nLab
M-complete category

Context

Category theory

Limits and colimits

MM-complete and EE-cocomplete categories

Idea

A category is MM-complete or EE-cocomplete if has certain limits or colimits of morphisms in a given class MM or EE.

Not to be confused with an M-category.

Definitions

Let CC be a category and let MM be a class of monomorphisms in CC. (Often, MM will be the right class in an orthogonal factorization system.) We say that CC is MM-complete if it admits all (even large) intersections of MM-subobjects. This means that it admits all (even large) wide pullbacks of families of MM-morphisms, and such pullbacks are again in MM. (If MM is the right class of an OFS, then any intersection of MM-morphisms which exists is automatically in MM.)

If MM is the class of all monomorphisms, we may say mono-complete for MM-complete.

Dually, if EE is a class of epimorphisms, we say CC is EE-cocomplete if it admits all cointersection?s of EE-morphisms, and epi-cocomplete if EE is the class of all epimorphisms.

Examples

  • If CC is MM-well-powered, then no large limits are required in the definition of MM-completeness. Therefore, if CC is well-powered and complete, it is MM-complete whenever MM is the right class in an OFS. Dually, if CC is well-copowered and cocomplete, it is EE-cocomplete whenever EE is the left class in an OFS.

  • For similar reasons, the category FinSet is mono-complete and epi-cocomplete—although it is not complete or cocomplete, it is finitely complete and cocomplete, and its subobject lattices and quotient lattices are likewise essentially finite?.

  • If CC is a topological concrete category over a category DD which is mono-complete or epi-complete, then CC is also mono-complete or epi-complete. For the faithful forgetful functor U:CDU\colon C\to D preserves and reflects monos and epis, and so the initial CC-structure on an intersection of underlying monos in DD gives an intersection in CC and the final CC-structure on a cointersection? of underlying epis in DD gives a cointersection in CC.

Construction of factorization systems

MM-completeness is useful for constructing orthogonal factorization systems. The following is Lemma 3.1 in CHK.

Theorem

Let MM be a class of maps in a category CC, and assume that

  1. MM consists of monomorphisms,
  2. MM is closed under composition,
  3. all pullbacks of MM-morphisms exist in CC and are again in MM, and
  4. CC is MM-complete.

Then there is an orthogonal factorization system (E,M)(E,M), with E= ME = {}^\perp M.

Proof

Given f:ABf\colon A\to B, let mm be the intersection of all MM-morphisms n:XBn\colon X \to B through which ff factors. Then by the universal property of this intersection, we have f=mef = m e for some ee; thus it suffices to show eEe\in E. Suppose given a commutative square

A g Z e p Y h W\array{ A & \overset{g}{\to} & Z \\ ^e \downarrow & & \downarrow^p \\ Y & \underset{h}{\to} & W}

with pMp\in M. By pulling pp back to YY (since pullbacks of MM-morphisms exist), we may assume that Y=WY=W and hh is the identity. But now the composite mpm p is an MM-morphism through which ff factors, so by definition, mm factors through it. Thus pp is an isomorphism and so the lifting problem can be solved.

In fact, it is easy to see that the same proof constructs a factorization structure for sinks.

Note that if MM is already part of a prefactorization system, then any composite, pullback, or intersection of MM-morphisms which exists is automatically also in MM, since M=E M = E^\perp.

Corollary

Let (E,M)(E,M) be a prefactorization system on a category CC, and assume that

  1. MM consists of monomorphisms,
  2. All pullbacks of MM-morphisms exist in CC, and
  3. CC is MM-complete.

Then (E,M)(E,M) is an orthogonal factorization system.

The following is a slight generalization of Theorem 3.3 of CHK. There it is stated only for the case M=M= strong monomorphisms, in which case a finitely complete and MM-complete category is called finitely well-complete.

Theorem

Let S:AC:TS : A \rightleftarrows C : T be an adjunction, and assume that AA is finitely complete and MM-complete for some OFS (E,M)(E,M), where MM consists of monomorphisms and contains the split monics. Define E SE_S to be the class of maps inverted by SS, and M S=(E S) M_S = (E_S)^\perp; then (E S,M S)(E_S,M_S) is an OFS on AA.

