Cattheory - User contributions [en]

MediaWiki:Sitenotice

2024-09-15T05:46:04Z

Vipul:

Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]] 
Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]]

User:Vipul/Sandbox

2024-09-15T05:45:37Z

Vipul:

* <math>\sqrt{7 + 2}!! + 3 = 723</math>
* <math>2^{8 - 1} = 128</math>
* <math>2^7 - 1 = 127</math>
* <math>(7 - 4)!! + 4! = 744</math>
* <math>(3 + 4)^3 = 343</math>

MediaWiki:Sitenotice

2024-09-08T17:59:53Z

Vipul:

'''This site is in the process of being migrated to a new server. Edits made until this notice has been removed may be lost.''' 
Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]] 
Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]]

MediaWiki:Sitenotice

2024-08-11T17:51:15Z

Vipul:

Cattheory:429 Too Many Requests error

2024-08-11T17:49:40Z

Vipul: Created page with "This content is copied from Ref:Ref:429 Too Many Requests error. If you get a 429 Too Many Requests error when browsing this site, read on. You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work. If you are an actual h..."

This content is copied from [[Ref:Ref:429 Too Many Requests error]].

If you get a 429 Too Many Requests error when browsing this site, read on.

You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work.

If you are an actual human being with a legitimate reason to be browsing the site heavily, first, thank you and sorry about this! We set rate limits to prevent bots, spiders, spammers, and malicious actors from consuming too much of our server's resources so that our server's resources can be devoted to real humans like you. Consider writing to vipulnaik1@gmail.com with your IP address to have the IP address whitelisted. You can get your IP address by [https://www.google.com/search?q=my+ip+address Googling "my IP address"] (scroll down a little bit to where Google includes the IP address in a box). NOTE: If you have both an IPv4 address and an IPv6 address, you should send both; the server supports both IPv4 and IPv6, so either may end up getting used. To check if you have an IPv6 address, try visiting [https://ipv6.google.com/ ipv6.google.com].

If your IP address changes, or you are away from your home network, then you'll get rate-limited again. So if you find yourself getting rate-limited after already having been whitelisted, check if you are on a different IP address than the one for which you requested whitelisting.

File:Site search autocompletion working.png

2024-08-11T17:48:58Z

Vipul:

File:Site search autocompletion broken.png

2024-08-11T17:48:21Z

Vipul:

Cattheory:Enabling site search autocompletion

2024-08-11T17:44:49Z

Vipul: Created page with "Content copied from Ref:Ref:Enabling site search autocompletion. Images used are specific to this site (Cattheory). Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on. ==What's wrong with site search autocompletion and how to fix it== ===What's wrong=== When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in..."

Content copied from [[Ref:Ref:Enabling site search autocompletion]]. Images used are specific to this site (Cattheory).

Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on.

==What's wrong with site search autocompletion and how to fix it==

===What's wrong===

When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in the screenshot below:

[[File:Site search autocompletion broken.png]]

Note that the actual search is still working -- you just have to hit Enter after typing the search query and it'll go to the search results page. It's the autocompletion before you hit Enter that is broken.

===How to fix it===

To fix it, you need to follow these steps:

* Write to vipulnaik1@gmail.com asking for a login to the site. Please include the following with your request: preferred username, preferred initial password (you can change it after logging in), real name (if you want it entered), email address to use (if you want an actual email address by which you can be contacted), and whether you want edit access as well. You don't need edit access for enabling site search autocompletion.
* Log in to the site. Then go to [[Special:Preferences]]. Go to the Appearance section and switch the Skin from "Vector (2022)" to "Vector legacy (2010)".
* Make sure to hit "Save" at the bottom.
* Now you can reload the page or load a new page.

Site search autocompletion should now work. Here's an example:

[[File:Site search autocompletion working.png]]

==More background==

We've recently upgraded the MediaWiki version of this wiki from 1.35.13 to 1.41.2 (see [[Special:Version]]). The upgrade allows us to migrate the wiki to a more modern operating system version running PHP 8. With the current setup for MediaWiki 1.41.2, we're in this situation:

* The "Vector legacy (2010)" skin has site search autocompletion working, but it doesn't render well on small screens. Specifically, even on small mobile screens, it still shows the left menu, and doesn't properly use the MobileFrontend extension settings.
* The "Vector (2022)" skin doesn't have site search autocompletion working (see screenshots in preceding section) but it does render fine on mobile devices.

It is possible to set only one default skin (that is applicable to all non-logged-in users and is the default for logged-in users who have not configured a skin for themselves). So, the selection of default skin comes down to whether it's more important for casual users to have the mobile experience working or to have site search autocompletion working. Based on a general understanding of user behavior, we believe that having a usable mobile experience is more important for casual users than having site search autocompletion.

However, for power users who are using the site extensively, site search autocompletion may be important. That's why we've written this page giving guidance on how to set up site search autocompletion.

MediaWiki:Sitenotice

2024-08-11T17:43:40Z

Vipul: Blanked the page

User:Vipul/Sandbox

2024-08-11T17:42:06Z

Vipul:

* <math>\sqrt{7 + 2}!! + 3 = 723</math>
* <math>2^{8 - 1} = 128</math>
* <math>2^7 - 1 = 127</math>
* <math>(7 - 4)!! + 4! = 744</math>

User:Vipul/Sandbox

2024-08-11T17:37:26Z

Vipul:

* <math>\sqrt{7 + 2}!! + 3 = 723</math>
* <math>2^{8 - 1} = 128</math>
* <math>2^7 - 1 = 127</math>

User:Vipul/Sandbox

2024-08-11T17:34:25Z

Vipul:

* <math>\sqrt{7 + 2}!! + 3 = 723</math>
* <math>2^{8 - 1} = 128</math>

Template:Top notice

2024-08-11T17:33:54Z

Vipul:

{{quotation|Welcome to '''{{fullsitetitle}}'''. This is a pre-pre-alpha stage category theory wiki primarily managed by [[User:Vipul|Vipul Naik]], a Ph.D. in Mathematics at the University of Chicago. It is part of a broader subject wikis initiative -- see the [[Ref:Main Page|subject wikis reference guide]] for more details.}}

MediaWiki:Sitenotice

2024-08-11T17:27:44Z

Vipul:

'''This wiki is in the process of being upgraded. The site may go down intermittently. Please try to avoid editing until this notice has been removed.'''

User:Vipul/Sandbox

2024-07-20T07:10:54Z

Vipul: Created page with "* <math>\sqrt{7 + 2}!! + 3 = 723</math>"

* <math>\sqrt{7 + 2}!! + 3 = 723</math>

Cattheory:Privacy policy

2022-09-25T15:34:58Z

Vipul:

This privacy policy is common to subject wikis. For the original privacy policy, refer [[Ref:Ref:Privacy policy]].

==Privacy for readers==

If you are surfing this website, your actions are logged in our usage logs. These usage logs are accessible to:

* The site's administrators and technical support group. For a full list of administrators, contact [[User:Vipul|Vipul Naik]] by email: vipulnaik1@gmail.com.
* The service that hosts the data and servers, which is currently [http://www.linode.com Linode].
* Google Analytics, which has been integrated to collect site statistics. View Google's privacy policy here: http://www.google.com/intl/en_ALL/privacypolicy.html
* Other third-party JS scripts that collect user activity; none of these should collect any personally identifiable information (PII). For a list of all scripts running at the current time, contact [[User:Vipul|Vipul Naik]] by email: vipulnaik1@gmail.com.

==Privacy for editors==

Editing on subject wikis is generally permitted only for registered users. Registered users must, at the time of registration, provide their real name, and enter basic information about their reason for interest. ''No'' private information such as date of birth, social security or taxation number, or home address is sought.

