Presheaves of Sets are (finitely) Bi-Complete

As the title says, I want to show that for any topological space X, the category of set-valued presheaves PSh(X) on X has all finite limits and co-limits.  

 

First, PSh(X) has both initial and terminal objects.  With a bit of thought, these are (obviously) the constant functors  and (resp.) where, for all open subsets U of X we have \textbf{0}(U) = \emptyset and \textbf{1}(U) = \{ *\} and the morphisms for 0 and 1 are just the identities.  

Second, we need to show that PSh(X) has a pullback for every diagram A \overset{\varphi}{\to} C \overset{\psi}{\leftarrow} B and a pushout for every diagram A \overset{\alpha}{\leftarrow} C \overset{\beta}{\to} B.  

 

Indeed, let A \overset{\varphi}{\to} C \overset{\psi}{\leftarrow} B be a diagram of presheaves.  I claim that the presheaf defined via A \times_C B(U) = A(U) \times_{C(U)} B(U) for all U \subseteq X open.  Basically, we just define the pullback “element-wise” on the source category.  The morphisms are a bit tricky though.  Indeed, given a map i: V \hookrightarrow U of open sets, how do we define the “restriction” map A \times_C B(i) : A \times_C B(U) \to A \times_C B(V)?  One should never diagram chase in public, but let me assure you that it really ends up just being the map guaranteed by the universal property of the pullback.  

 

Image

 

(picture for clarity).  It’s a bit clumsy, but one gets a pushout in the same manner, defining it element-wise as a the quotient A(U) \sqcup B(U)/ \thicksim, where a \thicksim b iff there exists a c \in C(U) such that a = \alpha(c) and b = \beta(c).  This sometimes denoted as A \sqcup_C B.  

 

Since PSh(X) has a terminal object and pullbacks, it has a finite limits.  On the other hand, it has an initial object and pushouts, so it has all finite co-limits.  Easy.  

 

It turns out that the category of sheaves on XSh(X) is finitely bi-complete as well, but showing it for PSh(X) is really easy so I decided to just do that one.  
 
Next time, I want to look at the following case:  If there is pair of adjoint functors F: \textbf{C} \to \textbf{D} and G : \textbf{D} \to \textbf{C} with F \dashv G, do we have a pair of adjoint functors between sheaves with values in C and sheaves with values in D on some topological space X?  I’ve only looked at the case where the adjunction is the forgetful functor and free abelian group functor and the case of presheaves, where this statement does in fact hold.  Until next time.  🙂

What are Sheaves, and why should I care?

For anyone who has done a bit of work in modern geometry (primarily the notion of a (smooth) manifold), we want objects to be “locally” trivial, or easy to study.  The global structure might be this crazy awesome geometric shape, but locally it’s going to look like boring old \mathbb{R}^n or something like that.  How much it’s supposed to “look like” \mathbb{R}^n depends on what you want to study.  For example, a smooth manifold M is a set together with an atlas of “smooth” charts, such that for any point p \in M, there is an open neighborhood U of p that is diffeomorphic to an open subset of \mathbb{R}^n.

The idea is that although the global structure of some object might be hard to study, local behavior should be easy.  Think of looking at say… a torus (doughnut).  For any point on the torus, if you look close enough, it looks pretty much flat.  Even though the global shape is decidedly not flat.

Think now of something like a smooth function on a smooth manifold M, say f: M \to \mathbb{R}.  We don’t really have to define f everywhere, we just have to know that f behaves smoothly with respect to the atlas of M.  That is, for any point p \in M, there is a neighborhood U \ni p, and chart \varphi: \mathbb{R}^n \to U, such that f \circ \varphi is a smooth, real-valued function.

Most people don’t go this deep down the rabbit hole, but there is a unifying principle behind extending local data to global data.  This is given by the notion of a “sheaf.”  Most of the time, people first encounter these things in an algebraic geometry or algebraic topology class, in the context of “cohomology with local coefficients” which are usually abelian groups or something similar.

First, presheaves (of abelian groups) on a topological space X.  A presheaf F on X constists of the data of:

  • For every open set U \subseteq X, an abelian group F(U).
  • For every inclusion of open sets V \hookrightarrow U, a “restriction” homomorphism \rho_{UV} : F(U) \to F(V).
  • F(\emptyset) = 0, the trivial group.

