Wiedemann–Franz law: Difference between revisions
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In [[category theory]], a branch of [[mathematics]], a '''connected category''' is a [[category (category theory)|category]] in which, for every two objects ''X'' and ''Y'' there is a [[finite sequence]] of objects | |||
:<math>X = X_0, X_1, \ldots, X_{n-1}, X_n = Y</math> | |||
with morphisms | |||
:<math>f_i : X_i \to X_{i+1}</math> | |||
or | |||
:<math>f_i : X_{i+1} \to X_i</math> | |||
for each 0 ≤ ''i'' < ''n'' (both directions are allowed in the same sequence). Equivalently, a category ''J'' is connected if each [[functor]] from ''J'' to a [[discrete category]] is constant. In some cases it is convenient to not consider the empty category to be connected. | |||
A stronger notion of connectivity would be to require at least one morphism ''f'' between any pair of objects ''X'' and ''Y''. Clearly, any category which this property is connected in the above sense. | |||
A [[small category]] is connected [[if and only if]] its underlying graph is [[connected graph|weakly connected]]. | |||
Each category ''J'' can be written as a disjoint union (or [[coproduct]]) of a connected categories, which are called the '''connected components''' of ''J''. Each connected component is a [[full subcategory]] of ''J''. | |||
==References== | |||
*{{cite book | first = Saunders | last = Mac Lane | authorlink = Saunders Mac Lane | year = 1998 | title = [[Categories for the Working Mathematician]] | series = Graduate Texts in Mathematics '''5''' | edition = (2nd ed.) | publisher = Springer-Verlag | isbn = 0-387-98403-8}} | |||
[[Category:Category theory]] | |||
{{categorytheory-stub}} | |||
Revision as of 19:20, 2 July 2013
In category theory, a branch of mathematics, a connected category is a category in which, for every two objects X and Y there is a finite sequence of objects
with morphisms
or
for each 0 ≤ i < n (both directions are allowed in the same sequence). Equivalently, a category J is connected if each functor from J to a discrete category is constant. In some cases it is convenient to not consider the empty category to be connected.
A stronger notion of connectivity would be to require at least one morphism f between any pair of objects X and Y. Clearly, any category which this property is connected in the above sense.
A small category is connected if and only if its underlying graph is weakly connected.
Each category J can be written as a disjoint union (or coproduct) of a connected categories, which are called the connected components of J. Each connected component is a full subcategory of J.
References
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