Regarding personal information:

* The email IDs of registered users are visible to site administrators only. For information about site administrators, contact vipulnaik1@gmail.com with the particular subject wiki and the reason for request.
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* Passwords chosen by registered users are not humanly accessible, even to site administrators.

MediaWiki:Sitenotice

2013-06-24T18:44:54Z

Vipul:

[[Main Page|{{fullsitetitle}} ({{sitestatus}})]]

2-category

2009-12-29T15:22:08Z

Vipul: /* Raw definition */

==Definition==

The notion discussed here is that of a '''strict 2-category'''. For the somewhat more general notion that is also loosely referred to by the same name, refer [[bicategory]]. The main difference is the loosening of the requirement for strict composition of the 1-morphisms.
===Definition as a category enriched over categories===

A 2-category is an category [[defining ingredient::enriched category|enriched]] over the monoidal category whose objects are categories, morphisms are functors, and whose monoidal operation is the product of categories. (To ease foundational issues, we can restrict attention to the monoidal category of [[locally small category|locally small categories]] or [[small category|small categories]]).

===Raw definition===

A '''2-category''' <math>\mathcal{C}</math> is the following data:

* '''Objects''': A [[defining ingredient::collection]] <math>\operatorname{Ob}\mathcal{C}</math> of '''objects'''.
* '''1-morphisms''': For any objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, a collection <math>\mathcal{C}(A,B)</math> of '''morphisms'''. Every element in <math>\mathcal{C}_1(A,B)</math> is termed a ''morphism'' from <math>A</math> (i.e., with source or domain <math>A</math>) to <math>B</math> (i.e., with target or co-domain <math>B</math>). The morphism sets for different pairs of objects are disjoint. Note that <math>f \in \mathcal{C}(A,B)</math> is also written as <math>f:A \to B</math>. The collection <math>\mathcal{C}(A,B)</math> is sometimes also denoted <math>\operatorname{Hom}_{\mathcal{C}}(A,B)</math> or simply <math>\operatorname{Hom}(A,B)</math>.
* '''Identity 1-morphism''': For every object <math>A \in \operatorname{Ob}\mathcal{C}</math>, a distinguished morphism <math>\operatorname{id}_A \in \mathcal{C}(A,A)</math>. This is called the identity morphism of <math>A</math>.
* '''Composition rule''': For <math>A,B,C \in \operatorname{Ob}\mathcal{C}</math>, a map, called ''composition of morphisms'', from <math>\mathcal{C}(B,C) \times \mathcal{C}(A,B)</math> to <math>\mathcal{C}(A,C)</math>. This map is denoted by <math>\circ</math>.
* '''2-morphisms''': For any two objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math> and any two morphisms <math>f,g \in \mathcal{C}_1(A,B)</math>, a collection <math>\mathcal{C}_2(f,g)</math> of '''2-morphisms''' from <math>f</math> to <math>g</math>. If <math>\alpha</math> is such a morphism, we write <math>\alpha:f \implies g</math>. The collection of such 2-morphisms may be denoted <math>\operatorname{Hom}(f,g)</math>.
* '''Identity 2-morphism''': For any two objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math> and any morphism <math>f \in \operatorname{Hom}(A,B)</math>, a 2-morphism <math>\operatorname{id}_f \in \operatorname{Hom}(f,f)</math>.
* '''Composition of 2-morphisms''': For any two objects <math>A,B</math>, and any three morphisms <math>f,g,h \in \operatorname{Hom}(A,B)</math>, a map <math>\operatorname{Hom}(g,h) \times \operatorname{Hom}(f,g) \to \operatorname{Hom}(f,h)</math>.
* '''Horizontal composition of 2-morphisms''': For any three objects <math>A,B,C</math>, with morphisms <math>f_1,f_2:A \to B</math> and morphisms <math>g_1,g_2:B \to C</math>, an operator <math>\operatorname{Hom}(f_1,f_2) \times \operatorname{Hom}(g_1,g_2) \to \operatorname{Hom}(f_1 \circ g_1, f_2\ circ g_2)</math>.

satisfying the following conditions:

* '''Associativity of composition of 1-morphisms''': For <math>A,B,C,D \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B), g \in \mathcal{C}(B,C), h \in \mathcal{C}(C,D)</math>, we have <math>h \circ (g \circ f) = (h \circ g) \circ f</math>.
* '''Identity 1-morphism behaves as an identity''': For <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B)</math>, we have <math>f \circ \operatorname{id}_A = \operatorname{id}_B \circ f = f</math>.
* '''Associativity of composition of 2-morphisms'''
* '''Identity 2-morphism behaves as an identity'''
* '''Associativity of horizontal composition'''

===Definition in terms of category definition===

A 2-category <math>\mathcal{C}</math> is a [[defining ingredient::category]] (that we'll also call <math>\mathcal{C}</math>), along with, for every <math>A,B \in \mathcal{C}</math>, a category whose collection of objects is the collection of morphisms from <math>A</math> to <math>B</math>, along with a horizontal composition...{{fillin}}

2-category

2009-12-29T15:19:11Z

Vipul:

==Definition==

The notion discussed here is that of a '''strict 2-category'''. For the somewhat more general notion that is also loosely referred to by the same name, refer [[bicategory]]. The main difference is the loosening of the requirement for strict composition of the 1-morphisms.
===Definition as a category enriched over categories===

A 2-category is an category [[defining ingredient::enriched category|enriched]] over the monoidal category whose objects are categories, morphisms are functors, and whose monoidal operation is the product of categories. (To ease foundational issues, we can restrict attention to the monoidal category of [[locally small category|locally small categories]] or [[small category|small categories]]).

===Raw definition===

A '''2-category''' <math>\mathcal{C}</math> is the following data:

* '''Objects''': A [[defining ingredient::collection]] <math>\operatorname{Ob}\mathcal{C}</math> of '''objects'''.
* '''1-morphisms''': For any objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, a collection <math>\mathcal{C}(A,B)</math> of '''morphisms'''. Every element in <math>\mathcal{C}_1(A,B)</math> is termed a ''morphism'' from <math>A</math> (i.e., with source or domain <math>A</math>) to <math>B</math> (i.e., with target or co-domain <math>B</math>). The morphism sets for different pairs of objects are disjoint. Note that <math>f \in \mathcal{C}(A,B)</math> is also written as <math>f:A \to B</math>. The collection <math>\mathcal{C}(A,B)</math> is sometimes also denoted <math>\operatorname{Hom}_{\mathcal{C}}(A,B)</math> or simply <math>\operatorname{Hom}(A,B)</math>.
* '''Identity 1-morphism''': For every object <math>A \in \operatorname{Ob}\mathcal{C}</math>, a distinguished morphism <math>\operatorname{id}_A \in \mathcal{C}(A,A)</math>. This is called the identity morphism of <math>A</math>.
* '''Composition rule''': For <math>A,B,C \in \operatorname{Ob}\mathcal{C}</math>, a map, called ''composition of morphisms'', from <math>\mathcal{C}(B,C) \times \mathcal{C}(A,B)</math> to <math>\mathcal{C}(A,C)</math>. This map is denoted by <math>\circ</math>.
* '''2-morphisms''': For any two objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math> and any two morphisms <math>f,g \in \mathcal{C}_1(A,B)</math>, a collection <math>\mathcal{C}_2(f,g)</math> of '''2-morphisms''' from <math>f</math> to <math>g</math>. If <math>\alpha</math> is such a morphism, we write <math>\alpha:f \implies g</math>. The collection of such 2-morphisms may be denoted <math>\operatorname{Hom}(f,g)</math>.
* '''Identity 2-morphism''': For any two objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math> and any morphism <math>f \in \operatorname{Hom}(A,B)</math>, a 2-morphism <math>\operatorname{id}_f \in \operatorname{Hom}(f,f)</math>.
* '''Composition of 2-morphisms''': For any two objects <math>A,B</math>, and any three morphisms <math>f,g,h \in \operatorname{Hom}(A,B)</math>, a map <math>\operatorname{Hom}(g,h) \times \operatorname{Hom}(f,g) \to \operatorname{Hom}(f,h)</math>.
* '''Horizontal composition of 2-morphisms''': For any three objects <math>A,B,C</math>, with morphisms <math>f_1,f_2:A \to B</math> and morphisms <math>g_1,g_2:B \to C</math>, an operator <math>\operatorname{Hom}(f_1,f_2) \times operatorname{Hom}(g_1,g_2) \to \operatorname{Hom}(f_1 \circ g_1, f_2\ circ g_2)</math>.