A sheaf is all this, subject to a nice “gluing” condition.  That is:

  • For every open set U \subseteq X and open cover \{U_i\}_{i \in I} of U, if s \in F(U) is such that s|_{U_i} = 0 for all i \in I, then s = 0 \in F(U).
  • For every open set U \subseteq X and open cover \{U_i\}_{i \in I} of U, if s_i \in F(U_i) are sections such that for all i,j \in I we have s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}, then there exists a section s \in F(U) such that s|_{U_i} = s_i for all i \in I.

Note here that the former condition implies that the section s \in F(U) in the latter condition is unique.

That was a bit of a mouthful.  So complicated a definition.  I don’t really like this way of defining it, but it’s okay.

Let’s start again.  Let X be a topological space, and make the category X whose objects are the open sets of X and morphisms are those induced by the obvious poset structure.  Then a presheaf F of abelian groups is just a functor F: \textbf{X}^{op} \to \textbf{Ab}.  Simple!

F is a sheaf if, for every open set U and cover \{U_i\}_{i \in I},

F(U) \to \prod_{i \in I} F(U_i) \rightrightarrows \prod_{(i,j) \in I \times I} F(U_i \cap U_j)

is an equalizer diagram.

So now we have sheaves.  What are the maps?  Well, the sheaves are just functors, so the obvious choice is that they’re natural transformations of functors.  Hence, we have a category of sheaves!  Denote this by \textbf{Sh}_{\textbf{C}}(X) if the sheaves have values in a category C.  

Why should I care?

\textbf{Sh}_{\textbf{C}}(X) tends to retain a lot of the structure of the category C.  The most encountered example is that \textbf{Sh}_{\textbf{C}}(X) is an abelian category whenever C is (I’ll revisit these neat abelian categories in a later post.  They basically “behave like abelian groups” enough for us to do homological algebra.).   The example I want to pursue is that \textbf{Sh}_{\textbf{C}}(X) is a topos whenever C is (I’ll DEFINITELY do a post on these things later).

They come up everywhere in geometry.  Smooth function on a smooth manifold?  Sheaf.  Continous functions on a topological space?  Sheaf.  Measurable functions on a a measure space?  Sheaf.   Regular functions on a variety? Sheaf.

I’m still learning this stuff, and I’m continually amazed at how pervasive the idea is.  Turns out that you can also define sheaves on a category by giving a the category a certain “topology” called a “Grothendieck topology.”

Wherever there is the study of local vs. global behavior, there is sheaf theory.  Even in physics now, where one studies the structure of “quantum events” via covers of boolean reference frames, or where “locality and contextuality” is the cohomology of sheaves.  So. Fucking. Cool.

Until next time.

Adjoint Functors

So last post I gave a (hurried) description of why adjoint pairs of functors are linked to this notion of “similar structure” between two categories.  In this post, I want to relate adjunctions to universal properties, and ultimately why we like adjoint pairs so much.  

 

Say we’re working with the “free group on a set” functor, F: \textbf{Set} \to \textbf{Grp}.  We know that the free group satisfies a really nice universal property: Given any (set) function f: X \to G, where G is a group, there exists a unique group homomorphism \overline{f} : F(X) \to G extending f.   If you recall the notion of the “forgetful” functor U: \textbf{Grp} \to \textbf{Set} that takes a group and gives its underlying set, the universal property of the free group on X states that there is a bijection

\text{Hom}_{\textbf{Grp}}(F(X),G) \cong \text{Hom}_{\textbf{Set}}(X, U(G))

i.e. F \dashv U is an adjoint pair (of course, one should check the naturality of this bijection).  Hence the universal property here is really just this adjunction in hiding.  This idea is generalized often by saying that a category C has “free objects” if a suitably defined forgetful functor from C possesses a left adjoint.  

It’s for this reason that I like to think of a pair of adjoint functors F \dashv G as a sort of “globally defined” universal property, as expressed by the naturality of the bijection between hom-sets.  Indeed, we have that a functor G: \textbf{D} \to \textbf{C} has a left adjoint provided that we can find, for each object A of an object “FA” of D and a morphism \eta_A : A \to G(``FA") that is universal among morphisms form A to the image of G, i.e. for all f: A \to GB, there is a unique \overline{f} : ``FA" \to B satisfying f = G(\overline{f}) \circ \eta_A.  I use the quotations here only to emphasize the fact that we really want the object “FA” to be “the component of some suitable functor F at A.”  

