nForum - Search Results Feed (Tag: category-theory) 2021-09-23T23:14:33-04:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher internal hom https://nforum.ncatlab.org/discussion/3372/ 2011-12-07T17:51:11-05:00 2021-09-07T15:44:53-04:00 Urs https://nforum.ncatlab.org/account/4/ at internal hom the following discussion was sitting. I hereby move it from there to here Here's some discussion on notation: Ronnie: I have found it convenient in a number of categories to use ...

at internal hom the following discussion was sitting. I hereby move it from there to here

Here's some discussion on notation:

Ronnie: I have found it convenient in a number of categories to use the convention that if say the set of morphisms is $hom(x,y)$ then the internal hom when it exists is $HOM(x,y)$. In particular we have the exponential law for categories

$Cat(x \times y,z) \cong Cat(x,CAT(y,z)).$

Then one can get versions such as $CAT_a(y,z)$ if $y,z$ are objects over $a$.

Of course to use this the name of the category needs more than one letter. Also it obviates the use of those fonts which do not have upper and lower case, so I have tended to use mathsf, which does not work here!

How do people like this? Of course, panaceas do not exist.

Toby: I see, that fits with using $\CAT$ as the $2$-category of categories but $\Cat$ as the category of categories. (But I'm not sure if that's a good thing, since I never liked that convention much.) I only used ’Hom’ for the external hom here since Urs had already used ’hom’ for the internal hom.

Most of the time, I would actually use the same symbol for both, just as I use the same symbol for both a group and its underlying set. Every closed category is a concrete category (represented by $I$), and the underlying set of the internal hom is the external hom. So I would distinguish them only when looking at the theorems that relate them, much as I would bother parenthesising an expression like $a b c$ only when stating the associative law.

Ronnie: In the case of crossed complexes it would be possible to use $Crs_*(B,C)$ for the internal hom and then $Crs_0(B,C)$ is the actual set of morphisms, with $Crs_1(B,C)$ being the (left 1-) homotopies.

But if $G$ is a groupoid does $x \in G$ mean $x$ is an arrow or an object? The group example is special because a group has only one object.

If $G$ is a group I like to distinguish between the group $Aut(G)$ of automorphisms, and the crossed module $AUT(G)$, some people call it the actor, which is given by the inner automorphism map $G \to Aut(G)$, and this seems convenient. Similarly if $G$ is a groupoid we have a group $Aut(G)$ of automorphisms but also a group groupoid, and so crossed module, $AUT(G)$, which can be described as the maximal subgroup object of the monoid object $GPD(G,G)$ in the cartesian closed closed category of groupoids.

Toby: ’But if $G$ is a groupoid does $x \in G$ mean $x$ is an arrow or an object?’: I would take it to mean that $x$ is an object, but I also use $\mathbf{B}G$ for the pointed connected groupoid associated to a group $G$; I know that groupoid theorists descended from Brandt wouldn't like that. I would use $x \in \Arr(G)$, where $\Arr(G)$ is the arrow category (also a groupoid now) of $G$, if you want $x$ to be an arrow. (Actually I don't like to use $\in$ at all to introduce a variable, preferring the type theorist's colon. Then $x: G$ introduces $x$ as an object of the known groupoid $G$, $f: x \to y$ introduces $f$ as a morphism between the known objects $x$ and $y$, and $f: x \to y: G$ introduces all three variables. This generalises consistently to higher morphisms, and of course it invites a new notation for a hom-set: $x \to y$.)

continued in next comment…

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Grothendieck toposes as bridges https://nforum.ncatlab.org/discussion/12465/ 2021-03-19T05:50:47-04:00 2021-09-06T18:15:59-04:00 Korman https://nforum.ncatlab.org/account/2706/ Am not sure if this page exists on the nlab, but I think a nice presentation of this proof technique, would be useful in the development of “new ways of doing mathematics”. I would also like to ...

Am not sure if this page exists on the nlab, but I think a nice presentation of this proof technique, would be useful in the development of “new ways of doing mathematics”. I would also like to know of other proof techniques in category theory that are useful but happen to be unpopular, maybe a page can be created to handle such? I would appreciate an example that illustrates transformation of a proof in theory A to a proof in theory B(a concept we may have taken for granted? I don’t know). You can view Olivia Caramello’s general idea

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coequalizer https://nforum.ncatlab.org/discussion/7071/ 2016-04-21T05:27:23-04:00 2021-09-04T08:01:51-04:00 Urs https://nforum.ncatlab.org/account/4/ I have added to coequalizer basic statements about its relation to pushouts. In the course of this I brought the whole entry into better shape.

I have added to coequalizer basic statements about its relation to pushouts.