satisfying the following conditions:

* '''Associativity of composition of 1-morphisms''': For <math>A,B,C,D \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B), g \in \mathcal{C}(B,C), h \in \mathcal{C}(C,D)</math>, we have <math>h \circ (g \circ f) = (h \circ g) \circ f</math>.
* '''Identity 1-morphism behaves as an identity''': For <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B)</math>, we have <math>f \circ \operatorname{id}_A = \operatorname{id}_B \circ f = f</math>.
* '''Associativity of composition of 2-morphisms'''
* '''Identity 2-morphism behaves as an identity'''
* '''Associativity of horizontal composition'''

===Definition in terms of category definition===

A 2-category <math>\mathcal{C}</math> is a [[defining ingredient::category]] (that we'll also call <math>\mathcal{C}</math>), along with, for every <math>A,B \in \mathcal{C}</math>, a category whose collection of objects is the collection of morphisms from <math>A</math> to <math>B</math>, along with a horizontal composition...{{fillin}}

Enriched category

2009-12-29T15:08:59Z

Vipul: /* Definition */

==Definition==

Suppose <math>\mathcal{M}</math> is a [[defining ingredient::monoidal category]]. A '''category enriched in''' <math>\mathcal{M}</math> is defined as a <math>\mathcal{C}</math> equipped with the following data:

* A [[collection]] of objects <math>\operatorname{Ob}\mathcal{C}</math>.
* For any objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, an object <math>\mathcal{C}(A,B) \in \operatorname{Ob}\mathcal{M}</math>.
* For any object <math>A \in \operatorname{Ob}\mathcal{C}</math>, a ''unit map'' <math>u_A</math> from the unit object in <math>\mathcal{M}</math> to <math>\mathcal{C}(A,A)</math>.
* For any objects <math>A,B,C \in \operatorname{Ob}\mathcal{C}</math>, a morphism in <math>\mathcal{M}</math>:

<math>\circ: \mathcal{C}(B,C) \otimes \mathcal{C}(A,B) \to \mathcal{C}(A,C)</math>

where <math>\otimes</math> denotes the monoidal operation on <math>\mathcal{M}</math>.

satisfying the following conditions:

* For any <math>A,B,C,D \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B), g \in \mathcal{C}(B,C), h \in \mathcal{C}(C,D)</math>, we have <math>h \circ (g \circ f) = (h \circ g) \circ f</math>.
* For any <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, the map <math>\circ: \mathcal{C}(B,B) \otimes \mathcal{C}(A,B) \to \mathcal{C}(A,B)</math>, composed with <math>u_B</math> on the first coordinate, gives the identity map on the second coordinate. Similarly, the other way around. Need to make more precise -- {{fillin}}

2-category

2009-12-29T15:00:59Z

Vipul:

==Definition==

The notion discussed here is that of a '''strict 2-category'''. For the somewhat more general notion that is also loosely referred to by the same name, refer [[bicategory]]. The main difference is the loosening of the requirement for strict composition of the 1-morphisms.
===Definition as a category enriched over categories===

A 2-category is an category [[defining ingredient::enriched category|enriched]] over the [[category of categories]] (in particular cases, it is usually enough to enrich over the [[category of locally small categories]], which presents fewer foundational hassles).

===Raw definition===

A '''2-category''' <math>\mathcal{C}</math> is the following data:

* '''Objects''': A [[defining ingredient::collection]] <math>\operatorname{Ob}\mathcal{C}</math> of '''objects'''.
* '''1-morphisms''': For any objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, a collection <math>\mathcal{C}(A,B)</math> of '''morphisms'''. Every element in <math>\mathcal{C}_1(A,B)</math> is termed a ''morphism'' from <math>A</math> (i.e., with source or domain <math>A</math>) to <math>B</math> (i.e., with target or co-domain <math>B</math>). The morphism sets for different pairs of objects are disjoint. Note that <math>f \in \mathcal{C}(A,B)</math> is also written as <math>f:A \to B</math>. The collection <math>\mathcal{C}(A,B)</math> is sometimes also denoted <math>\operatorname{Hom}_{\mathcal{C}}(A,B)</math> or simply <math>\operatorname{Hom}(A,B)</math>.
* '''Identity 1-morphism''': For every object <math>A \in \operatorname{Ob}\mathcal{C}</math>, a distinguished morphism <math>\operatorname{id}_A \in \mathcal{C}(A,A)</math>. This is called the identity morphism of <math>A</math>.
* '''Composition rule''': For <math>A,B,C \in \operatorname{Ob}\mathcal{C}</math>, a map, called ''composition of morphisms'', from <math>\mathcal{C}(B,C) \times \mathcal{C}(A,B)</math> to <math>\mathcal{C}(A,C)</math>. This map is denoted by <math>\circ</math>.
* '''2-morphisms''': For any two objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math> and any two morphisms <math>f,g \in \mathcal{C}_1(A,B)</math>, a collection <math>\mathcal{C}_2(f,g)</math> of '''2-morphisms''' from <math>f</math> to <math>g</math>. If <math>\alpha</math> is such a morphism, we write <math>\alpha:f \implies g</math>. The collection of such 2-morphisms may be denoted <math>\operatorname{Hom}(f,g)</math>.
* '''Identity 2-morphism''': For any two objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math> and any morphism <math>f \in \operatorname{Hom}(A,B)</math>, a 2-morphism <math>\operatorname{id}_f \in \operatorname{Hom}(f,f)</math>.
* '''Composition of 2-morphisms''': For any two objects <math>A,B</math>, and any three morphisms <math>f,g,h \in \operatorname{Hom}(A,B)</math>, a map <math>\operatorname{Hom}(g,h) \times \operatorname{Hom}(f,g) \to \operatorname{Hom}(f,h)</math>.
* '''Horizontal composition of 2-morphisms''': For any three objects <math>A,B,C</math>, with morphisms <math>f_1,f_2:A \to B</math> and morphisms <math>g_1,g_2:B \to C</math>, an operator <math>\operatorname{Hom}(f_1,f_2) \times operatorname{Hom}(g_1,g_2) \to \operatorname{Hom}(f_1 \circ g_1, f_2\ circ g_2)</math>.

satisfying the following conditions:

* '''Associativity of composition of 1-morphisms''': For <math>A,B,C,D \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B), g \in \mathcal{C}(B,C), h \in \mathcal{C}(C,D)</math>, we have <math>h \circ (g \circ f) = (h \circ g) \circ f</math>.
* '''Identity 1-morphism behaves as an identity''': For <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B)</math>, we have <math>f \circ \operatorname{id}_A = \operatorname{id}_B \circ f = f</math>.
* '''Associativity of composition of 2-morphisms'''
* '''Identity 2-morphism behaves as an identity'''
* '''Associativity of horizontal composition'''

===Definition in terms of category definition===

A 2-category <math>\mathcal{C}</math> is a [[defining ingredient::category]] (that we'll also call <math>\mathcal{C}</math>), along with, for every <math>A,B \in \mathcal{C}</math>, a category whose collection of objects is the collection of morphisms from <math>A</math> to <math>B</math>, along with a horizontal composition...{{fillin}}

Group

2009-12-29T14:57:37Z

Vipul:

==Definition==

===Category-theoretic definition===

A '''group''' is a [[small category]] with one object where all morphisms are isomorphisms. (note that the ''small'' assumption is to ensure that the morphisms form a set; however, it can be dropped for some purposes).