I don’t know about you, but this sound a lot like: “For any functor S: \textbf{1} \to \textbf{C}, the comma category (S(*) \downarrow G) has an initial object.” (Remember, a functor S: \textbf{1} \to \textbf{C} simply selects an element of C.)  That is, this sounds A LOT like how we initially (haha, pun) constructed universal properties from comma categories.  The statement of the above paragraph then says that if we can find such an initial object for any choice of functor S: \textbf{1} \to \textbf{C}, then G has a left adjoint.  Pretty neat, right?

 

 

Isomorphism, Equivalence, and Adjunction

Isomorphism

As mathematicians are wont to do, whenever we have a collection of algebraic “objects,” we want to know how to “relate” them.  In the case of categories, we saw earlier that maps called functors are what we want to examine.

The next step after defining these structure preserving maps is to wonder what it means for two objects to be “essentially the same.”  In Set, bijections do this.  In Grp, group isomorphisms do this.  In Top, homeomorphisms do this.  Etc, etc.  If we were to try an analogous procedure for categories, we would say two categories C and D are isomorphic if there exist two functors: F: \textbf{C} \to \textbf{D} and G: \textbf{D} \to \textbf{C} such that F \circ G = id_{\textbf{D}} and G \circ F = id_{\textbf{C}}.  Saying that two categories are isomorphic means that they are, for all intents and purposes, the same (maybe they differ in notation or something).

Equivalence

As it happens, this tends to be too restrictive a condition (i.e. categories that behave more or less the same tend to not actually be isomorphic).  What if, instead of requiring that FG = id_\textbf{D} and GF = id_\textbf{C}, we require that these functors are naturally isomorphic to the appropriate identities?  We would then say that C and  are “equivalent” categories.

The first time I ever saw this phenomenon was in algebraic geometry, where one sees that the category of finitely generated reduced k-algebras is equivalent to the category of (affine) algebraic varieties over k (here k is a field).  Later on we saw that the category of commutative rings with unity is equivalent to the category of affine schemes equipped with their structure sheaves.  Another cool example is that the category of quasi-coherent sheaves over an affine scheme \text{Spec}(R) is equivalent to the category of R-modules.

For a simpler example, take any poset (X,\leq) and consider it as a category.  Then reversing the direction of the arrows gives an equivalent category (X, \geq).  Obviously, this works for any category and its opposite category.  I just like the poset case because one can visualize it quite easily.

Adjunction

Simply put: adjunctions are ubiquitous.  It took me a long time to see that, and I’m still wading through the ramifications.  I gave a (brief) blurb about them in the last post, but let’s up the scrutiny.  We say F: \textbf{C} \to \textbf{D} and G: \textbf{D} \to \textbf{C} are an adjoint pair (written F \dashv G) if there is a natural bijection between maps f: A \to GB in C and \overline{f}: FA \to B in D.

Note that, in D, we have the map id_{FA} : FA \to FA.  The adjunction gives a unique map \eta_A: A \to GFA, and likewise we have a unique map id_{GB} : GB \to GB yielding \epsilon_B : FGB \to B.  These maps are called the “unit” and “counit” of the adjuction at A or B.  In fact, the adjunction yields a pair of natural transformations \eta: id_\textbf{C} \to GF and \epsilon: FG \to id_\textbf{D}.

That’s pretty neat.  It explicitly shows the “descending chain of equivalence” from isomorphism of categories, equivalence of categories, and adjunction of functors between categories.  The naturality of the unit and counit transformations from an adjunction F \dashv G actually implies the “bijection” criterion, so we can really just take the unit-counit thing as a starting point.

I’ll do more on this later.  I have class to go to 🙂

Limits and co-Limits: Some Cool Things

I’m not going to reiterate the definitions of limits and co-limits from the last post, so just look them up if you’re new here.  They’re not too hard.

This post is mainly just about some random cool things I’ve noticed/ “remembered” / come across whilst playing with the notions of limit and co-limit in various categories.

Thing 1

Say we’re working in R-Mod for some ring R, and let A,B be R-modules.  Taking the limit of the “discrete” diagram A,B gives us the “product diagram: ” A \overset{p_A}{\leftarrow} A \times B \overset{p_B}{\to} B.  We can then take the co-limit of this diagram, which is the quotient A \oplus B / K, where K is the module generated by elements of the form (p_A((a,b)), -p_B((a,b)) for (a,b) \in A \times B.  It then follows that the pushout is trivial.  Strange.

