nLab
colimit

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

Context

Category theory

category theory

Concepts

Universal constructions

Theorems

Extensions

Applications

Limits and colimits

limits and colimits

1-Categorical

2-Categorical

(∞,1)-Categorical

Model-categorical

Contents

Idea

The concept of colimit is that dual to a limit:

a colimit of a diagram in a category is, if it exists, the co-classifying space for morphisms out of that diagram.

The intuitive general idea of a colimit is that it defines an object obtained by sewing together the objects of the diagram, according to the instructions given by the morphisms of the diagram.

We have

  • the notion of colimit generalizes the notion of direct sum;

  • the notion of weighted colimit generalizes the notion of weighted (direct) sum.

Sometimes colimits (or some colimits) are called inductive limits or direct limits; see the discussion of terminology at limit.

Definition

A colimit in a category C is the same as a limit in the opposite category, C op.

More in detail, for F:D op→C op a functor, its limit limF is the colimit of F op:D→C.

Examples

Here are some important examples of colimits:

Weighted colimits

A weighted colimit in C is a weighted limit in C op.

Properties

The properties of colimits are of course dual to those of limits. It is still worthwhile to make some of them explicit.

Contravariant Hom sends colimits to limits

For C a locally small category, for F:D→C a functor, for c∈C and object and writing C(F(−),c):C→Set, we have

C(colimF,c)≃limC(F(−),c).C(colim F, c) \simeq lim C(F(-), c) \,.

Depending on how one introduces limits this holds by definition or is an easy consequence. In fact, this is just rewriting the respect of the covariant Hom of limits (as described there) in C op in terms of C:

C(colimF,c) ≃C op(c,colimF) ≃C op(c,limF op) ≃limC op(c,F op(−)) ≃limC(F(−),c)\begin{aligned} C(colim F, c) & \simeq C^{op}(c, colim F) \\ & \simeq C^{op}(c, lim F^{op}) \\ & \simeq lim C^{op}(c, F^{op}(-)) \\ & \simeq lim C(F(-), c) \end{aligned}

Notice that this actually says that C(−,−):C op×C→Set is a continuous functor in both variables: in the first it sends limits in C op and hence equivalently colimits in C to limits in Set.

Proposition – left adjoints commute with colimits

Let L:C→C′ be a functor that is left adjoint to some functor R:C′→C. Let D be a small category such that C admits limits of shape D. Then L commutes with D-shaped colimits in C in that

for F:D→C some diagram, we have

L(colimF)≃colim(L∘F).L(colim F) \simeq colim (L \circ F) \,.
Proof

Using the adjunction isomorphism and the above fact that commutes with limits in both arguments, one obtains for every c′∈C′

C′(L(colimF),c) ≃C(colimF,R(c′)) ≃limC(F(−),R(c′)) ≃limC′(L∘F(−),c′) ≃C′(colim(L∘F),c′)..\begin{aligned} C'(L (colim F), c) & \simeq C(colim F, R(c')) \\ & \simeq lim C(F(-), R(c')) \\ & \simeq lim C'(L \circ F(-), c') \\ & \simeq C'(colim (L \circ F), c') \,. \end{aligned} \,.

Since this holds naturally for every c′, the Yoneda lemma, corollary II on uniqueness of representing objects implies that R(limF)≃lim(R∘F).