===Universal algebra definition===

A '''group''' is a set <math>G</math> equipped with a binary operation <math>*</math>, a unary operation <math>{}^{-1}</math>, and a constant <math>e</math> such that the following hold:

* '''Associativity''': <math>a * (b * c) = (a * b) * c \ \forall \ a,b,c \in G</math>.
* '''Neutral element''' or '''identity element''': <math>a * e = e * a = a \ \forall \ a \in G</math>.
* '''Inverse element''': <math>a * a^{-1} = a^{-1} * a = e</math>.

(there are other slightly different formulations of this definition).

===Definition building on monoid===

A group is a [[monoid]] in which every element has a two-sided inverse.

===Equivalence of definitions===

The two definitions are equivalent in the following sense: given any group in the category-theoretic set, the morphisms, under composition, form a group in the set-theoretic sense. Conversely, given any group in the set-theoretic sense, we can construct a category with one object and with the morphisms corresponding to the elements of the group.

Further, this equivalence gives an equivalence of categories, where the first category is the subcategory corresponding to groups of the [[category of locally small categories]] (so the morphisms are [[functor]]s), and the second category is the [[category of groups]] in the conventional sense.

==Category of groups==

The collection of groups, with a suitably defined notion of homomorphism, forms a category called the [[category of groups]]. This is a [[concrete category]], and corresponds to the standard way of obtaining a category from the universal algebra definition of group.

==External links==

===Other subject wikis===

* [[Groupprops:Group|Group on the Group Properties Wiki]] -- the most detailed and canonical reference.
* [[Topospaces:Group|Group on the Topology Wiki]]

===Other resources===

* [[nlab:group|Group on nlab]]
* [[Wikipedia:Group|Group on Wikipedia]]
* [[Mathworld:Group|Group on Wolfram Mathworld]]
* [[Planetmath:Group|Group on Planetmath]]

Abelian group

2009-12-29T14:55:27Z

Vipul: Created page with '==Definition== An '''abelian group''' is a group in which any two elements commute. The collection of abelian groups is often studied as a category, called the [[category o…'

==Definition==

An '''abelian group''' is a [[group]] in which any two elements commute.

The collection of abelian groups is often studied as a category, called the [[category of abelian groups]].

2-category

2009-12-29T14:52:16Z

Vipul:

==Definition==

===Definition as a category enriched over categories===

A 2-category is an category [[defining ingredient::enriched category|enriched]] over the [[category of categories]] (in particular cases, it is usually enough to enrich over the [[category of locally small categories]], which presents fewer foundational hassles).

===Raw definition===

A '''2-category''' <math>\mathcal{C}</math> is the following data:

* '''Objects''': A [[defining ingredient::collection]] <math>\operatorname{Ob}\mathcal{C}</math> of '''objects'''.
* '''Morphisms''': For any objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, a collection <math>\mathcal{C}(A,B)</math> of '''morphisms'''. Every element in <math>\mathcal{C}_1(A,B)</math> is termed a ''morphism'' from <math>A</math> (i.e., with source or domain <math>A</math>) to <math>B</math> (i.e., with target or co-domain <math>B</math>). The morphism sets for different pairs of objects are disjoint. Note that <math>f \in \mathcal{C}(A,B)</math> is also written as <math>f:A \to B</math>. The collection <math>\mathcal{C}(A,B)</math> is sometimes also denoted <math>\operatorname{Hom}_{\mathcal{C}}(A,B)</math> or simply <math>\operatorname{Hom}(A,B)</math>.
* '''Identity morphism''': For every object <math>A \in \operatorname{Ob}\mathcal{C}</math>, a distinguished morphism <math>\operatorname{id}_A \in \mathcal{C}(A,A)</math>. This is called the identity morphism of <math>A</math>.
* '''Composition rule''': For <math>A,B,C \in \operatorname{Ob}\mathcal{C}</math>, a map, called ''composition of morphisms'', from <math>\mathcal{C}(B,C) \times \mathcal{C}(A,B)</math> to <math>\mathcal{C}(A,C)</math>. This map is denoted by <math>\circ</math>.
* '''2-morphisms''': For any two objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math> and any two morphisms <math>f,g \in \mathcal{C}_1(A,B)</math>, a collection <math>\mathcal{C}_2(f,g)</math> of '''2-morphisms''' from <math>f</math> to <math>g</math>. If <math>\alpha</math> is such a morphism, we write <math>\alpha:f \implies g</math>. The collection of such 2-morphisms may be denoted <math>\operatorname{Hom}(f,g)</math>.
* '''Identity 2-morphism''': For any two objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math> and any morphism <math>f \in \operatorname{Hom}(A,B)</math>, a 2-morphism <math>\operatorname{id}_f \in \operatorname{Hom}(f,f)</math>.
* '''Composition of 2-morphisms''': For any two objects <math>A,B</math>, and any three morphisms <math>f,g,h \in \operatorname{Hom}(A,B)</math>, a map <math>\operatorname{Hom}(g,h) \times \operatorname{Hom}(f,g) \to \operatorname{Hom}(f,h)</math>.
* '''Horizontal composition of 2-morphisms''': For any three objects <math>A,B,C</math>, with morphisms <math>f_1,f_2:A \to B</math> and morphisms <math>g_1,g_2:B \to C</math>, an operator <math>\operatorname{Hom}(f_1,f_2) \times operatorname{Hom}(g_1,g_2) \to \operatorname{Hom}(f_1 \circ g_1, f_2\ circ g_2)</math>.

satisfying the following conditions:

* '''Associativity of composition of morphisms''': For <math>A,B,C,D \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B), g \in \mathcal{C}(B,C), h \in \mathcal{C}(C,D)</math>, we have <math>h \circ (g \circ f) = (h \circ g) \circ f</math>.
* '''Identity morphism behaves as an identity''': For <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B)</math>, we have <math>f \circ \operatorname{id}_A = \operatorname{id}_B \circ f = f</math>.
* '''Associativity of composition of 2-morphisms'''
* '''Identity 2-morphism behaves as an identity'''
* '''Associativity of horizontal composition'''

===Definition in terms of category definition===

A 2-category <math>\mathcal{C}</math> is a [[defining ingredient::category]] (that we'll also call <math>\mathcal{C}</math>), along with, for every <math>A,B \in \mathcal{C}</math>, a category whose collection of objects is the collection of morphisms from <math>A</math> to <math>B</math>, along with a horizontal composition...{{fillin}}

2-category

2009-12-29T14:50:39Z

Vipul: Created page with '==Definition== A '''2-category''' <math>\mathcal{C}</math> is the following data: * '''Objects''': A defining ingredient::collection <math>\operatorname{Ob}\mathcal{C}</mat…'

==Definition==

A '''2-category''' <math>\mathcal{C}</math> is the following data:

