Gospel of John

29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church.

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:

• Group with a partial function replacing the binary operation;
• Category in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation, called inverse by analogy with group theory.
• Oriented graph

Special cases include:

• Setoids, that is: sets that come with an equivalence relation;
• G-sets, sets equipped with an action of a group G.

Groupoids are often used to reason about geometrical objects such as manifolds. Template:Harvs introduced groupoids implicitly via Brandt semigroups.

Definitions

A groupoid is an algebraic structure (G,$\ast$) consisting of a non-empty set G and a binary operation '$\ast$' defined on G.

Algebraic

A groupoid is a set G with a unary operation $^{-1}:G\to G,$ and a partial function $*:G\times G\rightharpoonup G.$.  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.

$\ast$ and ⁻¹ have the following axiomatic properties.  Let a, b, and c be elements of G.  Then:

1. Associativity: If a * b and b * c are defined, then (a * b) * c and a * (b * c) are defined and equal.  Conversely, if either of these last two expressions is defined, then so is the other (and again they are equal).
2. Inverse: a⁻¹ * a and a * a⁻¹ are always defined.
3. Identity: If a * b is defined, then a * b * b⁻¹ = a, and a⁻¹ * a * b = b.  (The previous two axioms already show that these expressions are defined and unambiguous.)

From these axioms, two easy and convenient properties follow:

• (a⁻¹)⁻¹ = a;
• If a * b is defined, then (a * b)⁻¹ = b⁻¹ * a⁻¹.

Proof of first property: from 2. and 3. we obtain (a⁻¹)⁻¹ = (a⁻¹)⁻¹ * a⁻¹ * a = a.  ✓
Proof of second property: since a * b is defined, so is (a * b)⁻¹ * a * b.  Therefore (a * b)⁻¹ * a * b * b⁻¹ = (a * b)⁻¹ * a is also defined.  Moreover since a * b is defined, so is a * b * b⁻¹ = a.  Therefore a * b * b⁻¹ * a⁻¹ is also defined.  From 3. we obtain (a * b)⁻¹ = (a * b)⁻¹ * a * a⁻¹ = (a * b)⁻¹ * a * b * b⁻¹ * a⁻¹ = b⁻¹ * a⁻¹.  ✓

Category theoretic

A groupoid is a small category in which every morphism is an isomorphism, and hence invertible. More precisely, a groupoid G is:

• A set G₀ of objects;
• For each pair of objects x and y in G₀, there exists a (possibly empty) set G(x,y) of morphisms (or arrows) from x to y. We write f : x → y to indicate that f is an element of G(x,y).

The objects and morphisms have the properties:

• For every object x, there exists the element $\mathrm {id} _{x}$ of G(x,x);
• For each triple of objects x, y, and z, there exists the function $\mathrm {comp} _{x,y,z}:$ G(x,y)$\times$G(y,z) → G(x,z). We write gf for $\mathrm {comp} _{x,y,z}(f,g)$, where f$\in$G(x,y), and g$\in$G(y,z);
• There exists the function $\mathrm {inv} _{x,y}:$ G(x,y) → G(y,x).

Moreover, if f : x → y, g : y → z, and h : z → w, then:

• $f\mathrm {id} _{x}=f$ and $\mathrm {id} _{y}f=f$;
• (hg)f = h(gf);
• $f\mathrm {inv} (f)=\mathrm {id} _{y}$ and $\mathrm {inv} (f)f=\mathrm {id} _{x}$.

If f is an element of G(x,y) then x is called the source of f, written s(f), and y the target of f (written t(f)).

Comparing the definitions

The algebraic and category-theoretic definitions are equivalent, as follows. Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G(x,y) (i.e. the sets of morphisms from x to y). Then $\mathrm {comp}$ and $\mathrm {inv}$ become partially defined operations on G, and $\mathrm {inv}$ will in fact be defined everywhere; so we define * to be $\mathrm {comp}$ and $^{-1}$ to be $\mathrm {inv}$. Thus we have a groupoid in the algebraic sense. Explicit reference to G₀ (and hence to $\mathrm {id}$) can be dropped.

