Brian Hepler, Ph.D.

Sheaves on my mind

So 5am rolls around, and I’m still not asleep. Of course. It’s not like I have to be up in 2.5 hours or anything. My brain is conspiring against me.

Whilst rolling around in bed NOT SLEEPING, my thoughts turned to sheaves: just what are they? I posted a bit ago (longer than I like) about sheaves, but it was kind of a lame post, caught up in the definition of the objects. This is a way of remedying that, and a bit more of some dry definitions (you really can’t escape them, unfortunately).

Put simply, I think of a sheaf (of sets) as a means of “observing” or “exploring” a space $X$ . A topological space is a mathematical object, independent of our puny reality and ordinary means of observation. Think of a scientist walking around “in/ on” $X$ . Let’s call him monsieur Faisceau. Monsieur Faisceau walks around $X$ , recording what he sees in every open set $U \subseteq X$ . He starts noticing patterns in his observations…like $X$ happens to look red at every open set. Being a good scientist and having dutifully recorded all of his observations, he sees that the observation of $X$ looking red in every open set agrees “on overlaps”. That is, whenever he compares the results of his observations, they agree. He then deduces (and maybe even publishes a paper) that “ $X$ looks red”.

This is obviously a vast simplification of what scientists can do (i.e. serve us better than acting as locally constant functions). Monsieur Faisceau could be taking temperature readings on every open set of $X$ , i.e. compiling a family of continuous functions (or smooth, depending on what he wants/ if $X$ is a smooth manifold) $T_i : U \to \mathbb{R}$ that give the give the temperature of $U$ at every point. Suppose, for the sake of the argument, that $\cup U_i = X$ . Again, he notices that his results agree on overlaps. In mathspeak, $T_i |_{U_i \cap U_j} = T_j |_{U_i \cap U_j}$ , for any $i,j$ (in whatever indexing set we’re dealing with). Being a good lazy intellectual, he doesn’t like keeping the data of each function $T_i$ , so he instead defines a function $T : X \to \mathbb{R}$ via $T(x) = T_i(x)$ if $x \in U_i$ . This is well defined, since $T_i(x) = T_j(x)$ if $x \in U_i$ and $x \in U_j$ . He then, of course, publishes a paper about his findings.

There’s then a theorem that says specifying a family of sheaves on an open cover $\{U_i\}_{i \in I}$ (i.e. $F_i$ is a sheaf on $U_i$ for each $i$ ) subject to some nice “gluing” conditions amounts to specifying a unique sheaf defined on all of $X$ (see Hartshorne’s Algebraic Geometry, ex. 22 in section 2.1). This can then be thought of as a collection of scientists who work together observing $X$ . They each work on their own open set, and talk to each other whenever they work on overlapping interests. That is, scientist $F_i$ specializes in studying the open set $U_i$ , and scientist $F_j$ specializes in studying $U_j$ . Thus they both specialize in studying $U_i \cap U_j$ as well, and any observation made by one scientist on $U_i \cap U_j$ is communicated “isomorphically” to the other scientist about that area. Obviously, any scientist communicates with himself about his work without having to do any extra work. Papers published about global results then amount to “joint work” by the scientists.

Another fundamental process to understand is “sheafification” (an AWESOME word, btw). To understand this, we must understand why presheaves are so lacking. These can be thought of as people with really poor memory who are walking around in $X$ . They’re just as smart as a scientist, and thus see all the same things when they look at each open set $U \subseteq X$ . They just have a hard time remembering their results/ observations, and can’t infer global data by “gluing” local data. This puts the “forgetful functor $\textbf{Sh}(X) \to \textbf{Psh}(X)$ (now aptly named) in a new perspective: it takes a scientist and hits him in the head, rendering him unable to remember anything he sees.

Hopefully this makes some sense to other people. I’ll return with sheafification soon.

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 0 and 1 (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.

(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 $X$ , Sh(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 C 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 D 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 I 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.

Universal Properties IV: Cones and a first look at Limits

Sorry for the delay since my last post (to those who actually read this…)

So I stumbled across a really nice way of looking at universal properties that is equivalent to specifying them as a terminal (or initial) object in a suitable comma category, but it has a much nicer “intuitive feel.”