Revised on August 26, 2012 23:53:09 by Urs Schreiber (89.204.139.66)
Edit | Back in time (15 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: category theory, groupoid, isomorphism, internalization, enriched category theory, sheaf and topos theory, Day convolution, anafunctor, morphism, category, object, group, equivalence, natural isomorphism, natural transformation, functor, topos, composition, subcategory, (-1)-category, discrete category, epimorphism, hypercover, sheaf, presheaf, skeleton, locally small category, fundamental groupoid, Lie infinity-algebroid, smooth algebra, monoid, Cat, Set, set, function, Ab, forgetful functor, faithful functor, full functor, category algebra, small category, functor category, monoidal category, enriched category, end, symmetric monoidal category, braided monoidal category, pseudomonic functor, endomorphism, Kan extension, automorphism, Nonabelian Algebraic Topology, (infinity,1)-category, hom-set, terminal object, finitely complete category, representable functor, subobject, diagram, limit, cone, multicategory, span, opposite category, hom-functor, global element, logic, cartesian monoidal category, extensive category, cover, sieve, dinatural transformation, weak factorization system, simplicial object, parallel morphisms, zero object, initial object, cartesian closed category, locally cartesian closed category, CW complex, simplex category, homotopy coherent category theory, (bo, ff) factorization system, orthogonal factorization system, essentially surjective functor, category of fibrant objects, full and faithful functor, factorization system, internal category, arrow category, free functor, projective object, power, Lawvere theory, Rel, nerve, image, allegory, geometric realization, regular category, geometric morphism, inhabited set, regular epimorphism, Yoneda lemma, interval object, fundamental category, coherent category, Heyting category, kernel pair, exact category, pretopos, balanced category, adjoint functor, internally projective object, Yoneda embedding, Cauchy complete category, pointed set, under category, bialgebra, Hopf algebra, strict n-category, coproduct, well-powered category, small object argument, fibration, Grothendieck fibration, concrete category, endofunctor, monomorphism, orthogonality, Galois connection, strong epimorphism, split epimorphism, regular monomorphism, internal logic, (infinity,n)-category of cobordisms, finite object, context, complete category, colimit, cocomplete category, pushout, intersection, span trace, hyperstructure, reflective subcategory, trace, isofibration, string diagram, conservative functor, homotopy n-type, 2-vector space, bimodule, profunctor, crossed square, cat-2-group, subterminal object, homotopy limit, Grpd, retract, idempotent, 2009 January changes, free cocompletion, dagger-category, SimpSet, comma category, overcategory, split monomorphism, higher van Kampen theorem, large site, coequalizer, weak equivalence, equivalence of categories, homotopy type, connection on a cubical set, weighted limit, action of a category on a set, 1-groupoid, 1-category, limit in a quasi-category, join of quasi-categories, ASSet, internal hom, constant morphism, partial order, disjoint union, quotient object, full subcategory, filtered category, pro-object, ind-object, category of elements, dense functor, FinSet, Isbell envelope, wedge sum, smash product, congruence, adjoint functor theorem, total category, large category, accessible category, sketch, locally presentable category, Categories and Sheaves, dense subcategory, associative unital algebra, cogenerator, compact object, Grothendieck universe, finite category, stuff, structure, property, left adjoint, right adjoint, projective limit, generalized element, zero morphism, kernel, cokernel, universal construction, bimorphism, exact functor, cocone, Thomason model structure, strong monomorphism, strict epimorphism, pre-abelian category, additive and abelian categories, biproduct, direct sum, direct product, mapping cone, local isomorphism, base change, category of sheaves, sheafification, IPC-property, group object, 2009 March changes, wide subcategory, injective object, inverse image, closed monoidal structure on presheaves, functors and comma categories, cohomology, Dold-Thom theorem, subsequential space, Godement product, natural numbers object, compact object in an (infinity,1)-category, type theory, diffeological space, final functor, co-Yoneda lemma, exact sequence, directed colimit, flat module, sequent calculus, limits and colimits by example, subquotient, derived stack, coreflective subcategory, topologizing subcategory, Gabriel multiplication, dependent product, exponential object, calculus of fractions, monadic functor, distributive category, local object, descent, category of simplices, simplicial model category, (infinity,1)-category of (infinity,1)-functors, 2009 April changes, flat functor, locally finitely presentable category, initial algebra of an endofunctor, power object, algebra for an endofunctor, strict morphism, coimage, commutativity of limits and colimits, higher