Conversely, given a groupoid G in the algebraic sense, let G₀ be the set of all elements of the form x * x⁻¹ with x varying through G and define G(x*x ^-1,y*y ^-1) as the set of all elements f such that y * y ^-1 * f * x * x ^-1 exists. Given f∈G(x*x^-1,y*y ^-1) and g∈G(y*y ^-1,z*z ^-1), their composite is defined as g * f ∈ G(x*x ^-1,z*z ^-1). To see this is well defined, observe that since z*z ^-1 * g * y*y ^-1 and y*y^-1 * f * x*x^-1 exist, so does z*z^-1 * g * y*y^-1 * y*y^-1 * f * x*x ^-1 = z*z^-1 * g*f * x*x ^-1. The identity morphism on x*x⁻¹ is then x*x⁻¹ itself, and the category-theoretic inverse of f is f^-1.

Sets in the definitions above may be replaced with classes, as is generally the case in category theory.

Vertex groups

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G(x,x), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

Category of groupoids

A subgroupoid is a subcategory that is itself a groupoid. A groupoid morphism is simply a functor between two (category-theoretic) groupoids. The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the groupoid category, or the category of groupoids, denoted Grpd.

It is useful that this category is, like the category of small categories, cartesian closed. That is, we can construct for any groupoids $H,K$ a groupoid $GPD(H,K)$ whose objects are the morphisms $H\to K$ and whose arrows are the natural equivalences of morphisms. Thus if $H,K$ are just groups, then such arrows are the conjugacies of morphisms. The main result is that for any groupoids $G,H,K$ there is a natural bijection

$Grpd(G\times H,K)\cong Grpd(G,GPD(H,K)).$

This result is of interest even if all the groupoids $G,H,K$ are just groups.

Fibrations, Coverings

Particular kinds of morphisms of groupoids are of interest. A morphism $p:E\to B$ of groupoids is called a fibration if for each object $x$ of $E$ and each morphism $b$ of $B$ starting at $p(x)$ there is a morphism $e$ of $E$ starting at $x$ such that $p(e)=b$. A fibration is called a covering morphism or covering of groupoids if further such an $e$ is unique. The covering morphisms of groupoids are especially useful because they can be used to model covering maps of spaces.

It is also true that the category of covering morphisms of a given groupoid $B$ is equivalent to the category of actions of the groupoid $B$ on sets.

Examples

Linear algebra

Given a field K, the corresponding general linear groupoid GL_*(K) consists of all invertible matrices whose entries range over K. Matrix multiplication interprets composition. If G = GL_*(K), then the set of natural numbers is a proper subset of G₀, since for each natural number n, there is a corresponding identity matrix of dimension n. G(m,n) is empty unless m=n, in which case it is the set of all nxn invertible matrices.

Topology

Given a topological space X, let G₀ be the set X. The morphisms from the point p to the point q are equivalence classes of continuous paths from p to q, with two paths being equivalent if they are homotopic. Two such morphisms are composed by first following the first path, then the second; the homotopy equivalence guarantees that this composition is associative. This groupoid is called the fundamental groupoid of X, denoted $\pi _{1}$(X). The usual fundamental group $\pi _{1}(X,x)$ is then the vertex group for the point x.

An important extension of this idea is to consider the fundamental groupoid $\pi _{1}$(X,A) where A is a set of "base points" and a subset of X. Here, one considers only paths whose endpoints belong to A. $\pi _{1}$(X,A) is a sub-groupoid of $\pi _{1}$(X). The set A may be chosen according to the geometry of the situation at hand.

Equivalence relation

If X is a set with an equivalence relation denoted by infix $\sim$, then a groupoid "representing" this equivalence relation can be formed as follows:

• The objects of the groupoid are the elements of X;
• For any two elements x and y in X, there is a single morphism from x to y if and only if x~y.

