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| In mathematics, a '''concrete category''' is a [[category (category theory)|category]] that is equipped with a [[faithful functor]] to the [[category of sets]]. This functor makes it possible to think of the objects of the category as sets with additional [[mathematical structure|structure]], and of its [[morphism]]s as structure-preserving functions. Many important categories have obvious interpretations as concrete categories, for example the [[category of topological spaces]] and the [[category of groups]], and trivially also the category of sets itself. On the other hand, the [[homotopy category of topological spaces]] is not '''concretizable''', i.e. it does not admit a faithful functor to the category of sets.
| | It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you have any type of inquiries concerning where and how you can utilize [http://www.youtube.com/watch?v=90z1mmiwNS8 dentist DC], you can call us at our web page. |
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| A concrete category, when defined without reference to the notion of a category, consists of a [[class (set theory)|class]] of ''objects'', each equipped with an ''underlying set''; and for any two objects ''A'' and ''B'' a set of functions, called ''morphisms'', from the underlying set of ''A'' to the underlying set of ''B''. Furthermore, for every object ''A'', the identity function on the underlying set of ''A'' must be a morphism from ''A'' to ''A'', and the composition of a morphism from ''A'' to ''B'' followed by a morphism from ''B'' to ''C'' must be a morphism from ''A'' to ''C''.<ref>{{Citation | last1=Mac Lane | first1=Saunders | author1-link=Saunders Mac Lane | last2=Birkhoff | first2=Garrett | author2-link=Garrett Birkhoff | title=Algebra | publisher=AMS Chelsea | edition=3rd | isbn=978-0-8218-1646-2 | year=1999}}</ref>
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| == Definition ==
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| A '''concrete category''' is a pair (''C'',''U'') such that
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| *''C'' is a category, and
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| *''U'' is a [[faithful functor]] ''C'' → '''Set''' (the category of sets and functions).
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| The functor ''U'' is to be thought of as a [[forgetful functor]], which assigns to every object of ''C'' its "underlying set", and to every morphism in ''C'' its "underlying function".
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| A category ''C'' is '''concretizable''' if there exists a concrete category (''C'',''U'');
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| i.e., if there exists a faithful functor ''U'':''C'' → '''Set'''. All small categories are concretizable: define ''U'' so that its object part maps each object ''b'' of ''C'' to the set of all morphisms of ''C'' whose [[codomain]] is ''b'' (i.e. all morphisms of the form ''f'': ''a'' → ''b'' for any object ''a'' of ''C''), and its morphism part maps each morphism ''g'': ''b'' → ''c'' of ''C'' to the function ''U''(''g''): ''U''(''b'') → ''U''(''c'') which maps each member ''f'': ''a'' → ''b'' of ''U''(''b'') to the composition ''gf'': ''a'' → ''c'', a member of ''U''(''c''). (Item 6 under '''Further examples''' expresses the same ''U'' in less elementary language via presheaves.) The '''Counter-examples''' section exhibits two large categories that are not concretizable.
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| == Remarks ==
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| It is important to note that, contrary to intuition, concreteness is not a [[property (philosophy)|property]] which a category may or may not satisfy, but rather a structure with which a category may or may not be equipped. In particular, a category ''C'' may admit several faithful functors into '''Set'''. Hence there may be several concrete categories (''C'',''U'') all corresponding to the same category ''C''.
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| In practice, however, the choice of faithful functor is often clear and in this case we simply speak of the "concrete category ''C''". For example, "the concrete category '''Set'''" means the pair ('''Set''',''I'') where ''I'' denotes the [[identity functor]] '''Set''' → '''Set'''. | |
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| The requirement that ''U'' be faithful means that it maps different morphisms between the same objects to different functions. However, ''U'' may map different objects to the same set and, if this occurs, it will also map different morphisms to the same function.
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| For example, if ''S'' and ''T'' are two different topologies on the same set ''X'', then
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| (''X'',''S'') and (''X'',''T'') are distinct objects in the category '''Top''' of topological spaces and continuous maps, but mapped to the same set ''X'' by the forgetful functor '''Top''' → '''Set'''. Moreover, the identity morphism (''X'',''S'') → (''X'',''S'') and the identity morphism (''X'',''T'') → (''X'',''T'') are considered distinct morphisms in '''Top''', but they have the same underlying function, namely the identity function on ''X''. | |
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| Similarly, any set with 4 elements can be given two non-isomorphic group structures: one isomorphic to <math>\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}</math>; the other isomorphic to <math>\mathbb{Z}/4\mathbb{Z}</math>.
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| == Further examples ==
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| # Any group ''G'' may be regarded as an "abstract" category with one object, <math>\ast</math>, and one morphism for each element of the group. This would not be counted as concrete according to the intuitive notion described at the top of this article. But every faithful [[group action|''G''-set]] (equivalently, every representation of ''G'' as a [[permutation group|group of permutations]]) determines a faithful functor ''G'' → '''Set'''. Since every group acts faithfully on itself, ''G'' can be made into a concrete category in at least one way.
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| # Similarly, any [[poset]] ''P'' may be regarded as an abstract category with a unique arrow ''x'' → ''y'' whenever ''x'' ≤ ''y''. This can be made concrete by defining a functor ''D'' : ''P'' → '''Set''' which maps each object ''x'' to <math>D(x)=\{a \in P : a \leq x\}</math> and each arrow ''x'' → ''y'' to the inclusion map <math>D(x) \hookrightarrow D(y)</math>.