category theory and physics, rigid monoidal category, split idempotent, coalgebra for an endofunctor, path category, Grothendieck construction, quasitopos, finitely cocomplete category, codiscrete groupoid, exponential ideal, (1,0)-category, replete subcategory, locally compact space, computad, topological concrete category, core, An Exercise in Kantization, weighted colimit, computer science, triangle identities, 2009 July changes, groupoid object in an (infinity,1)-category, model structure on chain complexes, essential image, adjoint equivalence, cosmos, nerve and realization, cocontinuous functor, comonad, cofibrantly generated model category, transfinite composition, extremal monomorphism, generalized multicategory, effective epimorphism, simplicial resolution, Reedy model structure, Kan fibrant replacement, exterior algebra, (infinity,1)-categorical hom-space, symmetric function, noetherian category, germ, property sup, semisimple object, Stone Spaces, universal colimit, Karoubi envelope, Cartesian morphism, quantum group, quasitriangular bialgebra, variety of algebras, solution set condition, database of categories, small object, infinitesimal object, subfunctor, semicartesian monoidal category, 2009 September changes, internal relation, single-sorted definition of a category, localic topos, Paine on a Category Theory Demonstrations program, diagonal morphism, A Survey of Elliptic Cohomology - E-infinity rings and derived schemes, graph of a functor, cograph of a functor, adjoint (infinity,1)-functor, inverse limit, direct limit, inductive limit, category theory - contents, type, Q-category, free groupoid, ringed topos, A Survey of Elliptic Cohomology - elliptic curves, connected object, smooth topos, injective hull, amazing right adjoint, strict category, preset, microlinear space, Quillen bifunctor, cocomplete well-pointed topos, constant functor, tiny object, 2-rig, tensor category, completion, propositions as types, category theory vs order theory, syntax - semantics duality, regular and exact completions, finitely generated object, proposition, extranatural transformation, Bousfield localization of model categories, saturated class of maps, filtered limit, small presheaf, monadicity theorem, nPOV, bifibration, codomain fibration, Beck-Chevalley condition, species, sink, cokernel pair, internal category in a monoidal category, cograph of a profunctor, cofibration, cleavage, algebraic weak factorization system, inhabited object, Euclidean-topological infinity-groupoid, covering, Grothendieck-Maltsiniotis infinity-category, Thom spectrum, adjoint triple, semifunctor, theory, (infinity,n)-module, free object, dependent type, essentially unique, closed functor, applications of (higher) category theory, connected category, solid functor, universal element, model structure on algebraic fibrant objects, Tannaka duality, Como, canonical morphism, Topological Quantum Field Theories from Compact Lie Groups, idempotent adjunction, preserved limit, semicategory, (infinity,1)-Grothendieck construction, category of representations, W-type, model structure for left fibrations, category of presheaves, An Introduction to Homological Algebra, clique, type theory - contents, sequential limit, algebra for a profunctor, first-order hyperdoctrine, tripos, Isbell duality, Street fibration, internal logic of an (infinity,1)-topos, identity type, model structure on operads, calculus of constructions, finite quantum mechanics in terms of dagger-compact categories, direct category, Conduche functor, final (infinity,1)-functor, Mod, tangent category, algebrad, essentially small (infinity,1)-category, accessible functor, Hall algebra, connected topos, forcing, symmetric algebra, smooth manifold, created limit, parametric right adjoint, BDR 2-vector bundle, zigzag, AT category, locally connected topos, composable pair, exact (infinity,1)-functor, absolute colimit, two-out-of-six property, exact square, Cisinski model structure, geometric homotopy groups in an (infinity,1)-topos, C-star-category, sifted colimit, Gabriel-Ulmer duality, embedding, Chern-Weil homomorphism, Freyd cover, effective monomorphism, sifted category, adjoint monad, thin category, Freyd-Mitchell embedding theorem, indecomposable object, model structure on cosimplicial abelian groups, model structure on monoids in a monoidal model category, indexed functor, Lawvere distribution, algebraic model for modal logics, geometric realization of simplicial topological spaces, poset of subobjects, augmented simplicial set, algebra over a Lawvere theory, comonadic functor, regular functor, associativity, type-theoretic definition of category, zigzag category, essential fiber, codiagonal, pure type system, cartesian functor, topological groupoid, Topos, derived geometry, progroup, local geometric morphism, trivial model structure, strongly connected site, cohesive topos, the logic S4(m), cohesive site, locally connected site, modal logic, Haefliger groupoid, conical limit, cofinal diagrams, higher homotopy van Kampen theorem, category of monoids, blob n-category, adjoint quadruple, diagonal