* '''Objects''': A [[defining ingredient::collection]] <math>\operatorname{Ob}\mathcal{C}</math> of '''objects'''.
* '''Morphisms''': For any objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, a collection <math>\mathcal{C}(A,B)</math> of '''morphisms'''. Every element in <math>\mathcal{C}_1(A,B)</math> is termed a ''morphism'' from <math>A</math> (i.e., with source or domain <math>A</math>) to <math>B</math> (i.e., with target or co-domain <math>B</math>). The morphism sets for different pairs of objects are disjoint. Note that <math>f \in \mathcal{C}(A,B)</math> is also written as <math>f:A \to B</math>. The collection <math>\mathcal{C}(A,B)</math> is sometimes also denoted <math>\operatorname{Hom}_{\mathcal{C}}(A,B)</math> or simply <math>\operatorname{Hom}(A,B)</math>.
* '''Identity morphism''': For every object <math>A \in \operatorname{Ob}\mathcal{C}</math>, a distinguished morphism <math>\operatorname{id}_A \in \mathcal{C}(A,A)</math>. This is called the identity morphism of <math>A</math>.
* '''Composition rule''': For <math>A,B,C \in \operatorname{Ob}\mathcal{C}</math>, a map, called ''composition of morphisms'', from <math>\mathcal{C}(B,C) \times \mathcal{C}(A,B)</math> to <math>\mathcal{C}(A,C)</math>. This map is denoted by <math>\circ</math>.
* '''2-morphisms''': For any two objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math> and any two morphisms <math>f,g \in \mathcal{C}_1(A,B)</math>, a collection <math>\mathcal{C}_2(f,g)</math> of '''2-morphisms''' from <math>f</math> to <math>g</math>. If <math>\alpha</math> is such a morphism, we write <math>\alpha:f \implies g</math>. The collection of such 2-morphisms may be denoted <math>\operatorname{Hom}(f,g)</math>.
* '''Identity 2-morphism''': For any two objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math> and any morphism <math>f \in \operatorname{Hom}(A,B)</math>, a 2-morphism <math>\operatorname{id}_f \in \operatorname{Hom}(f,f)</math>.
* '''Composition of 2-morphisms''': For any two objects <math>A,B</math>, and any three morphisms <math>f,g,h \in \operatorname{Hom}(A,B)</math>, a map <math>\operatorname{Hom}(g,h) \times \operatorname{Hom}(f,g) \to \operatorname{Hom}(f,h)</math>.
* '''Horizontal composition of 2-morphisms''': For any three objects <math>A,B,C</math>, with morphisms <math>f_1,f_2:A \to B</math> and morphisms <math>g_1,g_2:B \to C</math>, an operator <math>\operatorname{Hom}(f_1,f_2) \times operatorname{Hom}(g_1,g_2) \to \operatorname{Hom}(f_1 \circ g_1, f_2\ circ g_2)</math>.

satisfying the following conditions:

* '''Associativity of composition of morphisms''': For <math>A,B,C,D \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B), g \in \mathcal{C}(B,C), h \in \mathcal{C}(C,D)</math>, we have <math>h \circ (g \circ f) = (h \circ g) \circ f</math>.
* '''Identity morphism behaves as an identity''': For <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B)</math>, we have <math>f \circ \operatorname{id}_A = \operatorname{id}_B \circ f = f</math>.
* '''Associativity of composition of 2-morphisms'''
* '''Identity 2-morphism behaves as an identity'''
* '''Associativity of horizontal composition'''

===Definition in terms of category definition===

A 2-category <math>\mathcal{C}</math> is a [[defining ingredient::category]] (that we'll also call <math>\mathcal{C}</math>), along with, for every <math>A,B \in \mathcal{C}</math>, a category whose collection of objects is the collection of morphisms from <math>A</math> to <math>B</math>, along with a horizontal composition...{{fillin}}

Group

2009-12-29T13:19:49Z

Vipul: /* External links */

==Definition==

===Category-theoretic definition===

A '''group''' is a [[small category]] with one object where all morphisms are isomorphisms. (note that the ''small'' assumption is to ensure that the morphisms form a set; however, it can be dropped for some purposes).

===Ordinary definition===

A '''group''' is a set <math>G</math> equipped with a binary operation <math>*</math>, a unary operation <math>{}^{-1}</math>, and a constant <math>e</math> such that the following hold:

* '''Associativity''': <math>a * (b * c) = (a * b) * c \ \forall \ a,b,c \in G</math>.
* '''Neutral element''' or '''identity element''': <math>a * e = e * a = a \ \forall \ a \in G</math>.
* '''Inverse element''': <math>a * a^{-1} = a^{-1} * a = e</math>.

(there are other slightly different formulations of this definition).

===Definition building on monoid===

A group is a [[monoid]] in which every element has a two-sided inverse.

===Equivalence of definitions===

The two definitions are equivalent in the following sense: given any group in the category-theoretic set, the morphisms, under composition, form a group in the set-theoretic sense. Conversely, given any group in the set-theoretic sense, we can construct a category with one object and with the morphisms corresponding to the elements of the group.

Further, this equivalence gives an equivalence of categories, where the first category is the subcategory corresponding to groups of the [[category of locally small categories]] (so the morphisms are [[functor]]s), and the second category is the [[category of groups]] in the conventional sense.

==External links==

===Other subject wikis===

* [[Groupprops:Group|Group on the Group Properties Wiki]] -- the most detailed and canonical reference.
* [[Topospaces:Group|Group on the Topology Wiki]]

===Other resources===

* [[nlab:group|Group on nlab]]
* [[Wikipedia:Group|Group on Wikipedia]]
* [[Mathworld:Group|Group on Wolfram Mathworld]]
* [[Planetmath:Group|Group on Planetmath]]

Group

2009-12-29T13:01:09Z

Vipul: /* Other resources */

==Definition==

===Category-theoretic definition===

A '''group''' is a [[small category]] with one object where all morphisms are isomorphisms. (note that the ''small'' assumption is to ensure that the morphisms form a set; however, it can be dropped for some purposes).

===Ordinary definition===

A '''group''' is a set <math>G</math> equipped with a binary operation <math>*</math>, a unary operation <math>{}^{-1}</math>, and a constant <math>e</math> such that the following hold:

* '''Associativity''': <math>a * (b * c) = (a * b) * c \ \forall \ a,b,c \in G</math>.
* '''Neutral element''' or '''identity element''': <math>a * e = e * a = a \ \forall \ a \in G</math>.
* '''Inverse element''': <math>a * a^{-1} = a^{-1} * a = e</math>.

(there are other slightly different formulations of this definition).

===Definition building on monoid===

A group is a [[monoid]] in which every element has a two-sided inverse.

===Equivalence of definitions===

The two definitions are equivalent in the following sense: given any group in the category-theoretic set, the morphisms, under composition, form a group in the set-theoretic sense. Conversely, given any group in the set-theoretic sense, we can construct a category with one object and with the morphisms corresponding to the elements of the group.

Further, this equivalence gives an equivalence of categories, where the first category is the subcategory corresponding to groups of the [[category of locally small categories]] (so the morphisms are [[functor]]s), and the second category is the [[category of groups]] in the conventional sense.

==External links==

===Other subject wikis===

* [[Groupprops:Group|Group on the Group Properties Wiki]] -- the most detailed and canonical reference.
* [[Topospaces:Group|Group on the Topology Wiki]]

===Other resources===

* [[nlab:group|group on nlab]]
* [[Wikipedia:Group|Group on Wikipedia]]

Group

2009-12-29T13:00:38Z

Vipul: /* Other resources */

==Definition==

===Category-theoretic definition===

A '''group''' is a [[small category]] with one object where all morphisms are isomorphisms. (note that the ''small'' assumption is to ensure that the morphisms form a set; however, it can be dropped for some purposes).

