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Topic: Colimits

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Limit (category theory)

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Pullback (category theory)

Category of topological spaces

Pullback

	Limit (category theory) - Wikipedia, the free encyclopedia
		Limits and colimits have strong relationships to the categorial concepts of universal morphisms and adjoint functors.
		A covariant functor that commutes with the construction of colimits is said to be cocontinuous or colimit preserving.
		The free product of groups is an example of a colimit construction, and it follows that the free product of a family of free groups is free.
	en.wikipedia.org /wiki/Colimit (1907 words)

	Category of topological spaces - Wikipedia, the free encyclopedia
		The category Top is both complete and cocomplete, which means that all small limits and colimits exist in Top.
		The forgetful functor U : Top → Set has a left adjoint which equips a given set with the discrete topology and a right adjoint which equips a given set with the trivial topology.
		The empty set (considered as a topological space) is the initial object of Top; any singleton topological space is a terminal object.
	en.wikipedia.org /wiki/Category_of_topological_spaces (520 words)

	SUO: Re: Enchoiry on Colimits and Diagrams of Theories (Site not responding. Last check: 2007-08-19)
		Also in general, the colimit construction builds a "sum" or "fusion" of the diagram D. As usual, the most immediate examples appear in the category Set, whose objects are sets and whose functions are between sets.
		The colimit of this diagram is called the coequalizer of the "projection" function p : A2 --> A2/E on the quotient set of A2 by the equivalence relation E \subset A2xA2 generated by the endorelation {(f1(a1), f2(a1))
		Form the colimit L* = col(L) of this diagram of languages > > with associated colimit injection language morphisms > > {l_n : L_n --> L*}.
	0-suo.ieee.org.csulib.ctstateu.edu /email/msg10164.html (1455 words)

	Colimit (Site not responding. Last check: 2007-08-19)
		In category theory the colimit of a functor, also known as a direct limit, is dual to the notion of a limit (or inductive limit).
		The general definition of a colimit is given on the limit page.
		Special cases of colimits, each dual to a special case of limits, include coproduct[?], coequaliser[?], pushout[?].
	www.eurofreehost.com /co/Colimit.html (122 words)

	[No title]
		We now remark that colimits over a Segal category J can be transformed into colimits over a 1-category J0; thus, in the above discussion, there would be no loss of generality in considering the indexing category J to be a 1- category.
		This is a colimit indexed by an ordinal fi (where the ordered set fi is considered as a category with morphisms going in the increasing direction) with the additional property that if i 2 fi is a limit element then the i-th object Xi is equivalent to the colimit of the Xj for j
		In effect, a coproduct or sequential colimit of cofibrations, calculated in M, is a homotopy colimit (cf [8] [15] [16]), in other words it is a colimit in bC; and the -trivial morphis* ms in Cb are stable under coproduct and sequential colimit* because preserves colimits by hypothesis.
	hopf.math.purdue.edu /Simpson/giraudH.txt (11010 words)

	[No title]
		Con* siderable care has to be taken in formulating the construction for topological monoids, b ut the outcome clarifies the status of the original colimits* when K is flag; flag complexes ar* e precisely those for which the colimit* and the homotopy colimit coincide.
		Our main result is the* refore that DJ (K) is modelled by the homotopy colimit* of the relevant diagram of topologi* cal groups, in all three cases and for arbitrary K. When K is flag, the natural projection onto the original colimit* is a homotopy equivalence, and is compatible with the two model maps.
		Theorem 2.8 asserts that standard limits and colimits may themselves be enric* hed in the presence of tensors and cotensors, since they are special cases of indexed limi ts and colimits*.
	hopf.math.purdue.edu /Panov-Ray-Vogt/0202081.txt (9710 words)

	[No title]
		C, such that the classifying space BI belongs to C, the unpointed homotopy colimit hocolimIF belongs to C. The purpose of this paper is to understand to what extent a closed class is closed under extensions by fibrations and under taking unpointed homotopy colimits.
		The homotopy colimit In this section we describe the notion of a diagram indexed by a simplici* al set and define the homotopy colimit* of such a diagram.
		K of simplicial sets, can be reconstructed (up to a weak equivalence) from the homotopy colimit of a diagram indexed essentially by the range of K. The constituents of this diagram are the analogues (in the simplicial category) of the point inverse images of f.
	www.math.purdue.edu /research/atopology/Chacholski/barc.txt (2943 words)

	Practical Foundations of Mathematics (Site not responding. Last check: 2007-08-19)
		Since most of the interesting phenomena may be observed more clearly in the concrete cases of pullback, equaliser, pushout and coequaliser, we postpone the abstract definition to Section 7.3.
		The diversity of the behaviour of finite limits and colimits is striking: the basic features of groups, rings, vector spaces and topology may often be discovered just by looking for the coproducts in these categories.
		Limits and colimits also interact in that we can try to construct one from the other as in Theorem 3.6.9.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s50.html (340 words)