In the course of this I brought the whole entry into better shape.

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monoidal category https://nforum.ncatlab.org/discussion/4226/ 2012-09-11T11:36:39-04:00 2021-08-29T02:06:13-04:00 Urs https://nforum.ncatlab.org/account/4/ Todd, when you see this here and have a minute, would you mind having a look at monoidal category to see if you can remove the query-box discussion there and maybe replace it by some crisp ...

Todd,

when you see this here and have a minute, would you mind having a look at monoidal category to see if you can remove the query-box discussion there and maybe replace it by some crisp statement?

Thanks!

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internal category https://nforum.ncatlab.org/discussion/621/ 2010-01-11T09:03:56-05:00 2021-08-28T13:54:36-04:00 Urs https://nforum.ncatlab.org/account/4/ I edited the formatting of [[internal category]] a bit and added a link to [[internal infinity-groupoid]] it looks like the first query box discussion there has been resolved. Maybe we can remove ...

I edited the formatting of [[internal category]] a bit and added a link to [[internal infinity-groupoid]]

it looks like the first query box discussion there has been resolved. Maybe we can remove that box now?

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Diagram of Examples of Categories https://nforum.ncatlab.org/discussion/13381/ 2021-08-24T19:36:06-04:00 2021-08-25T03:03:51-04:00 Leopold https://nforum.ncatlab.org/account/2969/ I was thinking about making a diagram of examples of categories, organized by which category is a subcategory of another (up to isomorphism). Then I realized that this might lend itself nicely to a ... I was thinking about making a diagram of examples of categories, organized by which category is a subcategory of another (up to isomorphism). Then I realized that this might lend itself nicely to a collaborative effort, and also might be a nice addition to the "database of categories" entry. I'm drawing the diagram with TikZ's automated graph drawing with LuaLaTeX, saving any fine-tuning for the end (so right now it's somewhat messy). If you like this idea or would like to contribute, I suggest that the tex file just be reposted with changes. If this is the wrong place for this discussion, please let me know too, since I'm knew here. Thanks!

Current tex file (surely with some mistakes): (output: https://imgur.com/a/BxcuqGE )

% !TeX engine = lualatex
\documentclass[tikz]{standalone}
\usetikzlibrary{cd, arrows, graphs, graphdrawing}
\usegdlibrary{layered}
\newcommand{\cat}{\ensuremath{\mathbf{#1}}}
\begin{document}
\begin{tikzpicture}[>={Stealth}, rounded corners]
\graph [layered layout, nodes={font=\bfseries}]{
Vect -> Mod -> Ab -> {Grp, CMon -> Mon -> Cat};
Grp -> {Loop, ISGrp, Mon};
SGrp -> SGrpd;
Grp -> Grpd -> {SGrpd, pSet, Cat -> SGrpd -> Set};
{UMag, Loop} -> pSet -> Set;
{{ISGrp, Mon} -> SGrp, {Loop, ISGrp} -> QGrp, {Loop, Mon} -> UMag} -> Mag -> Set;
Field -> CRing -> Ring -> Grp;
CRing -> Ab;
Poset -> Set;
Hilb -> IPS -> Norm -> TVect -> Top -> Set;
Hilb -> Ban -> CMet -> Met -> "Top\textsubscript{Haus}" -> Top;
Ban -> Norm -> Met;
TVect -> Vect;
Set -> Rel;
};
\end{tikzpicture}
\end{document} ]]>
Quillen equivalence https://nforum.ncatlab.org/discussion/7165/ 2016-06-23T09:58:16-04:00 2021-08-20T04:33:56-04:00 Urs https://nforum.ncatlab.org/account/4/ I have added the characterization of Quillen equivalences in the case that the right adjoint creates weak equivalences, here.

I have added the characterization of Quillen equivalences in the case that the right adjoint creates weak equivalences, here.

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polynomial functor https://nforum.ncatlab.org/discussion/5596/ 2014-01-10T10:05:55-05:00 2021-08-11T02:52:45-04:00 Urs https://nforum.ncatlab.org/account/4/ added to polynomial functor the evident but previously missing remark why it is called a “polynomial”, here.

added to polynomial functor the evident but previously missing remark why it is called a “polynomial”, here.