Group action

If the group G acts on the set X, then we can form the action groupoid representing this group action as follows:

• The objects are the elements of X;
• For any two elements x and y in X, there is a morphism from x to y corresponding to every element g of G such that gx = y;
• Composition of morphisms interprets the binary operation of G.

More explicitly, the action groupoid is the set $G\times X$ with source and target maps s(g,x) = x and t(g,x) = gx. It is often denoted $G\ltimes X$ (or $X\rtimes G$). Multiplication (or composition) in the groupoid is then $(h,y)(g,x)=(hg,x)$ which is defined provided y=gx.

For x in X, the vertex group consists of those (g,x) with gx = x, which is just the isotropy subgroup at x for the given action (which is why vertex groups are also called isotropy groups).

Another way to describe G-sets is the functor category $[\mathrm {Gr} ,\mathrm {Set} ]$, where $\mathrm {Gr}$ is the groupoid (category) with one element and isomorphic to the group G. Indeed, every functor F of this category defines a set X=F$(\mathrm {Gr} )$ and for every g in G (i.e. for every morphism in $\mathrm {Gr}$) induces a bijection F_g : X→X. The categorical structure of the functor F assures us that F defines a G-action on the set X. The (unique) representable functor F : $\mathrm {Gr}$→$\mathrm {Set}$ is the Cayley representation of G. In fact, this functor is isomorphic to $\mathrm {Hom} (\mathrm {Gr} ,-)$ and so sends $\mathrm {ob} (\mathrm {Gr} )$ to the set $\mathrm {Hom} (\mathrm {Gr} ,\mathrm {Gr} )$ which is by definition the "set" G and the morphism g of $\mathrm {Gr}$ (i.e. the element g of G) to the permutation F_g of the set G. We deduce from the Yoneda embedding that the group G is isomorphic to the group {F_g | g∈G}, a subgroup of the group of permutations of G.

Fifteen puzzle

The symmetries of the fifteen puzzle form a groupoid (not a group, as not all moves can be composed). This groupoid acts on configurations.

Mathieu groupoid

The Mathieu groupoid is a groupoid introduced by John Horton Conway acting on 13 points such that the elements fixing a point form a copy of the Mathieu group M₁₂.

Relation to groups

Template:Group-like structures If a groupoid has only one object, then the set of its morphisms forms a group. Using the algebraic definition, such a groupoid is literally just a group. Many concepts of group theory generalize to groupoids, with the notion of functor replacing that of group homomorphism.

If x is an object of the groupoid G, then the set of all morphisms from x to x forms a group G(x). If there is a morphism f from x to y, then the groups G(x) and G(y) are isomorphic, with an isomorphism given by the mapping g → fgf ⁻¹.

Every connected groupoid (that is, one in which any two objects are connected by at least one morphism) is isomorphic to an action groupoid (as defined above) (G, X) [by connectedness, there will only be one orbit under the action]. If the groupoid is not connected, then it is isomorphic to a disjoint union of groupoids of the above type (possibly with different groups G and sets X for each connected component).

Note that the isomorphism described above is not unique, and there is no natural choice. Choosing such an isomorphism for a connected groupoid essentially amounts to picking one object x₀, a group isomorphism h from G(x₀) to G, and for each x other than x₀, a morphism in G from x₀ to x.

In category-theoretic terms, each connected component of a groupoid is equivalent (but not isomorphic) to a groupoid with a single object, that is, a single group. Thus any groupoid is equivalent to a multiset of unrelated groups. In other words, for equivalence instead of isomorphism, one need not specify the sets X, only the groups G.

Consider the examples in the previous section. The general linear groupoid is both equivalent and isomorphic to the disjoint union of the various general linear groups GL_n(F). On the other hand:

• The fundamental groupoid of X is equivalent to the collection of the fundamental groups of each path-connected component of X, but an isomorphism requires specifying the set of points in each component;
• The set X with the equivalence relation $\sim$ is equivalent (as a groupoid) to one copy of the trivial group for each equivalence class, but an isomorphism requires specifying what each equivalence class is:
• The set X equipped with an action of the group G is equivalent (as a groupoid) to one copy of G for each orbit of the action, but an isomorphism requires specifying what set each orbit is.