Proof

First of all, since M SM_S belongs to a prefactorization system, it is closed under composites, pullbacks, and any intersections which exist. Therefore, if we define MMM SM' \coloneqq M \cap M_S, then MM' satisfies the hypotheses of Theorem 1, and so we have an OFS (E,M)(E',M').

Moreover, it is useful to notice that E S= T(hom(C))E_S={}^\perp T(\hom(C)): this is an easy consequence of the fact that if STS\dashv T, then SabaTbS a\perp b\iff a\perp T b, since fTuSfuf\perp T u\iff S f\perp u for each uhom(C)u\in\hom(C), so that SfS f is an isomorphism.

Now suppose given f:ABf\colon A\to B; we want to construct an (E S,M S)(E_S,M_S)-factorization. Let vv be the pullback of TSfT S f along the unit η B:BTSB\eta_B \colon B \to T S B. The naturality square for η\eta at ff shows that ff factors through vv, say f=vwf = v w.

A w P u TSA v TSf B η B TSB\begin{array}{ccccc} A & \\ &\overset{w}\searrow\\ && P &\overset{u}\to& T S A \\ && {}^v\downarrow && \downarrow^{T S f}\\ && B &\underset{\eta_B}\to& T S B \end{array}

Since TSfT S f is evidently in M S=( T(hom(C))) T(hom(C))M_S=({}^\perp T(\hom(C)))^\perp\supseteq T(\hom (C)), so is vv; thus it suffices to find an (E S,M S)(E_S,M_S)-factorization of ww.

Let w=ngw = n g be the (E,M)(E',M')-factorization of ww. Since MM SM' \subseteq M_S, it suffices to show that gE Sg\in E_S. Note also that since ww is a first factor of the unit η A\eta_A, by passing to adjuncts we find that SwS w is split monic: in the former diagram we have uw=η Au w=\eta_A, so that the adjunct ϵ SASuSnSg=1\epsilon_{S A} \cdot S u\cdot S n\cdot S g=1, hence also SgS g is a split monic. But TSgT S g is then also split monic, hence belongs to MM and thus also to MM' (since it obviously belong to M S=( T(hom(C))) T(hom(C))M_S=({}^\perp T(\hom(C)))^\perp\supseteq T(\hom (C))). Therefore, since gEg\in E', the naturality square for η\eta at gg contains a lift: there is an α:XTSA\alpha\colon X\to T S A such that in the diagram

A η A TSA g TSg X η X TSX\begin{array}{ccc} A & \overset{\eta_A}\to & T S A \\ {}^g\downarrow && \downarrow^{T S g} \\ X &\overset{\eta_X}\to& T S X \end{array}

αg=η A\alpha\cdot g=\eta_A and TSgα=η XT S g \cdot \alpha=\eta_X. Passing to adjuncts again, we find that SgS g is also split epic, since we can consider the diagram

SA Sη A STSA ϵ SA SA Sg aaa STSg Sg SX Sη X STSX ϵ SX SX\begin{array}{ccccc} S A & \overset{S\eta_A}\longrightarrow & S T S A &\overset{\epsilon_{S A}}\longrightarrow & S A\\ {}^{S g}\downarrow && \phantom{aaa}\downarrow_{S T S g} &&\downarrow^{S g}\\ S X &\underset{S\eta_X}\longrightarrow & S T S X &\underset{\epsilon_{S X}}\longrightarrow & S X \end{array}

and the commutativity

Sgϵ SASα=ϵ SXSTSgSα=ϵ SXSη X=1 S g \cdot \epsilon_{S A} \cdot S\alpha = \epsilon_{S X} \cdot S T S g \cdot S\alpha = \epsilon_{S X}\cdot S\eta_X = 1

Hence SgS g is an isomorphism; thus gE Sg\in E_S as desired.

This is useful in the construction of reflective factorization systems.

References

  • Cassidy and Hébert and Kelly, “Reflective subcategories, localizations, and factorization systems”. J. Austral. Math Soc. (Series A) 38 (1985), 287–329

Revised on April 20, 2014 22:08:48 by Fosco Loregian (79.1.178.57)