===Ordinary definition===

A '''group''' is a set <math>G</math> equipped with a binary operation <math>*</math>, a unary operation <math>{}^{-1}</math>, and a constant <math>e</math> such that the following hold:

* '''Associativity''': <math>a * (b * c) = (a * b) * c \ \forall \ a,b,c \in G</math>.
* '''Neutral element''' or '''identity element''': <math>a * e = e * a = a \ \forall \ a \in G</math>.
* '''Inverse element''': <math>a * a^{-1} = a^{-1} * a = e</math>.

(there are other slightly different formulations of this definition).

===Definition building on monoid===

A group is a [[monoid]] in which every element has a two-sided inverse.

===Equivalence of definitions===

The two definitions are equivalent in the following sense: given any group in the category-theoretic set, the morphisms, under composition, form a group in the set-theoretic sense. Conversely, given any group in the set-theoretic sense, we can construct a category with one object and with the morphisms corresponding to the elements of the group.

Further, this equivalence gives an equivalence of categories, where the first category is the subcategory corresponding to groups of the [[category of locally small categories]] (so the morphisms are [[functor]]s), and the second category is the [[category of groups]] in the conventional sense.

==External links==

===Other subject wikis===

* [[Groupprops:Group|Group on the Group Properties Wiki]] -- the most detailed and canonical reference.
* [[Topospaces:Group|Group on the Topology Wiki]]

===Other resources===

* [[nlab:group|group on nlab]]
* [[Wikipedia:Group|Group on Wikipedia]]
* [[Pm:Group|Group on Planetmath]]
* [[Mw:Group|Group on Mathworld]]

Group

2009-12-29T13:00:10Z

Vipul: /* External links */

==Definition==

===Category-theoretic definition===

A '''group''' is a [[small category]] with one object where all morphisms are isomorphisms. (note that the ''small'' assumption is to ensure that the morphisms form a set; however, it can be dropped for some purposes).

===Ordinary definition===

A '''group''' is a set <math>G</math> equipped with a binary operation <math>*</math>, a unary operation <math>{}^{-1}</math>, and a constant <math>e</math> such that the following hold:

* '''Associativity''': <math>a * (b * c) = (a * b) * c \ \forall \ a,b,c \in G</math>.
* '''Neutral element''' or '''identity element''': <math>a * e = e * a = a \ \forall \ a \in G</math>.
* '''Inverse element''': <math>a * a^{-1} = a^{-1} * a = e</math>.

(there are other slightly different formulations of this definition).

===Definition building on monoid===

A group is a [[monoid]] in which every element has a two-sided inverse.

===Equivalence of definitions===

The two definitions are equivalent in the following sense: given any group in the category-theoretic set, the morphisms, under composition, form a group in the set-theoretic sense. Conversely, given any group in the set-theoretic sense, we can construct a category with one object and with the morphisms corresponding to the elements of the group.

Further, this equivalence gives an equivalence of categories, where the first category is the subcategory corresponding to groups of the [[category of locally small categories]] (so the morphisms are [[functor]]s), and the second category is the [[category of groups]] in the conventional sense.

==External links==

===Other subject wikis===

* [[Groupprops:Group|Group on the Group Properties Wiki]] -- the most detailed and canonical reference.
* [[Topospaces:Group|Group on the Topology Wiki]]

===Other resources===

* [[nlab:group|group on nlab]]
* [[Wikipedia:Group|Group on Wikipedia]]
* [[Planetmath:Group|Group on Planetmath]]
* [[Mathworld:Group|Group on Mathworld]]

Group

2009-12-29T12:58:50Z

Vipul:

==Definition==

===Category-theoretic definition===

A '''group''' is a [[small category]] with one object where all morphisms are isomorphisms. (note that the ''small'' assumption is to ensure that the morphisms form a set; however, it can be dropped for some purposes).

===Ordinary definition===

A '''group''' is a set <math>G</math> equipped with a binary operation <math>*</math>, a unary operation <math>{}^{-1}</math>, and a constant <math>e</math> such that the following hold:

* '''Associativity''': <math>a * (b * c) = (a * b) * c \ \forall \ a,b,c \in G</math>.
* '''Neutral element''' or '''identity element''': <math>a * e = e * a = a \ \forall \ a \in G</math>.
* '''Inverse element''': <math>a * a^{-1} = a^{-1} * a = e</math>.

(there are other slightly different formulations of this definition).

===Definition building on monoid===

A group is a [[monoid]] in which every element has a two-sided inverse.

===Equivalence of definitions===

The two definitions are equivalent in the following sense: given any group in the category-theoretic set, the morphisms, under composition, form a group in the set-theoretic sense. Conversely, given any group in the set-theoretic sense, we can construct a category with one object and with the morphisms corresponding to the elements of the group.

Further, this equivalence gives an equivalence of categories, where the first category is the subcategory corresponding to groups of the [[category of locally small categories]] (so the morphisms are [[functor]]s), and the second category is the [[category of groups]] in the conventional sense.

==External links==

===Other subject wiki links===

* [[Groupprops:Group|Group on the Group Properties Wiki]] -- the most detailed and canonical reference.
* [[Topospaces:Group|Group on the Topology Wiki]]

===Other links===

* [[nlab:group|group on nlab]]

Group

2009-12-29T12:51:43Z

Vipul: Created page with '==Definition== ===Category-theoretic definition=== A '''group''' is a small category with one object where all morphisms are isomorphisms. (note that the ''small'' assumpti…'

==Definition==

===Category-theoretic definition===

A '''group''' is a [[small category]] with one object where all morphisms are isomorphisms. (note that the ''small'' assumption is to ensure that the morphisms form a set; however, it can be dropped for some purposes).

===Ordinary definition===

A '''group''' is a set <math>G</math> equipped with a binary operation <math>*</math>, a unary operation <math>{}^{-1}</math>, and a constant <math>e</math> such that the following hold:

* '''Associativity''': <math>a * (b * c) = (a * b) * c \ \forall \ a,b,c \in G</math>.
* '''Neutral element''' or '''identity element''': <math>a * e = e * a = a \ \forall \ a \in G</math>.
* '''Inverse element''': <math>a * a^{-1} = a^{-1} * a = e</math>.

(there are other slightly different formulations of this definition).

===Equivalence of definitions===

The two definitions are equivalent in the following sense: given any group in the category-theoretic set, the morphisms, under composition, form a group in the set-theoretic sense. Conversely, given any group in the set-theoretic sense, we can construct a category with one object and with the morphisms corresponding to the elements of the group.

Further, this equivalence gives an equivalence of categories, where the first category is the subcategory corresponding to groups of the [[category of locally small categories]] (so the morphisms are [[functor]]s), and the second category is the [[category of groups]] in the conventional sense.

==External links==

===Other subject wiki links===

* [[Groupprops:Group|Group on the Group Properties Wiki]] -- the most detailed and canonical reference.
* [[Topospaces:Group|Group on the Topology Wiki]]

Category of Abelian groups

2009-12-29T12:45:00Z

Vipul: moved Category of Abelian groups to Category of abelian groups

#REDIRECT [[Category of abelian groups]]

Category of abelian groups

2009-12-29T12:45:00Z

Vipul: moved Category of Abelian groups to Category of abelian groups

{{particular category}}

==Definition==

The '''category of Abelian groups''', sometimes denoted <math>\operatorname{Ab}</math>, is defined as follows:

* Its objects are [[Abelian group]]s.
* Its morphisms are homomorphisms of groups.
* The identity morphism is defined as the identity map.
* The composition of morphisms is defined by function composition.