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| # The category '''[[Category of relations|Rel]]''' whose objects are [[Set (mathematics)|sets]] and whose morphisms are [[Relation (mathematics)|relations]] can be made concrete by taking ''U'' to map each set ''X'' to its power set <math>2^X</math> and each relation <math>R \subseteq X \times Y</math> to the function <math>\rho: 2^X \rightarrow 2^Y</math> defined by <math>\rho(A)=\{y \in Y : \exists x{\in}A~.~xRy\}</math>. Noting that power sets are [[complete lattice]]s under inclusion, those functions between them arising from some relation ''R'' in this way are exactly the [[Complete lattice#Morphisms of complete lattices|supremum-preserving maps]]. Hence '''Rel''' is equivalent to a full subcategory of the category '''Sup''' of [[complete lattices]] and their sup-preserving maps. Conversely, starting from this equivalence we can recover ''U'' as the composite '''Rel''' → '''Sup''' → '''Set''' of the forgetful functor for '''Sup''' with this embedding of '''Rel''' in '''Sup'''.
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| # The category '''Set'''<sup>op</sup> can be embedded into '''Rel''' by representing each set as itself and each function ''f'': ''X'' → ''Y'' as the relation from ''Y'' to ''X'' formed as the set of pairs (''f''(''x''),''x'') for all ''x'' ∈ ''X''; hence '''Set'''<sup>op</sup> is concretizable. The forgetful functor which arises in this way is the [[Functor#Examples|contravariant powerset functor]] '''Set'''<sup>op</sup> → '''Set'''.
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| # It follows from the previous example that the opposite of any concretizable category ''C'' is again concretizable, since if ''U'' is a faithful functor ''C'' → '''Set''' then ''C''<sup>op</sup> may be equipped with the composite ''C''<sup>op</sup> → '''Set'''<sup>op</sup> → '''Set'''.
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| # If ''C'' is any small category, then there exists a faithful functor ''P'' : '''Set'''<sup>''C''<sup>op</sup></sup> → '''Set''' which maps a presheaf ''X'' to the coproduct <math>\coprod_{c \in \mathrm{ob}C} X(c)</math>. By composing this with the [[Yoneda embedding]] ''Y'':''C'' → '''Set'''<sup>''C''<sup>op</sup></sup> one obtains a faithful functor ''C'' → '''Set'''.
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| # For technical reasons, the category '''Ban'''<sub>1</sub> of [[Banach spaces]] and [[contraction (operator theory)|linear contractions]] is often equipped not with the "obvious" forgetful functor but the functor ''U''<sub>1</sub> : '''Ban'''<sub>1</sub> → '''Set''' which maps a Banach space to its (closed) [[unit ball]].
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| == Counter-examples ==
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| The category '''[[homotopy category of topological spaces|hTop]]''', where the objects are [[topological space]]s and the morphisms are [[homotopy|homotopy classes]] of continuous functions, is an example of a category that is not concretizable.
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| While the objects are sets (with additional structure), the morphisms are not actual functions between them, but rather classes of functions.
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| The fact that there does not exist ''any'' faithful functor from '''hTop''' to '''Set''' was first proven by [[Peter Freyd]].
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| In the same article, Freyd cites an earlier result that the category of "small categories and [[natural equivalence]]-classes of functors" also fails to be concretizable.
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| == Implicit structure of concrete categories ==
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| Given a concrete category (''C'',''U'') and a [[cardinal number]] ''N'', let ''U<sup>N</sup>'' be the functor ''C'' → '''Set''' determined by ''U<sup>N</sup>(c) = (U(c))<sup>N</sup>''.
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| Then a [[subfunctor]] of ''U<sup>N</sup>'' is called an ''N-ary predicate'' and a
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| [[natural transformation]] ''U<sup>N</sup>'' → ''U'' an ''N-ary operation''.
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| The class of all ''N''-ary predicates and ''N''-ary operations of a concrete category (''C'',''U''), with ''N'' ranging over the class of all cardinal numbers, forms a [[proper class|large]] [[signature (logic)|signature]]. The category of models for this signature then contains a full subcategory which is [[equivalence of categories|equivalent]] to ''C''. | |
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| == Relative concreteness ==
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| In some parts of category theory, most notably [[topos theory]], it is common to replace the category '''Set''' with a different category ''X'', often called a ''base category''. | |
| For this reason, it makes sense to call a pair (''C'',''U'') where ''C'' is a category and ''U'' a faithful functor ''C'' → ''X'' a '''concrete category over''' ''X''.
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| For example, it may be useful to think of the models of a theory [[Structure (mathematical logic)#Many-sorted structures|with ''N'' sorts]] as forming a concrete category over '''Set'''<sup>''N''</sup>.
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| In this context, a concrete category over '''Set''' is sometimes called a ''construct''.
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| == Notes ==
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| <references />
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| == References ==
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| * Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990). [http://katmat.math.uni-bremen.de/acc/acc.pdf ''Abstract and Concrete Categories''] (4.2MB PDF). Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition).
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| * Freyd, Peter; (1970). [http://www.tac.mta.ca/tac/reprints/articles/6/tr6abs.html ''Homotopy is not concrete'']. Originally published in: The Steenrod Algebra and its Applications, Springer Lecture Notes in Mathematics Vol. 168. Republished in a free on-line journal: Reprints in Theory and Applications of Categories, No. 6 (2004), with the permission of Springer-Verlag.
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| * Rosický, Jiří; (1981). ''Concrete categories and infinitary languages''. [http://www.sciencedirect.com/science/journal/00224049 ''Journal of Pure and Applied Algebra''], Volume 22, Issue 3.
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| [[Category:Category theory]]
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It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.
Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.
At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.
Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.
Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.
Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.
Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.
Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.
In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.
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