	[No title] (Site not responding. Last check: 2007-08-19)
		Their interconnection and composition is defined using the categorical technique of diagrams and colimits.
		The main technical result of the paper is that the categories of algebra transformation sytems have colimits.
		Constructions of special colimits are then interpreted in terms of component interconnection, and their meaning and relevance is discussed by examples.
	tfs.cs.tu-berlin.de /~mgr/abstracts/seqtrans (166 words)

	Limit (category theory)
		The category C is called co-complete if every functor F : J → C with small J has a colimit.
		As an example in the category of groups, Grp, the functor F : Set → Grp which assigns to every set S the free group over S has a right adjoint (the forgetful functor Grp → Set) and is therefore cocontinuous.
		Limits and colimits are related as follows: A functor F : J → C has a colimit iff for every object N of C, the contravariant functor G : J → Set defined by G(X) = Mor
	www.sciencedaily.com /encyclopedia/limit__category_theory_ (1858 words)

	[No title] (Site not responding. Last check: 2007-08-19)
		Tuesday, 10 April 2001 2:30 - 4:00 Richard Wood Decomposing Regularity Abstract: Many important classes of categories are specified by certain types of colimits, certain types of limits, and exactness conditions relating these.
		If the colimits are given by a KZ-doctrine R and the limits by a co-KZ-doctrine L then it makes sense to enquire about the existence of a distributive law LR--->RL in the sense of Beck.
		The talk will report on joint work with Claudia Centazzo directed towards the problem of solving D=RL, for R, where D is the doctrine for regular categories and L is the doctrine for categories with finite limits.
	www.math.mcgill.ca /~rags/seminar/rwood.txt (147 words)

	categories: Re: colimits of categories (Site not responding. Last check: 2007-08-19)
		Thanks for all the responses to my colimits question; I greatly appreciate them, and it will take me a while to digest them.
		I think I should be a bit more precise as to what I'm asking for, so let me try again: First, here's what I mean by a colimit of categories: let I be a 1-category, trivially extended to a weak 2-category with only identity 2-morphisms.
		Also, we can generalize these questions to the setting of F be a functor from I to nCat, by which I mean the weak (n+1)-category of all n-categories; then we have a functor of n-categories that we want to represent.
	north.ecc.edu /alsani/ct02(1-2)/msg00053.html (494 words)

	An isomorphism between Bredon and Quinn homology via homotopy colimits, by Robert N. Talbert (Site not responding. Last check: 2007-08-19)
		In this paper, we show an isomorphism between the Bredon homology of X with a certain coefficient system and the Quinn homology of X/G with coefficients in the spectral sheaf S(p) where S is any homotopy-invariant functor from spaces to spectra (e.g., the K-theory spectrum functor).
		We then show that Bredon homology may be expressed as the homology of a category when G is discrete and acting cellularly on a simplicial complex, and then this homology may be computed as the homotopy groups of a homotopy colimit of a spectrum-valued functor.
		The homology isomorphism is obtained on the homotopy colimit level.
	www.math.uiuc.edu /K-theory/0260 (202 words)

	Abstract of: Weighted colimits and formal balls in generalized metric spaces (Site not responding. Last check: 2007-08-19)
		Abstract of: Weighted colimits and formal balls in generalized metric spaces
		(a) Limits of Cauchy sequences in a (possibly non-symmetric) metric space are shown to be weighted colimits (a notion introduced by Borceux and Kelly, 1975).
		As a consequence, further insights from enriched category theory are applicable to the theory of metric spaces, thus continuing Lawvere's (1973) approach.
	db.cwi.nl /rapporten/abstract.php?abstractnr=1093 (174 words)

	Limit (category theory) (Site not responding. Last check: 2007-08-19)
		Limits and colimits are defined via [[universal propertyuniversal properties]] and as such provide many examples of adjoint functors.
		Colimits are defined analogous to limits: A colimit of the functor F : J
		D is cocontinuous if it transforms colimits into colimits.
	www.theezine.net /l/limit-category-theory-.html (701 words)

	TTT30 (Site not responding. Last check: 2007-08-19)
		The cohomology of the latter colimit space is the Stanley-Reisner face ring of K appearing in the commutative algebra of simplicial complexes.
		This implies, that in the case of flag complex colimits of the diagrams of tori and classifying spaces for tori determined by K agree (in the sence that the classifying space for the colimit group is the colimit space).
		Finally, in the case of general K the inconsistence between the diagram of groups and the diagram of classifying spaces can be remedied by replacing colimits with homotopy colimits.
	www.shef.ac.uk /~pm1jg/ttt/ttt30.html (483 words)

	Practical Foundations of Mathematics
		Limits and colimits are perhaps the most important cases of universal properties, so we devote the next three sections to them and to how they relate to adjunctions in general.
		For the graph on the right, a limit is a pullback, but the colimit is simply the value of the diagram at the corner.
		For the opposite of this diagram-shape, a colimit is a pushout.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s73.html (1987 words)