==Relation with other categories==

===Functors from this category===

* [[Category of groups]]: The category of Abelian groups embeds as a full subcategory of the category of groups.
* [[Category of monoids]]
* [[Category of pointed sets]]
* [[Category of sets]]

===Functors to this category===

{{fillin}}

==Additional structure==

===Monoidal structure===

* [[Monoidal category of Abelian groups]]: This is a monoidal category where the monoidal operation is the tensor product of Abelian groups. Note that the tensor product is neither a product nor a coproduct in the category of Abelian groups.

Monoidal category of Abelian groups

2009-12-29T12:44:16Z

Vipul: moved Monoidal category of Abelian groups to Monoidal category of abelian groups

#REDIRECT [[Monoidal category of abelian groups]]

Monoidal category of abelian groups

2009-12-29T12:44:16Z

Vipul: moved Monoidal category of Abelian groups to Monoidal category of abelian groups

{{particular monoidal category}}

==Definition==

The '''monoidal category of Abelian groups''' is defined as follows:

* The underlying category is the [[category of Abelian groups]].
* The monoidal operation is the [[tensor product of Abelian groups]].
* The unit object for the monoidal operation is the group of integers.

Template:Top notice

2009-03-27T11:32:43Z

Vipul:

{{quotation|Welcome to '''{{fullsitetitle}}'''. This is a pre-pre-alpha stage category theory wiki primarily managed by [[User:Vipul|Vipul Naik]], a Ph.D. student in Mathematics at the University of Chicago. It is part of a broader subject wikis initiative -- see the [[Ref:Main Page|subject wikis reference guide]] for more details.}}

Main Page

2009-03-27T11:31:44Z

Vipul: Replaced content with '{{top notice}}'

User talk:Jon Awbrey

2009-02-25T20:09:41Z

Vipul: Welcome!

'''Welcome to ''Cattheory''!''' We hope you will contribute much and well.
You will probably want to read the [[Help:Contents|help pages]]. Again, welcome and have fun! [[User:Vipul|Vipul]] 20:09, 25 February 2009 (UTC)

User:Jon Awbrey

2009-02-25T20:09:40Z

Vipul: Creating user page with biography of new user.

See [http://www.mywikibiz.com/Directory:Jon_Awbrey Web Vita]

MediaWiki:Sidebar

2009-01-16T15:51:17Z

Vipul:

Cattheory:General disclaimer

2009-01-16T15:41:50Z

Vipul: New page: This is a general disclaimer common to all subject wikis, with the source at Ref:Ref:General disclaimer. For specific hazards of using this particular subject wiki, refer [[{{SITENAME}...

This is a general disclaimer common to all subject wikis, with the source at [[Ref:Ref:General disclaimer]]. For specific hazards of using this particular subject wiki, refer [[{{SITENAME}}:Hazards]].

'''SUBJECT WIKIS MAKE NO GUARANTEE OF VALIDITY'''

Content on individual subject wikis need ''not'', in general, be correct or useful. There are the following general hazards:

* Specific content pages may have wrong or misleading information.
* Content pages may use terminology that is not standard or generally accepted, or differs from terminology or notation in other sources.
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Also note that:

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Cattheory:Privacy policy

2009-01-16T01:46:30Z

Vipul: New page: This privacy policy is common to subject wikis. For the original privacy policy, refer Ref:Ref:Privacy policy. ==Privacy for readers== If you are surfing this website, your actions a...

This privacy policy is common to subject wikis. For the original privacy policy, refer [[Ref:Ref:Privacy policy]].

==Privacy for readers==

If you are surfing this website, your actions are logged in our usage logs. These usage logs are accessible to:

* The site's administrators and technical support group. For a full list of administrators, contact [[User:Vipul|Vipul Naik]] by email: vipul@math.uchicago.edu or vipul.wikis@gmail.com. The technical support is [http://www.4am.co.in 4AM].
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Your usage logs are not made available to other parties. Aggregated data from logs, such as general usage patterns, may be used by the MedaWiki software as well as by site administrators in decision making. For instance, MediaWiki keeps track of the number of times each page is viewed.

==Privacy for editors==

Editing on subject wikis is generally permitted only for registered users. Registered users must, at the time of registration, provide their real name, and enter basic information about their reason for interest. ''No'' private information such as date of birth, social security or taxation number, or home address is sought.

Regarding personal information:

* The email IDs of registered users are visible to site administrators only. For information about site administrators, contact vipul.wikis@gmail.com or vipul@math.uchicago.edu with the particular subject wiki and the reason for request.
* All editing activity by registered users is recorded on the site and is visible to all site users. However, this information is not indexed by search engines that follows robots.txt.
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MediaWiki:Sitenotice

2009-01-15T22:40:24Z

Vipul:

[[Image:Logo.jpg|thumb|75px|right|[http://www.4am.co.in Visit]]]
[[Main Page|{{fullsitetitle}} ({{sitestatus}})]]

'''ALSO CHECK OUT''': <random>[[Groupprops:Main Page|Groupprops]]: The Group Properties Wiki@@@[[Commalg:Main Page|Commalg]]: The Commutative Algebra Wiki@@@[[Diffgeom:Main Page|Diffgeom]]: The Differential Geometry Wiki@@@[[Topospaces:Main Page|Topospaces]]: The Topology Wiki</random>

Template:Sitestatus

2009-01-15T22:35:24Z

Vipul: New page: pre-pre-alpha

pre-pre-alpha

Cattheory:Copyrights

2009-01-15T22:26:24Z

Vipul: New page: This is a common copyright notice to all subject wikis. Original notice available at Ref:Ref:Copyrights. ==General license information== All content is put up under the [http://creat...

This is a common copyright notice to all subject wikis. Original notice available at [[Ref:Ref:Copyrights]].

==General license information==

All content is put up under the [http://creativecommons.org/licenses/by-sa/3.0/ Creative Commons Attribute Share-Alike 3.0 license (unported)] (shorthand: CC-by-SA). Read the [http://creativehttp://ref.subwiki.org/w/skins/common/images/button_media.pngcommons.org/licenses/by-sa/3.0/legalcode full legal code here].

===Exceptions===

The software engine that runs the wiki is MediaWiki, and this software is licensed under the GNU General Public License, which is distinct from and currently incompatible with the Creative Commons license. Similarly, some of the extensions to the software engine are licensed by their developers using the GPL, which is incompatible with Creative Commons licenses.

More information on the version of MediaWiki and extensions: [[Special:Version]].

In addition, some templates may have been copied or adapted from Wikipedia or other wiki-based websites that use licenses other than Creative Commons licenses. Wikipedia uses the GNU Free Documentation License (GFDL) that, as of now, is incompatible with CC licenses. In such cases, it is clearly indicated on the page that it is licensed differently.

===Short description of the license===

The CC-by-SA license has the following features:

# Content can be used and reused for any purposes, commercial or noncommercial.
# Any public use or reuse of the material requires attribution of the original source (for attribution formats, see later in the document).
# In case of adaptation or creation of derivative works, the newly created works must also be licensed under the same license (CC-by-SA) unless the creator of the derivative work takes ''explicit'' permission from the creator of the original work.

===Fair use and other rights===

The license cannot and does not limit fair use rights. Fair use may include quoting or copying select passages for the purposes of criticism and review.

The license also cannot limit the use of data or information contained in the content on subject wikis. This means that anybody can use or reuse the facts or knowledge they gain from the subject wiki in any way they want.