functor, final subcategory, ring object, evaluation map, normal modal logic, monoidal Quillen adjunction, indexed category, cosheaf, Boardman-Vogt resolution, integral transforms on sheaves, Drinfel'd double, Leinster2010, two-sided fibration, proper topological groupoid, universally closed morphism, geometric realization of categories, Tannaka duality for Lie groupoids, over-topos, pure morphism, adjoint lifting theorem, coherent functor, infinity-connected (infinity,1)-site, locally infinity-connected (infinity,1)-site, Tannaka duality for geometric stacks, sheaf semantics of concurrent interacting objects, factorization category, term, projection, connected site, generalisation as an adjunction, variable, Frobenius reciprocity, Handbook of Categorical Algebra, geometric category, cartesian category, cartesian logic, complete small category, hyperdoctrine, canonical presentation, locally contractible topological infinity-groupoid, two-out-of-three, cohesive (infinity,1)-topos -- infinitesimal cohesion, categorical semantics, Poset, syntactic category, cartesian theory, unital category, proadjoint, axiom, Frobenius functor, homotopy type theory, (infinity,1)-vector bundle, fiber functor, umbrella category, p-divisible group, sigma-model -- exposition of a general abstract formulation, internal subcategory, type of propositions, theorem, truncated object, unit type, pure subobject, type of types, types and logic - table, axiom UIP, totalization, contents of contents, locally presentable categories - introduction, syntax, product type, quotient type, indexed adjoint functor theorem, concrete object, descent object, cohesive homotopy type theory, abstract general, concrete general and concrete particular, factorization system over a subcategory, sesquialgebra, reflective factorization system, tensor, Mac Lane's proof of the coherence theorem for monoidal categories, M-complete category, saturated class of limits, stable factorization system, factorization structure, regular epimorphism in an (infinity,1)-category, homotopy level, category of descent data, function type, bracket type, (epi, mono) factorization system, (eso+full, faithful) factorization system, (eso, fully faithful) factorization system, dependent product type, filtered object, Coq, empty type, quantum lattice system, inductive type, Agda, higher inductive type, final lift, existential quantifier, universal quantifier, n-image, coend in a derivator, proofs as programs, generalized Reedy model structure, locally cartesian closed functor, proof, HoTT methods for homotopy theorists, univalence axiom, intensional type theory, atomic category, strong adjoint functor, pushout-product, small limit, monad with arities, judgmental equality, dependent type theory, contractible type, extensional type theory, sum type, cartesian closed functor, codiscrete object, co-concrete object, semantics, h-set, decidability, h-proposition, equivalence in homotopy type theory, dual object in a closed category, Cartan-Eilenberg category, canonical form, presentation of a category by generators and relations, polymorphism, relative adjoint functor, model structure for dendroidal complete Segal spaces, adhesive category, coherent hyperdoctrine, symmetric multicategory, projectively cofibrant diagram, atom, type formation, compact element, Joyal-Tierney calculus, dependent sum type, exactness property, Segal condition, exhaustive category, substitution, fixed-point combinator, relation between type theory and category theory, M-type, categorical wreath product, locally bounded category, categorical model of dependent types, sequent, first-order theory, morphism of sites, Haskell, permutative category, logical framework, Burali-Forti's paradox, trialgebra, judgment, Russell's paradox, distributivity pullback, projective presentation, context extension, Baer's criterion, tower, bifunctor, injective presentation, abelianization, numeral, definition, program, gaunt category, elementary function arithmetic, Brouwer-Heyting-Kolmogorov interpretation, second-order arithmetic, computational trinitarianism, arity class, monad (in computer science), computation, programming language, natural deduction, geometry of physics, transfinite construction of free algebras, motivation for higher differential geometry, metalanguage, theory of presheaf type, Practical Foundations for Programming Languages, term in context, Tannakian category, contradiction, inconsistency, modal type, type of h-sets, Bishop set, antecedent, succedent, consequent, looping combinator, deductive system, pointed endofunctor, dependent sum, h-groupoid, coinductive type, practical foundation of mathematics, adjoint triangle theorem, separator, modal type theory, differential homotopy type theory, strict initial object, axiom K, mapping stack, observational type theory, meaning explanation, natural numbers type, hopfish algebra, triangular Hopf algebra, structure on algebras and their module categories - table, geometry of physics - modules, equifibered natural transformation, van Kampen colimit, Drinfeld center, Hopf monoidal category