nForum - Search Results Feed (Tag: definitions) 2021-01-22T01:53:36-05:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher path https://nforum.ncatlab.org/discussion/7920/ 2017-07-21T11:51:38-04:00 2017-07-22T12:05:04-04:00 Peter Heinig https://nforum.ncatlab.org/account/1588/ (New thread since, after a semi-cursory search, no LatestChanges thread for [path] was found.) Added to [path] a definition of “inverse path”. Also tried to make the definition of “Moore ...

(New thread since, after a semi-cursory search, no LatestChanges thread for [path] was found.)

Added to [path] a definition of “inverse path”.

Also tried to make the definition of “Moore path” clearer. Quibblingly speaking, this term used to be defined by saying what it has, without relating it to the initial definition of “path”. I was tempted to change the definition of “path” to the one given by tom Dieck in “Algebraic Topology”, having $a$ and $b$ for the endpoints of an artbitrary interval, which in particular would make it possible to simply say “for Moore path take $a=0$, $b=n$”, but then refrained, suspecting that whoever wrote it this way set store by having path to be always a space-modelled-on $[0,1]$, which for several reasons seems more simple and systematic indeed.

Motivation is that I try to concentrate on writing an exposition of a theorem of J.A. Power, and for this, I have resolved to use a —mildly—topological writing style, and for this I in particular need to get serious about paths, and I need Moore paths.

[Incidentally, in the nLab there lives Moore path category which has much to do with a “Moore path” of the type that lives on path since its creation on September 16, 2011. Maybe one should harmonize the two “Moore path”s a little more, saying a few situating thins on either page. Yes, path already links to Moore path category, but, it seems, not the other way round. Nothing urgent here, though.]

[Incidentally, I had recourse to a footnote in path. I did not forget the advice given recently, it just seemed right here to, simultaneously,

• give a reference

• warn readers of some notational issues

• not clutter the main text with this

and I found my hand forced by this. If this is inacceptable, you might even just say “make it into this or that format” and I’ll hopefully do so soon. Now back to pasting schemes.]

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Acirclic categories? https://nforum.ncatlab.org/discussion/7901/ 2017-07-13T02:56:06-04:00 2017-07-13T07:18:43-04:00 Peter Heinig https://nforum.ncatlab.org/account/1588/ Is there any notion and technical term in category theory for the concept of acirclic category described below ? The most strict definition of the term, i.e. category C\mathsf{C} such that the ...

Is there any notion and technical term in category theory for the concept of acirclic category described below ?

The most strict definition of the term, i.e. category $\mathsf{C}$ such that the quiver obtained from $\mathsf{C}$ by applying the forgetful functor $Cat\rightarrow Quiv$ does not contain any directed-cycle-of-length-at- least-two-without-repeated-vertices is probably much too combinatorial and strict (use of equality, use of negations…), but what about

acirclic category $:=$ category $\mathsf{C}$ such that there does not exist any finite sequence of objects $O_0,...,O_{\ell-1}$ such that

• no two of the $O_i$ are isomorphic in $\mathsf{C}$
• the hom-set $\mathsf{C}(O_i,O_{i+1})$ is non-empty for each $i\in\ell$, with index-calculations modulo $\ell$.

This at least does not speak of equality of objects.

I never saw this being an issue in category theory anywhere. In this case I am not saying this should be studied.

At least according to some definitions of importance, it evidently is unimportant from a category-theoretical perspective. Possible this has to do with the definition making too much use of negatives. Another aspect is that acirclic categories are intuitively very far from groupoids.

One instructive aspect of this might be to try to characterize acirclicity in categorical terms.

Terminological note: chose “acirclic” since it is more distinctive and different from the many standard uses of “acyclic”

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Technical term for the category having morphism precisely the families of parallel morphisms? https://nforum.ncatlab.org/discussion/7827/ 2017-06-07T08:12:51-04:00 2017-06-10T06:52:14-04:00 Peter Heinig https://nforum.ncatlab.org/account/1588/ If &Dscr;\mathcal{D} is a category, one may define a category with the same objects but having morphisms precisely all possible families of parallel morphisms of &Dscr;\mathcal{D}, in the ...

If $\mathcal{D}$ is a category, one may define a category with the same objects but having morphisms precisely all possible families of parallel morphisms of $\mathcal{D}$, in the obvious way.

It appears practically certain that this is a standard construction with a usual technical name; would you please tell what it is?

In short: it is possible to define compositions of one hom-set with another hom-set, and thus get categories having the hom-sets themselves as the morphisms; is there a usual technical term for this?

(In a sense, it is the straightforward generalization of what are called “Minkowski sums” or “sum-sets” in the special situation of commutative monoids, but this question concerns the general construction for any category. It can probably also made into an endofunctor of the category of all small categories.)

Perhaps needlessly, details:

Suppose $\mathcal{D}$ is a category.

Let $\mathcal{D}^+$ denote the category which has

• precisely the same objects as $\mathcal{D}$
• for arbitrary objects $O,O'$ of $\mathcal{D}^+$ we define the hom-‘set’ to be

$\mathcal{D}^+(O,O') :=$ class of all class-indexed families $\mathcal{M}=\{ h_i\colon i\in I\}$,

with $I$ a class and each $h_i\in\mathcal{D}(O,O')$.

(We note that each morphism is a class of parallel morphisms.)

Composition is defined in the obvious way: if $O,O',O''$ are objects,

and if $\{ h_i\colon i\in I\}\in\mathcal{D}^+(O,O')$ and $\{ h_i'\colon i\in I'\}\in\mathcal{D}^+(O',O'')$, then

$\{ h_i'\colon i\in I'\}\circ\{ h_j\colon j\in I\} := \{ h_i'\circ h_j\colon (i,j)\in I'\times I\}$

(We note that $\mathrm{dom}(h_i')=\mathrm{cod}(h_j)$ for all indices, so all the compositions are defined.)

(We note that the identity morphism at an object of $\mathcal{D}^+$ is the singleton-indexed class containing only the singleton-morphism of $\mathcal{D}$ at that object.)

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How to call the full subcategory of convergent diagrams within a functor category? https://nforum.ncatlab.org/discussion/7833/ 2017-06-10T05:56:27-04:00 2017-06-10T05:56:27-04:00 Peter Heinig https://nforum.ncatlab.org/account/1588/ How would you call the full subcategory of a functor category [J,C] consisting only of those functors which have a limit in C ? Of course, the description just given seems to do just fine, but do ...

How would you call the full subcategory of a functor category [J,C] consisting only of those functors which have a limit in C ?

Of course, the description just given seems to do just fine, but do you prefer a technical term for that?

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