EDIT:  for some clarification, I want to show that for any diagram of the form A \overset{f}{\leftarrow} C \overset{g}{\to} B, then the colimit is the object A \oplus B/ K, where K is the submodule generated by the elements \{(f(c),-g(c)\}_{c \in C}.   Clearly, it does fit into the appropriate diagram.   Now let D be any other module with maps j_A: A \to D, j_B: B \to D such that j_A \circ f = j_B \circ g.  Then, via the universal property of the direct sum, there is a unique map F: A \oplus B \to D such that j_A = F \circ i_A and j_B = F \circ i_B (where i_A: A \to A \oplus B and i_B : B \to A \oplus B are the canonical inclusions).  Then we have that F \circ i_A \circ f = F \circ j_B \circ i_B, so F((f(c),0) = F(0,g(c)) for any c \in C.  Hence F((f(c),-g(c)) = 0, i.e. K \subseteq \text{ Ker} F.  Via the universal property of the quotient, there is a unique homomorphism \overline{F} : A \oplus B/ K \to D that makes the whole diagram commute.

In the case where C = A \times B and f = p_A and g = p_B, K = \langle (p_A(a,b),-p_B(a,b)) \rangle_{a \in A, b \in B} = \langle i_A(A) + i_B(B) \rangle \cong A \oplus B, so that the quotient A \oplus B/ (A \oplus B) = \{0\}.

The dual construction is (obviously) similar, where we first take the co-limit of the discrete diagram, then the limit of the “co-product diagram.”  It is also the zero module.

Thing 2

Adjunction spaces in Top, the category of topological spaces.  Let X,Y be topological spaces, A \subseteq Y (represented as a monic i: A \to Y) be a subspace.  Let f: A \to X be a continuous function.  Then the adjunction space obtained by gluing X to Y along f is just the co-limit of the diagram Y \overset{i}{\leftarrow} A \overset{f}{\to} X.

Thing 3

Let X be a topological space, which has a natural poset structure on its collection of open sets.  Formally, we turn X into a category X whose objects are the open sets of X and the morphisms are determined via \text{Hom}(V,U) \neq \{ \emptyset \} iff V \subseteq U.  Let V,U,W be elements of X such that V \subseteq W and U \subseteq W.  Then the limit of the diagram U \to W \leftarrow V is just the intersection U \cap V.

Thing 4

Limits as functors.  Turns out you can replace the notion of a “diagram in C” (where C is the category we’re looking at) with a functor \mathcal{A} : \textbf{I} \to \textbf{C}, where is a small category.  Think about it!  The limit of such a diagram is denoted by \varprojlim \mathcal{A}.

Quick note: Adjoint Pairs of Functors

Say we have two functors F: \textbf{C} \to \textbf{D} and G: \textbf{D} \to \textbf{C}.  We say F,G form an adjoint pair if, for all X \in \textbf{C}, Y \in \textbf{D}, we have a bijection \text{Hom}_{\textbf{D}}(F(X),Y) \cong \text{Hom}_{\textbf{C}}(X,G(Y)) that is natural in X and Y.  Furthermore, we say F is left adjoint to G, and similarly G is right adjoint to F.  Also, G is a right adjoint functor if it has a left adjoint, and likewise for left adjoint functors.

Thing 5

Right adjoint functors commute with Limits.  Let F: \textbf{C} \to \textbf{D} and G: \textbf{D} \to \textbf{C} be an adjoint pair of functors, and let \mathcal{A} : \textbf{I} \to \textbf{D} be a diagram.  The statement is then that

G(\varprojlim \mathcal{A}) = \varprojlim G \circ \mathcal{A}

Awesome.  The proof is actually pretty straightforward abstract nonsense, just take the definition of \varprojlim \mathcal{A} as a limiting cone, apply G, get a map G(\varprojlim \mathcal{A}) \to \varprojlim G \circ \mathcal{A}.  Then, use adjunction to get a map F (\varprojlim G \circ \mathcal{A} ) \to \mathcal{A}(I) for all objects I in I. The universal property of \varprojlim \mathcal{A} gives us a map \varprojlim G \circ \mathcal{A} \to G(\varprojlim \mathcal{A}) by applying adjunction again.  These maps are quickly seen to be inverses of each other (keep looking through universal properties and such).  A good outline of this proof can be found in Paolo Aluffi’s “Algebra: Chapter 0”

Similarly, we have that Left adjoints commute with co-limits.

Math is awesome.