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ambidextrous adjunction https://nforum.ncatlab.org/discussion/5594/ 2014-01-10T08:05:52-05:00 2021-07-26T09:20:38-04:00 Urs https://nforum.ncatlab.org/account/4/ started a stub for ambidextrous adjunction, but not much there yet

started a stub for ambidextrous adjunction, but not much there yet

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sylleptic 3-group https://nforum.ncatlab.org/discussion/4442/ 2012-10-26T02:11:48-04:00 2021-07-21T12:56:09-04:00 Urs https://nforum.ncatlab.org/account/4/ sylleptic 3-group … … whereby the periodic table is fianlly un-grayed

… whereby the periodic table is fianlly un-grayed

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braided 2-group https://nforum.ncatlab.org/discussion/4438/ 2012-10-25T23:25:16-04:00 2021-07-21T11:31:20-04:00 Urs https://nforum.ncatlab.org/account/4/ braided 2-group

braided 2-group

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symmetric 2-group https://nforum.ncatlab.org/discussion/4441/ 2012-10-26T02:02:20-04:00 2021-07-21T10:49:34-04:00 Urs https://nforum.ncatlab.org/account/4/ symmetric 2-group

symmetric 2-group

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reflective subcategory https://nforum.ncatlab.org/discussion/446/ 2009-11-26T11:13:51-05:00 2021-07-20T16:49:09-04:00 Urs https://nforum.ncatlab.org/account/4/ edited [[reflective subcategory]] and expanded a bit the beginning

edited [[reflective subcategory]] and expanded a bit the beginning

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Dold-Kan correspondence https://nforum.ncatlab.org/discussion/394/ 2009-11-17T16:13:21-05:00 2021-07-14T10:49:14-04:00 Urs https://nforum.ncatlab.org/account/4/ added reference to dendroidal version of Dold-Kan correspondence

added reference to dendroidal version of Dold-Kan correspondence

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cylinder object https://nforum.ncatlab.org/discussion/7040/ 2016-04-08T11:29:58-04:00 2021-07-13T07:28:37-04:00 Urs https://nforum.ncatlab.org/account/4/ I added to cylinder object a pointer to a reference that goes through the trouble of spelling out the precise proof that for XX a CW-complex, then the standard cyclinder X&times;IX \times I is ...

I added to cylinder object a pointer to a reference that goes through the trouble of spelling out the precise proof that for $X$ a CW-complex, then the standard cyclinder $X \times I$ is again a cell complex (and the inclusion $X \sqcup X \to X\times I$ a relative cell complex).

What would be a text that features a graphics which illustrates the simple idea of the proof, visualizing the induction step where we have the cylinder over $X_n$, then the cells of $X_{n+1}$ glued in at top and bottom, then the further $(n+1)$-cells glued into all the resulting hollow cylinders? (I’d like to grab such graphics to put it in the entry, too lazy to do it myself. )

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monoidal model category https://nforum.ncatlab.org/discussion/7034/ 2016-04-04T13:01:48-04:00 2021-07-11T07:51:34-04:00 Urs https://nforum.ncatlab.org/account/4/ I have added to monoidal model category statement and proof (here) of the basic statement: Let (&Cscr;,&otimes;)(\mathcal{C}, \otimes) be a monoidal model category. Then 1) the left ...

I have added to monoidal model category statement and proof (here) of the basic statement:

Let $(\mathcal{C}, \otimes)$ be a monoidal model category. Then 1) the left derived functor of the tensor product exsists and makes the homotopy category into a monoidal category $(Ho(\mathcal{C}), \otimes^L, \gamma(I))$. If in in addition $(\mathcal{C}, \otimes)$ satisfies the monoid axiom, then 2) the localization functor $\gamma\colon \mathcal{C}\to Ho(\mathcal{C})$ carries the structure of a lax monoidal functor

$\gamma \;\colon\; (\mathcal{C}, \otimes, I) \longrightarrow (Ho(\mathcal{C}), \otimes^L , \gamma(I)) \,.$

The first part is immediate and is what all authors mention. But this is useful in practice typically only with the second part.

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category of fibrant objects https://nforum.ncatlab.org/discussion/6090/ 2014-07-15T11:27:12-04:00 2021-07-06T23:38:39-04:00 Urs https://nforum.ncatlab.org/account/4/ in reaction to an email discussion I had, I have finally added to the section Derived hom-spaces at category of fibrant objects the definition and theorem that had been alluded to there all along.

in reaction to an email discussion I had, I have finally added to the section Derived hom-spaces at category of fibrant objects the definition and theorem that had been alluded to there all along.

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cyclic set https://nforum.ncatlab.org/discussion/5822/ 2014-03-31T12:25:03-04:00 2021-06-29T03:27:18-04:00 Urs https://nforum.ncatlab.org/account/4/ Have added to cyclic set a pointer to notes from 1996 by Ieke Moerdijk where the theory classified by the topos of cyclic sets is identified (abstract circles). This is an unpublished note, but on ...

