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...y help export some of the ideas from [[algebraic topology]] and [[homotopy theory]] into other domains, such as in [[commutative algebra]] and [[algebraic ge In a category <math>\mathcal{C}</math> with a product <math>\times</math> and initial obj ...

...bject of morphisms'—together with a collection of morphisms in the ambient category satisfying certain identities. [[Group object]]s, are common examples of in ...make the collection of internal categories in a fixed category into a [[2-category]]. ...

In [[category theory]], a branch of abstract mathematics, a '''tower''' is defined as follows. ...erse order, regarded as a category. A '''(countable) tower of objects in a category <math>\mathcal A</math>''' is a [[functor]] from <math>\mathcal I</math> to ...

...any finite coproduct category can be thought of as a cocartesian monoidal category. ...ries]].<ref>{{nlab|id=cartesian+monoidal+category|title=Cartesian monoidal category}}</ref> ...

...y, in [[category theory]], a '''2-functor''' is a [[morphism]] between [[2-category|2-categories]].<ref>{{cite conference | title = Category Seminar: Proceedings of the Sydney Category Theory Seminar, 1972/1973 ...

...een [[Bicategory|bicategories]] generalizes that of [[functor]]s between [[Category (mathematics)|categories]]. * for each pair of objects ''x,y ∈ C'' a functor on morphism-categories, <math> P_{x,y}: C(x,y)\to D(P ...

{{Short description|Category-theoretic construction}} ...by <math>H(X, Y)</math>. (It may also be defined using the language of [[2-category]].) ...

...tal number of nodes, i.e. the sum of the number of nodes of the individual objects. The EGF <math>f_n(z)</math> of the number of different configurations unde ...whose size is less than <math>|G|</math>.) As mentioned, the nodes of the objects are re-labelled when they are distributed into the slots. Say an object of ...

...were emphasised,<ref>Kubovy, Michael: Concurrent-Pitch Segregation and the Theory of Indispensable Attributes. In: Kubovy, Michael/Pomeranz, James R. (Eds.): ...d is stressed.<ref>Kubovy, Michael/Valkenburg, D. Van: ''In defense of the theory of indispensable attributes.'' Cognition 87, 2003, pp. 225–233, here: p. 22 ...

...mford criterion which characterises stable points in [[geometric invariant theory]], the trajectory of <math>\tilde{x}</math> is looked at along the flow of ...ot minimal amount of symmetry), '''polystable''' (being made out of stable objects), or '''unstable''' (having too much symmetry, the opposite of stable). ...

{{Short description|Category theory concept}} ...or keeping track of data surrounding a fixed object <math>X</math> in some category <math>\mathcal{C}</math>. There is a dual notion of undercategory, which is ...

...erre subcategories. They are strongly linked to the notion of a [[quotient category]]. Let <math>\mathcal{A}</math> be an [[abelian category]]. A non-empty full [[subcategory]] <math>\mathcal{C}</math> is called a '' ...

...is a [[category (mathematics)|category]] whose [[Object (category theory)|objects]] are certain [[Lie algebra representation|representations]] of a [[Semisim == Definition of category O == ...

...ightarrow Y</math> that "behaves" like [[Surjection|surjections]] on the [[category of sets]]. ...nd cartesian closed categories with author commentary |journal=Reprints in Theory and Applications of Categories |date=2006 |issue=15 |pages=1–13 |url=http:/ ...

In [[category theory]], a branch of [[mathematics]], a '''groupoid object''' is both a generaliz ...theory)|fiber product]]s consists of a pair of [[Object (category theory)|objects]] <math>R, U</math> together with five [[morphism]]s ...

...s]] <math>\mathbf{E}</math> over any one of its [[object (category theory)|objects]] <math>X</math> is itself a topos. Moreover, if there is a [[morphism]] <m ...over B</math> denote an object of it, where ''X'' is an object of the base category. Then <math>B^*</math> is a functor which maps: <math> - \mapsto {B \times ...

{{Short description|Type of mathematical category}} ...''<ref>{{harvnb|Trimble|n.d.|loc=§ 1}}</ref> is a [[category (mathematics)|category]] where ...

In [[category theory]], a branch of [[mathematics]], the '''categorical trace''' is a generaliza ...[tensor product]].) An [[object (category theory)|object]] ''X'' in such a category ''C'' is called [[dualizable object|dualizable]] if there is another object ...

...em. These combinatorial models will provide, through the [[combinatorics]] theory, the operations needed to solve the problem. ...of a set of ''n'' elements. It is needed to know if the order in which the objects are selected matters and whether an object can be selected more than once o ...

{{Short description|Category theory}} ...ory]], a '''highest-weight category''' is a [[linear category|''k''-linear category]] '''C''' (here ''k'' is a [[field (mathematics)|field]]) that ...