The collapse of a groupoid into a mere collection of groups loses some information, even from a category-theoretic point of view, because it is not natural. Thus when groupoids arise in terms of other structures, as in the above examples, it can be helpful to maintain the full groupoid. Otherwise, one must choose a way to view each G(x) in terms of a single group, and this choice can be arbitrary. In our example from topology, you would have to make a coherent choice of paths (or equivalence classes of paths) from each point p to each point q in the same path-connected component.

As a more illuminating example, the classification of groupoids with one endomorphism does not reduce to purely group theoretic considerations. This is analogous to the fact that the classification of vector spaces with one endomorphism is nontrivial.

Morphisms of groupoids come in more kinds than those of groups: we have, for example, fibrations, covering morphisms, universal morphisms, and quotient morphisms. Thus a subgroup H of a group G yields an action of G on the set of cosets of H in G and hence a covering morphism p from, say, K to G, where K is a groupoid with vertex groups isomorphic to H. In this way, presentations of the group G can be "lifted" to presentations of the groupoid K, and this is a useful way of obtaining information about presentations of the subgroup H. For further information, see the books by Higgins and by Brown in the References.

Lie groupoids and Lie algebroids

When studying geometrical objects, the arising groupoids often carry some differentiable structure, turning them into Lie groupoids. These can be studied in terms of Lie algebroids, in analogy to the relation between Lie groups and Lie algebras.

See also

• ∞-groupoid

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Brown, Ronald, 1987, "From groups to groupoids: a brief survey," Bull. London Math. Soc. 19: 113-34. Reviews the history of groupoids up to 1987, starting with the work of Brandt on quadratic forms. The downloadable version updates the many references.
• —, 2006. Topology and groupoids. Booksurge. Revised and extended edition of a book previously published in 1968 and 1988. Groupoids are introduced in the context of their topological application.
• —, Higher dimensional group theory Explains how the groupoid concept has to led to higher dimensional homotopy groupoids, having applications in homotopy theory and in group cohomology. Many references.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• F. Borceux, G. Janelidze, 2001, Galois theories. Cambridge Univ. Press. Shows how generalisations of Galois theory lead to Galois groupoids.
• Cannas da Silva, A., and A. Weinstein, Geometric Models for Noncommutative Algebras. Especially Part VI.
• Golubitsky, M., Ian Stewart, 2006, "Nonlinear dynamics of networks: the groupoid formalism", Bull. Amer. Math. Soc. 43: 305-64
• Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.
  
  my web-site http://himerka.com/
• Higgins, P. J., "The fundamental groupoid of a graph of groups", J. London Math. Soc. (2) 13 (1976) 145—149.
• Higgins, P. J. and Taylor, J., "The fundamental groupoid and the homotopy crossed complex of an orbit space", in Category theory (Gummersbach, 1981), Lecture Notes in Math., Volume 962. Springer, Berlin (1982), 115—122.
• Higgins, P. J., 1971. Categories and groupoids. Van Nostrand Notes in Mathematics. Republished in Reprints in Theory and Applications of Categories, No. 7 (2005) pp. 1–195; freely downloadable. Substantial introduction to category theory with special emphasis on groupoids. Presents applications of groupoids in group theory, for example to a generalisation of Grushko's theorem, and in topology, e.g. fundamental groupoid.
• Mackenzie, K. C. H., 2005. General theory of Lie groupoids and Lie algebroids. Cambridge Univ. Press.
• Weinstein, Alan, "Groupoids: unifying internal and external symmetry — A tour through some examples." Also available in Postscript., Notices of AMS, July 1996, pp. 744–752.
• Weinstein, Alan, "The Geometry of Momentum" (2002)
• R.T. Zivaljevic. "Groupoids in combinatorics—applications of a theory of local symmetries". In Algebraic and geometric combinatorics, volume 423 of Contemp. Math., 305–324. Amer. Math. Soc., Providence, RI (2006)

• Template:Nlab