of topology in terms of filters), so I think we’d better have good terminology for working with limits and colimits of filtered things.

   I hope you will agree with me that this is not an ideal situation. Definitions have two opposing requirements, namely that they should be as clear and consistent as possible while also not being so detailed as to be cumbersome. My proposed definitions, as I see it, are more clear and consistent (both internally and with the usage outside category theory), and they are no more cumbersome than the existing ones. In summary, my suggested definitions are:

   Filtered category: a category in which every finite diagram has a cone

   Directed category: a category in which every finite diagram has a co-cone (equivalently, a category whose opposite is filtered)

   Filtered limit: limit of a functor whose domain is a filtered category

   Filtered colimit: colimit of a functor whose domain is a filtered category

   Directed limit: limit of a functor whose domain is a directed category

   Directed colimit: colimit of a functor whose domain is a directed category

   Notice also that this does not require us to define the phrases “cofiltered limit”, “cofiltered colimit”, “codirected limit”, and “codirected colimit”. Since directedness and filteredness are now dual notions, “cofiltered” just means “directed” and “codirected” means “filtered”.

   (PS: A remark about the definition of a filter in a poset. A filter in a poset $P$ is defined to be a subset $F$ which is downward-directed, upward-closed, non-empty, and proper (i.e. is not equal to the whole power set $\mathcal{P}(P)$). Thus, using my definition, a filtered subcategory of $P$ is not necessarily a filter. A filter is a filtered subcategory which is additionally upward-closed, non-empty, and proper. It seems to me that these three extra conditions are not really “fundamental” to the intuitive notion of limits and convergence, while downward-directedness is really what we are interested in when, for instance, we visualise a principal neighbourhood filter “filtering down” to a point. Requiring non-emptiness and properness is merely to exclude degenerate cases (the improper filter $\mathcal{P}(X)$ of subsets of a topological space $X$ converges to every point), and a prefilter (a filter without the condition of upward-closure) converges to a point if and only if the filter it generates (i.e., its upward closure) does. Also, a function between topological spaces is continuous if and only if, whenever a prefilter in the domain converges to a point, the image of that prefilter converges to the image of the limit point. The image of a filter under a function is always a prefilter, but never a filter unless the function is surjective, while the image of a prefilter is always a prefilter. This is pretty convincing evidence for the argument that it is really downward-closure that is important for understanding limits and convergence in this context, so requiring extra conditions to get a filter on a poset from the definition of a filtered category is not a major loss.)

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTime5 days ago
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItex> Definitions have two opposing requirements, namely that they should be as clear and consistent as possible while also not being so detailed as to be cumbersome. My proposed definitions, as I see it, are more clear and consistent (both internally and with the usage outside category theory), and they are no more cumbersome than the existing ones. There is also a third requirement: ability to connect to a vast body of the existing literature. The unfortunate term “filtered colimit” is entrenched in the literature. I think it is impossible to change it now. The other notion is “cofiltered limit”, which is relatively unambiguous. “Filtered limit” makes no sense to me. As for “directed colimit” and “directed limit”, these appear to be obscure terms, primarily used by Rosický. The more traditional names are “direct limit” and “inverse limit”. These were in use from the time of Bourbaki. So perhaps the nLab should get rid of “directed colimit” and “directed limit” altogether, since these terms are quite esoteric.

   Definitions have two opposing requirements, namely that they should be as clear and consistent as possible while also not being so detailed as to be cumbersome. My proposed definitions, as I see it, are more clear and consistent (both internally and with the usage outside category theory), and they are no more cumbersome than the existing ones.

   There is also a third requirement: ability to connect to a vast body of the existing literature.

   The unfortunate term “filtered colimit” is entrenched in the literature. I think it is impossible to change it now. The other notion is “cofiltered limit”, which is relatively unambiguous. “Filtered limit” makes no sense to me.

   As for “directed colimit” and “directed limit”, these appear to be obscure terms, primarily used by Rosický.

   The more traditional names are “direct limit” and “inverse limit”. These were in use from the time of Bourbaki.

   So perhaps the nLab should get rid of “directed colimit” and “directed limit” altogether, since these terms are quite esoteric.