Groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
Special cases include:
Groupoids are often used to reason about geometrical objects such as manifolds. Heinrich Brandt (1927) introduced groupoids implicitly via Brandt semigroups.
Definitions
A groupoid is an algebraic structure (G,
) consisting of a non-empty set G and a binary partial function '' defined on G.
Algebraic
A groupoid is a set G with a unary operation 
and a partial function 
. Here * is not a binary operation because it is not necessarily defined for all possible pairs of G-elements. The precise conditions under which * is defined are not articulated here and vary by situation.
∞-groupoid
In category theory, a branch of mathematics, an ∞-groupoid (also called Kan complex) is a fibrant object in the category of simplicial sets (with the standard model structure). It is an ∞-category generalization of a groupoid, a category in which every morphism is an isomorphism.
The homotopy hypothesis states that ∞-groupoids are spaces.
See also
External links
Magma (algebra)
In abstract algebra, a magma (or groupoid; not to be confused with groupoids in category theory) is a basic kind of algebraic structure. Specifically, a magma consists of a set, M, equipped with a single binary operation, M × M → M. The binary operation must be closed by definition but no other properties are imposed.
History and terminology
The term groupoid was introduced in 1926 by Heinrich Brandt describing his Brandt groupoid (translated from the German Gruppoid). The term was then appropriated by B. A. Hausmann and Øystein Ore (1937) in the sense (of a set with a binary operation) used in this article. In a couple of reviews of subsequent papers in Zentralblatt, Brandt strongly disagreed with this overloading of terminology. The Brandt groupoid is a groupoid in the sense used in category theory, but not in the sense used by Hausmann and Ore. Nevertheless, influential books in semigroup theory, including Clifford and Preston (1961) and Howie (1995) use groupoid in the sense of Hausmann and Ore. Hollings (2014) writes that the term groupoid is "perhaps most often used in modern mathematics" in the sense given to it in category theory.
Podcasts:
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Yessir
by: Kurupt(Pity the child... pity the child)
Now, why should I play around?
Lay it down like most of y'all and lay around?
I'd rather gather hollow heads and lay ya' down
But I'm a little too old to play around
You still bangin' Dr. Dre and Dogg Pound
Snoop is still the hottest motherfucker out
Yessir! We still around
Yessir! We still world renowned
I caught silence, I silence sound
I'm under the radar, stuff silent sound
I'm here, my dude, I am born again
We saber-toothed tigers, y'all Cornish hens
Y'all cornball ballas, we presidents
With presidential-acquired hidden residents
Yessir! We residents
And controllers of secret soldiers and regiments
Yessir! We ball out
Yessir! Till we fall out
We get high (get high), we don't have to try
We don't have to look and we don't have to buy
We don't have to make ends meet to get by
We don't have to try, I am that guy
Yessir! I am that guy
Yessir! I am that fly
I can't help it and I can't stop
And I won't stop and I don't stop
I'm a take the top and put it where the bottom drop
Then I'm a reverse it and make the bottom hit top
See this is what they all call the top spot
Then I'm a show the top where the bottom stop
I am everything that is anything
I can't sing a note so I let the semi sing
Send 'em out on a mission, see what the semi bring
Back home to your selected and newly-elected king
Now that's gangsta [x5]
See that's gangsta
Now that's gangsta
(Keepin it gangsta with'chall)
Now that's gangsta (gangsta)
Gangsta [x11] (gangsta)
Yessir! I am that guy
Yessir! I am that high
Rollercoaster, this whole movement's over
Givin' ya west coasters, the cold shoulder
I am ready, yes, I am here
I am back, Jack, the spinner of the year
Let's get one thing here real crystal clear
We reappear, their hopes disappear
Evaporate, dissipate, your folks disappear
And if you were smart you wouldn't be here
Now that's what I call a double whammy
I don't need the white folks, I get ghetto Grammys
Yessir! I get ghetto Grammys
Yessir! My women's eye candy
I can't help it and I can't stop
And I won't stop, that's why I don't stop
I'm similar to Biggie, similar to Pac
Similar to Snoop, similar to Doc
Similar because they all taught me something, ock
Firsthand, like cock and pop before a nigga talk
I'm airborne, you're grounded
Like you hid your report card and your mama found it
Yessir! I found it
My drrr-ream to go far as God allows me
The sky's the limit
It's in my arm's reach,
So I'm a reach as far as my arms reach
Gangsta [x15]
(Now that's... gangsta)