Cones (and co-Cones)

Let C be some category. Let $\{d_i\}_{i \in I}$ be a collection of objects of C indexed by some set $I$ , and let $\{g_{ij}: d_i \to d_j\}_{i,j \in I}$ be a collection of morphisms in C (we do not require that there is a morphism for any two $i,j$ , we only allow for the possibility of there being one). We call this collection of objects and morphisms a diagram in C.

Let $D$ be a diagram in C. A cone on $D$ is a C-object $c$ and collection of morphisms $f_i : c \to d_i$ , such that for all $i,j \in I$ , $f_j \circ g_{ij} = f_i$ . If we have two such “cones,” $c$ and $c'$ on this diagram, we say $h: c' \to c$ such that the appropriate diagram commutes (try to figure it out! It’s a good idea to get an intuitive feel for these things). It follows pretty quickly then that we have a category of cones over $D$ , call it $\textbf{C}_D$ . A limit of the diagram $D$ is then just a terminal object in $\textbf{C}_D$ .

Similarly, a co-cone is just a cone with all the arrows reversed (i.e. an object $c$ together with maps $f_i : d_i \to c$ for each $i$ ). A co-limit of such a co-cone is an initial object in the category of co-cones over the appropriate diagram.

Examples………………………

Say we’re working in the category R-Mod for some ring with unity $R$ .

Pullbacks: Let $A,B,C$ be three $R$ -modules, and consider the diagram $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ . The limit of this diagram is then just the ordinary pullback (or fiber product), the module $A \times_C B = \{ (a,b) | f(a) = g(b)\}$
Products: Let $A,B$ be $R-$ modules. Then the limit of the “diagram” consisting of just $A$ and $B$ and no morphisms between them is the product $A \times B$ .
co-Products: consider the same diagram used for the product. The co-limit of this is then the co-product of $A$ and $B$ , $A \oplus B$ .
Terminal objects: are just the limit of the “empty diagram.”
Initial objects: are just the co-limit of the “empty diagram.”

and so on.

Having “Finite (co-) Limits”

Notice that all the above limits and co-limits were taken over a “finite” diagram. That is, there were only finitely many objects and morphisms in each diagram. Such (co-)limits are referred to as “finite” (co-)limits (I wonder why…). It turns out that it’s a highly desirable property for a category to “have all (finite) (co-)limits.” It took me a lonnnnnggg time to grok this.

Remember when we first started talking about universal properties? When you specify that an object satisfies a certain universal property, it is unique up to unique isomorphism if it actually exists. These objects don’t have to exist. The property of having, say, all finite limits or co-limits means that whenever you specify a universal property for an object with a finite diagram, that object actually exists. It’s a theorem (that I don’t currently know how to prove) that a category C with a terminal object and all pullbacks has all finite limits. Dually, if C has an initial object and all pushouts, it has all finite co-limits. Is this so unreasonable? Look at the list of examples again.

Back? Good. Suppose we’ve got all pullbacks and a terminal object, call it 1. Then the product is just the limit of the pullback diagram $A \to \textbf{1} \leftarrow B$ . The equalizer of two parallel maps $f,g : A \to B$ is the pullback of $A \overset{f}{\to} B \overset{g}{\leftarrow} A$ . The kernel of $f: A \to B$ (we’re still working with $R-$ modules) is the pullback of $A \overset{f}{\to} B \overset{0}{\leftarrow} A$ . Get the picture?

A pretty good thing to try here would be to find out how these are equivalent to universal properties. So go try that. 🙂

Universal Properties III: Bringing it all together

So last time I mentioned that we could describe the kernel of a group homomorphism via a universal property. For example, let $\varphi: G \to H$ be a group homomorphism, and let D be the full subcategory of Grp consisting of all groups $K$ such that for any group homomorphism $f: K \to G$ we have $\varphi \circ f = 0_H$ is the zero homomorphism from $K$ to $H$ . Good. Now if A is the category with one element, and $S : \textbf{A} \to \textbf{Grp}$ is a functor with $S(*) = G$ , $U: \textbf{D} \to \textbf{Grp}$ the inclusion functor, then the terminal object in the comma category $(U \downarrow G)$ is the kernel of $\varphi$ ! Simple.