	Homotopy Colimits - Comparison Lemmas for Combinatorial Applications - Welker, Ziegler, Zivaljevi'c (ResearchIndex) (Site not responding. Last check: 2007-08-19)
		Abstract: We provide a "toolkit" of basic lemmas for the comparison of homotopy types of homotopy colimits of diagrams of spaces over small categories.
		We show how this toolkit can be used on quite different fields of applications.
		0.5: Fibrations And Homotopy Colimits Of Simplicial Sheaves - Charles Re Zk
	citeseer.lcs.mit.edu /welker97homotopy.html (881 words)

	Andy Tonks' publications and preprints (Site not responding. Last check: 2007-08-19)
		The Kan extension condition is shown to be automatic for cubical groups which have extra connection degeneracies; the cubical theory is then parallel to the simplicial.
		Theory and applications of crossed complexes: the Eilenberg-Zilber theorem and homotopy colimits.
		We prove an Eilenberg-Zilber theorem for crossed complexes, and use this to develop the theory of homotopy colimits of (homotopy coherent) diagrams in Crs.
	www.bangor.ac.uk /ma/research/tonks/pubs.html (489 words)

	W208
		We say that the iterated colimit can be reduced if it is possible to 'skip' the intermediate step represented by P and to directly represent N as the simple colimit of a 'large' pattern R whose objects are those of the various patterns P
		Conversely, if some links of P are not simple, the iterated colimit cannot be reduced to a simple colimit.
		As a concrete example, in the system modelling the occidental society, with its members and their various social groups, Europe constructed as a 'Europe of nations' would be modeled by a non-reducible 2-colimit, in which institutional links must be mediated by the nations.
	perso.wanadoo.fr /vbm-ehr/Ang/W208.htm (627 words)

	Talk:Limit (category theory) (Site not responding. Last check: 2007-08-19)
		I'm wondering if it would be better to separate out the stuff on colimits.
		That way there would be more space to discuss the special cases such as products, equalisers and pushouts (and their duals on the colimit) page.
		All is still licensed under the GNU FDL.
	www.termsdefined.net /ta/talk:limit-(category-theory).html (202 words)

	On Sifted Colimits And Generalized Varieties - Ad, Rosick (ResearchIndex) (Site not responding. Last check: 2007-08-19)
		Filtered colimits, i.e., colimits over schemes D such that D-colimits in Set commute with finite limits, have a natural generalization to sifted colimits: these are colimits over schemes D such that D-colimits in Set commute with finite products.
		Generalized varieties are defined as free completions of small categories under sifted-colimits (analogously to finitely accessible categories which are free filtered-colimit...
		Adamek and J. Rosicky, On sifted colimits and generalized varieties, submitted.
	citeseer.ist.psu.edu /418095.html (375 words)

	Citations: Theorie des Topos et Cohomologie Etale des Schemas - Artin, Grothendieck, Verdier (ResearchIndex) (Site not responding. Last check: 2007-08-19)
		....products and filtered colimits, that for any set D i (i I) of filtered diagrams in C the canonical morphism is an isomorphism.
		A generalization of the concept of accessible category, obtained by considering an arbitrary class of colimits, has been studied by Hongde Hu [14] Although this....
		For example, if A is a small category with finite colimits, then Ind A = A op ; Set] lex is the category of all presheaves on A op preserving finite limits, see [AGV] and, as we will show, Sind A = A op ; Set] fp is....
	citeseer.ifi.unizh.ch /context/214782/0 (4582 words)

	categories: Re: colimits of categories (Site not responding. Last check: 2007-08-19)
		Alternatively, observe that Cat is locally finitely presentable (lfp), so that if it preserved all colimits it would have a right adjoint, and prove that it does not have one.
		One could use the description of filtered colimits in Cat given in the Kelly-Lack paper to show that IR preserves filtered colimits, and deduce that R does so.
		Alternatively one could show that the ``free-living isomorphism'' (called C above) is finitely presentable in both Gpd and Cat, and constitutes a strong generator of Gpd, and deduce that I preserves finitely presentable objects.
	north.ecc.edu /alsani/ct02(1-2)/msg00049.html (502 words)

	Categorified Arithmetic
		Various kinds of categories with colimits Various ways of categorifying addition or multiplication as a property that a category might have, including on the additive side: a.
		Categories with finite colimits free category on 1 with finite colimits is FinSet The adjunction between Cat and CoLex c.
		Categories with all small colimits (cocomplete categories) free cocomplete category on 1 is Set The free cocomplete category on a category C: presheaves on C or $\hat{C}$ The Yoneda embedding is the unit of this adjunction; we can think of \hat as giving a pseudomonad on Cat!
	math.ucr.edu /home/baez/hda/categorified_arithmetic.html (959 words)

	SUO: Composing Ontologies using morphisms and colimits (Site not responding. Last check: 2007-08-19)
		Previous message: KIF: Re: SUO: Composing Ontologies using morphisms and colimits
		An extended KIF kernel compatible with CT (and that computes things such as colimits and colimit-preserving functors) would be a welcome addition to our tool set.
		To end this rambling note: If you or anybody on your cc list are interested in the comments I've received from the CT community, I'd be happy to send them along (all anonymous unless I receive permission to use names).
	grimpeur.tamu.edu /pipermail/kif/2001-January/000321.html (321 words)

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