==Applicability of license to the subject wiki==

===Examples of personal use that do not necessitate use of the license terms===

* Saving, making copies, or taking printouts of pages on the subject wiki, as long as these copies are meant for personal use and are not to be distributed to others.
* Learning facts from the subject wikis and using those facts as part of instruction or research.
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* Quoting short passages from the subject wiki in a criticism or review (it is preferable, though not legally enforceable, that a link to the original content being reviewed be provided).

===Examples of personal use that necessitate use of the license terms===

* Creating a mirror website by exporting pages or content in bulk: In this case, the mirror website must release the mirrored content under the same license terms and ''also'' attribute the original source with a link. Attribution formats are discussed below.
* Giving handouts of pages or collections of pages to a group of people, such as students in a course: In this case, the handouts should be accompanied by an attribution that makes clear both the source and the original license terms, as well as a link to the Uniform Resource Identifier (URI) of the subject wiki.

Attribution formats:

* For websites: ''This content is from [{{fullurl:Main Page}} {{SITENAME}}]'' or ''This content is from [{{fullurl:Main Page}} {{fullsitetitle}}]''. Preferably, also a link to the specific pages (if any) used. The language of the attribution may be modified somewhat to indicate whether the content has been used as such, or whether changes have been made.
* For print materials: ''This content is from {{SITENAME}}. The site URL is http://{{SITENAME}}.subwiki.org''

Attribution of individuals who have contributed content to the subject wiki entries is ''not'' necessary. (However, this list of individuals can be tracked down, as discussed in the next section).

==Uses not within the ambit of the license or fair use rights==

Those who have contributed all the content to a particular subject wiki entry can republish the same content anywhere else ''without restriction'' and can also release the content elsewhere under a different license, since they own the full copyright to their content.

To use the content on the wiki in a manner that is outside the terms of the license or beyond fair use rights, the individuals who have contributed that particular content need to be contacted. For any particular page, this information can be accessed using the history tab of the page. The users who have made edits to the page are the ones who need to be contacted.

In case of difficulty doing this, please contact vipul.wikis@gmail.com or vipul@math.uchicago.edu for more information.

==Copyright violations==

All users using subject wikis are requested not to violate copyright of other authors or publishers. This includes providing links that may aid and abet copyright violation. Examples of things that may be considered copyright violation:

* Putting up links to personal copies of restricted-access journal articles acquired through a personal or library subscription: This usually goes against the Terms of Service of the subscription. The exception is when the content of the articles is out of copyright.
* Giving links to personal copies, or copies on filesharing systems, of books that are under copyright and where either owning or sharing a copy in that manner is against the terms of copyright.

In case copyright violations are detected, please email vipul.wikis@gmail.com and vipul@math.uchicago.edu to have the matter looked into immediately.

Template:Fullsitetitle

2009-01-15T22:25:01Z

Vipul: New page: Cattheory, The Category Theory Wiki

Cattheory, The Category Theory Wiki

Template:Top notice

2009-01-15T22:02:21Z

Vipul: New page: {{quotation|Welcome to '''Cattheory (The Category Theory Wiki)'''. This is a pre-pre-alpha stage category theory wiki primarily managed by Vipul Naik, a Ph.D. student in Mat...

{{quotation|Welcome to '''Cattheory (The Category Theory Wiki)'''. This is a pre-pre-alpha stage category theory wiki primarily managed by [[User:Vipul|Vipul Naik]], a Ph.D. student in Mathematics at the University of Chicago. It is part of a broader subject wikis initiative -- see the [[Ref:Main Page|subject wikis reference guide]] for more details.}}

Category of algebras in a variety

2008-12-26T07:15:30Z

Vipul: New page: ==Definition== Let <math>\mathcal{V}</math> be a variety of algebras. In other words, <math>\mathcal{V}</math> is the collection of all algebras with a particular operator domain (eac...

==Definition==

Let <math>\mathcal{V}</math> be a [[variety of algebras]]. In other words, <math>\mathcal{V}</math> is the collection of all algebras with a particular operator domain (each algebra has the same collection of operation arities) satisfying a set of universal identities. In particular, <math>\mathcal{V}</math> is closed under taking subalgebras, quotient algebras, and arbitrary direct products.

The category of algebras in <math>\mathcal{V}</math> is defined as follows:

# The objects of the category are the algebras in <math>\mathcal{V}</math>.
# The set of morphisms between any two objects of the category is precisely the set of algebra homomorphisms between them.
# The identity map is the identity morphism.
# Composition of morphisms is done by function composition.

The category of algebras in a variety has the natural structure of a [[concrete category]] via the functor sending each algebra to its underlying set and sending each morphism to its underlying set function. In particular, it is a [[locally small category]].

==Examples==

* The [[category of sets]] is the category corresponding to the variety of sets: here, a set is a set with no operations and no universal identities.
* The [[category of sets]] is the category corresponding to the variety of pointed sets: here, a pointed set is a set with a single 0-ary operation that returns the distinguished point of the set. There are no universal identities.
* The [[category of groups]] is the category corresponding to the [[variety of groups]].
* The [[category of Abelian groups]] is the category corresponding to the [[variety of Abelian groups]].
* The [[category of unital rings]] is the category corresponding to the [[variety of unital rings]].

Normal category

2008-12-26T06:59:39Z

Vipul:

{{pointed sets-enriched category property}}
{{preadditive category property}}
==Definition==

===For a category enriched over pointed sets===

A '''normal category''' is a [[category]] [[enriched category|enriched]] over the [[monoidal category of pointed sets]] satisfying the condition that every [[monomorphism]] is [[normal monomorphism|normal]].

===For a preadditive category===

A [[preadditive category]] is termed a '''normal category''' if every [[monomorphism]] in it is [[normal monomorphism|normal]].

Template:Preadditive category property

2008-12-26T06:57:36Z

Vipul: New page: {{quotation|''This article defines a preadditive category property: a property that can be evaluated to true/false given a preadditive category.'' [[:Category:Preadditive catego...

{{quotation|''This article defines a [[preadditive category property]]: a property that can be evaluated to true/false given a [[preadditive category]].'' [[:Category:Preadditive category properties|View a complete list of preadditive category properties]]<nowiki>|</nowiki>[[Help:Preadditive category property lookup|Get preadditive category property lookup help]] <nowiki>|</nowiki>[[Help:Preadditive category property exploration|Get exploration suggestions]] '''VIEW RELATED''': {{#ask: [[Fact about::{{PAGENAME}}]][[Category:Preadditive category property implications]]|limit = 0 | searchlabel = Preadditive category property implications}} <nowiki>|</nowiki> {{#ask: [[Fact about::{{PAGENAME}}]][[Category:Preadditive category property non-implications]]|limit = 0 | searchlabel = Preadditive category property non-implications}} <nowiki>|</nowiki> {{#ask: [[Fact about::{{PAGENAME}}]][[Category:Preadditive category metaproperty satisfactions]]|limit = 0| searchlabel = Preadditive category metaproperty satisfactions}} <nowiki>|</nowiki> {{#ask: [[Fact about::{{PAGENAME}}]][[Category:Preadditive category metaproperty dissatisfactions]]|limit = 0|searchlabel = Preadditive category metaproperty dissatisfactions}} <nowiki>|</nowiki> {{#ask: [[Fact about::{{PAGENAME}}]][[Category:Preadditive category property satisfactions]]|limit = 0|searchlabel = Preadditive category property satisfactions}} <nowiki>|</nowiki>{{#ask: [[Fact about::{{PAGENAME}}]][[Category:Preadditive category property dissatisfactions]]|limit = 0|searchlabel = Preadditive category property dissatisfactions}}}}<includeonly>[[Category:Preadditive category properties]]</includeonly>