Have added to cyclic set a pointer to notes from 1996 by Ieke Moerdijk where the theory classified by the topos of cyclic sets is identified (abstract circles).

This is an unpublished note, but on request I have now uploaded it to the nLab

• Ieke Moerdijk, Cyclic sets as a classifying topos, 1996 (pdf)

I have also added a corresponding brief section to classifying topos.

By the way, there is an old query box with an exchange between Mike and Zoran at cyclic set. It seems to me that this has been resolved and the query box could be removed (to make the entry read more smoothly). Maybe Mike and/or Zoran could briefly look into this.

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monad https://nforum.ncatlab.org/discussion/1563/ 2010-07-08T16:31:03-04:00 2021-06-22T18:13:28-04:00 Urs https://nforum.ncatlab.org/account/4/ following Zoran’s suggestion I added to the beginning of the Idea-section at monad a few sentences on the general idea, leading then over to the Idea with respect to algebraic theories that used to ...

following Zoran’s suggestion I added to the beginning of the Idea-section at monad a few sentences on the general idea, leading then over to the Idea with respect to algebraic theories that used to be the only idea given there.

Also added a brief stub-subsection on monads in arbitrary 2-categories. This entry deserves a bit more atention.

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Segal condition https://nforum.ncatlab.org/discussion/3740/ 2012-04-17T00:55:01-04:00 2021-06-18T17:32:05-04:00 Urs https://nforum.ncatlab.org/account/4/ finally a stub for Segal condition. Just for completeness (and to have a sensible place to put the references about Segal conditions in terms of sheaf conditions).

finally a stub for Segal condition. Just for completeness (and to have a sensible place to put the references about Segal conditions in terms of sheaf conditions).

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cartesian model category https://nforum.ncatlab.org/discussion/5226/ 2013-09-04T18:09:18-04:00 2021-06-14T14:21:46-04:00 Zhen Lin https://nforum.ncatlab.org/account/318/ I added an explicit definition of cartesian model category to cartesian closed model category to highlight the convention that the terminal object is assumed cofibrant.

I added an explicit definition of cartesian model category to cartesian closed model category to highlight the convention that the terminal object is assumed cofibrant.

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semicategory https://nforum.ncatlab.org/discussion/4541/ 2012-11-20T11:56:41-05:00 2021-06-07T15:56:25-04:00 Urs https://nforum.ncatlab.org/account/4/ I have touched the following entries, trying to interlink them more closely by added sentences with cross-links that indicate how they relate to each other: semicategory, semi-simplicial set, ...

I have touched the following entries, trying to interlink them more closely by added sentences with cross-links that indicate how they relate to each other:

Also linked for instance to semicategory from category, etc.

Linked also to Delta space, but the entry doe not exist yet.

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accessible category https://nforum.ncatlab.org/discussion/620/ 2010-01-10T17:58:37-05:00 2021-06-01T16:22:02-04:00 Urs https://nforum.ncatlab.org/account/4/ quickly added at [[accessible category]] parts of the MO discussion here. Since Mike participated there, I am hoping he could add more, if necessary.

quickly added at [[accessible category]] parts of the MO discussion here. Since Mike participated there, I am hoping he could add more, if necessary.

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total category https://nforum.ncatlab.org/discussion/5396/ 2013-11-04T22:46:50-05:00 2021-05-31T15:38:13-04:00 Urs https://nforum.ncatlab.org/account/4/ at total category I have added after the definition and after the first remark these two further remarks: +– {: .num_remark} Remark Since the Yoneda embedding is a full and faithful functor, a ...

at total category I have added after the definition and after the first remark these two further remarks:

+– {: .num_remark}

###### Remark

Since the Yoneda embedding is a full and faithful functor, a total category $C$ induces an idempotent monad $Y \circ L$ on its category of presheaves, hence a modality. One says that $C$ is a totally distributive category if this modality is itself the right adjoint of an adjoint modality.

=–

+– {: .num_remark}

###### Remark

The $(L \dashv Y)$-adjunction of a total category is closely related to the $(\mathcal{O} \dashv Spec)$-adjunction discussed at Isbell duality and at function algebras on ∞-stacks. In that context the $L Y$-modality deserves to be called the affine modality.

=–

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conservative functor https://nforum.ncatlab.org/discussion/6747/ 2015-09-08T13:46:31-04:00 2021-05-26T16:48:30-04:00 Urs https://nforum.ncatlab.org/account/4/ added to conservative functor the proposition saying that pullback along strong epis is a conservative functor (if strong epis pull back). How about the &infin;\infty-version?

added to conservative functor the proposition saying that pullback along strong epis is a conservative functor (if strong epis pull back).

How about the $\infty$-version?

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