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...certain Serre subcategories. They are strongly linked to the notion of a [[quotient category]]. ...he object <math>A</math> is in <math>\mathcal{C}</math> if and only if the objects <math>A'</math> ...

...ot minimal amount of symmetry), '''polystable''' (being made out of stable objects), or '''unstable''' (having too much symmetry, the opposite of stable). ...by constructing [[moduli space]]s of them, because the symmetries of these objects cause the formation of [[Singularity_(mathematics)|singularities]], and obs ...

{{Short description|Objects between rings and their fields of fractions}} In [[mathematics]], '''almost modules''' and '''almost rings''' are certain objects interpolating between [[commutative ring| rings]] and their [[field of frac ...

...ory <math>\mathcal{B}</math> given by the <math>\mathcal{C}</math>-closed objects in <math>\mathcal{A}</math>. ...

...cs)|group]]: a quotient variety, say, would be a coarse approximation of a quotient stack. ...a stratification by quotient stacks (e.g., a [[Deligne–Mumford stack]].) A quotient stack is also used to construct other stacks like '''classifying stacks'''< ...

== Problem of constructing a quotient == ...set-theoretic group action, there is no straightforward way to construct a quotient for a group-scheme action. One exception is the case when the action is fre ...

...ing (i.e. treating as [[Zero-object|zero]]) all [[Object (category theory)|objects]] from <math>\mathcal{B}</math>. There is a canonical [[Exact functor|exac ...s of groups]]. Serre quotients are somewhat similar to [[Quotient category|quotient categories]], the difference being that with Serre quotients all involved c ...

...eory of [[impartial game]]s in [[misère]] play, an '''indistinguishability quotient''' is a [[commutative monoid]] Suppose the game of [[Nim]] is played as usual with heaps of objects, but that at the start of play, ...

...theory)|fiber product]]s consists of a pair of [[Object (category theory)|objects]] <math>R, U</math> together with five [[morphism]]s === Group objects === ...

...on the set of 2-dimensional objects. However, in a graph there are no such objects, so ''C''₂ is a trivial group. Therefore, the image of the secon ...x, y and z are equivalent - they are in the same equivalence class of the quotient. In other words, <math>H_0(X) </math> is generated by a single element (any ...

.../math> is a class of all morphisms whose ranges belong to a given class of objects <math>L</math> in <math>K</math> it is convenient to replace <math>\Omega</ .../math> is a class of all morphisms whose ranges belong to a given class of objects <math>M</math> in <math>K</math> it is convenient to replace <math>\Phi</ma ...

...tics and Physics |author-link2=Alan Weinstein|date=2008|chapter=Group-like objects in Poisson geometry and algebra |series=Contemporary Mathematics |url=http: ...</math>, as the [[subcategory]] of <math>\mathcal{C}</math> made up by all objects of <math>\mathcal{C}</math> lying over <math>U</math> and all morphisms of ...

* every (possibly infinite) family of objects in <math>\mathcal{A}</math> has a [[coproduct]] (also known as direct sum) ...ath>\mathcal{A}</math>, then both <math>\mathcal{C}</math> and the [[Serre quotient category]] <math>\mathcal{A}/\mathcal{C}</math> are Grothendieck categories ...

...operatorname{Quot}_F(X))</math> is the set of isomorphism classes of the [[quotient by an equivalence relation|quotients]] of <math>F \times_S T</math> that ar It is typically used to construct another scheme parametrizing geometric objects that are of interest such as a [[Hilbert scheme]]. (In fact, taking ''F'' t ...

...adjoint bundle]] <math> P \times_G \mathfrak{g}</math>. In conclusion, the quotient by <math> G</math> of the exact sequence above yields a short exact sequenc ...are points of ''<math> M</math>'', and whose morphisms are elements of the quotient of ''<math> P \times P</math>'' by the diagonal action of ''<math> G</math> ...

...(mathematics)|stack]] is a prestack with effective descents, meaning local objects may be patched together to become a global object. Given an object ''U'' in ''C'' and objects ''x'', ''y'' in <math>F(U) = p^{-1}(U)</math>, for each morphism <math>f: V ...

...them (e.g. by gluing along localizations or taking noncommutative [[stack quotient]]s). ...f the values of the field is that it also provides new techniques to study objects in commutative algebraic geometry such as [[Brauer group]]s. ...

...the {{nowrap|2-group}} Aut&nbsp;''C''. This is the monoidal category whose objects are the autoequivalences of ''C'' (i.e. [[equivalence of categories|equival ...at ''x'', written Π₂(''X'',''x''). As a monoidal category, the objects are [[loop (topology)|loop]]s at ''x'', with multiplication given by concat ...

...s in a microscope field, but shape factors also apply to three-dimensional objects. The particles could be the grains in a [[Metallography|metallurgical]] or Another very common shape factor is the circularity (or [[isoperimetric quotient]]), a function of the perimeter ''P'' and the area ''A'': ...

...ion classes are disjoint, their adjacencies constitute a new graph, a '''[[quotient graph]]''' <math>G/P</math>, whose vertices are the members of <math>P</mat ...tion. In the upper right diagram, the edges between these sets depict the quotient given by this partition, while the edges internal to the sets depict the co ...