If you can understand all that, then it shouldn’t be too hard to see that the cokernel has a similar description. Cokernels are a bit more annoying to deal with when we’re just talking about ordinary groups (the image of a homomorphism is not necessarily a normal subgroup of the target group). Let’s then just restrict our attention to Ab, where things are much nicer. Ordinarily, we would define the cokernel of a homomorphism $\varphi: G \to H$ as the quotient group $H/ Im(\varphi)$ . As before, let D be the full subcategory of Ab such that for any group $C$ in D, we have that for any homomorphism $f: H \to C$ , the composition $f \circ \varphi = 0_C$ , the homomorphism that sends everything in $G$ to $0$ in $C$ . Let $U : \textbf{D} \to \textbf{Ab}$ be the inclusion functor, and $S: \textbf{A} \to \textbf{Ab}$ the functor from the category with one element with value $S(*) = H$ . Then the cokernel is the initial object in the comma category $(H \downarrow U)$ .

If we’re in a “nice” category, like the category Ab of abelian groups. Then the image of a group homomorphism $\varphi: G \to H$ has a particularly cool “set free” definition. Recall that when we defined the cokernel of a homomorphism, the object is actually a pair $(C,f)$ , where $f: H \to C$ is a homomorphism and $C$ is the object that we normally think of as a cokernel. Since $f$ is a group homomorphism, we can ask “what is the kernel of $f$ ?” It’s the image of $f$ ! You should check this for yourself, but its pretty mechanical if you know the definition of what the cokernel is and have been following along. This property is often expressed as “The image is the kernel of the cokernel of $\varphi$ .”

Of course, there is a much more “involved” definition of the image of a morphism for when we don’t have things like kernels or cokernels to play with. I don’t really like it as much, but it follows the same basic idea of being an initial object in a certain comma category.

If we’re back in Ab, and have the same group homomorphism $\varphi: G \to H$ , what would the “cokernel of the kernel” be? What would it mean for the “cokernel of the kernel” and the “kernel of the cokernel” of $\varphi$ to be isomorphic, and how does this relate to the first isomorphism theorem for groups?

Universal Properties II: Comma Categories

In my last post, I spent a good bit trying to get you interested in looking at universal properties. Hopefully, you’ve read that post, and are still sufficiently interested to continue, because it’s only going to get harder before we see the light.

We left off at defining these special objects in some category C called “initial” and “terminal” objects. Go read the previous post now if you need a refresher on what they are.

Back now? Good. The next object of study is called a comma category, a category that, in a sense, examines a particular category by looking at certain kinds of morphisms in it. Take that with a grain of salt, please. Formally, if we have three categories A,B, C, and functors $S: \textbf{A} \to \textbf{C}$ and $T: \textbf{B} \to \textbf{C}$ , the comma category $(S \downarrow T)$ is the category where

The objects are triples $(\alpha, \beta, f)$ with $\alpha \in \text{Ob}(\textbf{A})$ , $\beta \in \text{Ob}(\textbf{B})$ , and $f: S(\alpha) \to T(\beta)$ is a morphism in C.
The morphisms are pairs $(g,h): (\alpha, \beta, f) \to (\alpha',\beta',f')$ with $g: \alpha \to \alpha'$ in A and $h: \beta \to \beta'$ in B, such that $T(h) \circ f = f' \circ S(g)$ .
Composition of morphisms is done component-wise. Thus if $(g,h) : (\alpha,\beta,f) \to (\alpha',\beta',f')$ and $(g',h') : (\alpha',\beta',f') \to (\alpha'',\beta'',f'')$ , then $(g',h') \circ (g,h) := (g' \circ g, h' \circ h)$ .

Now, as far as I’ve seen, one most often comes across comma category theory through a select few vast simplifications.

The Slice Category

Let A = 1, the category with only one object (usually denoted $*$ and one morphism, the identity map. Then a functor from 1 to any other category C simply “picks out” an object of C. That is, $S : \textbf{1} \to \textbf{C}$ is uniquely determined by the image of $*$ , say $S(*) = X$ .

In the definition of a comma category, we need three categories. Let C be any category, and suppose B = C. let $Id_C: \textbf{C} \to \textbf{C}$ be the indentity functor. Then our three categories are 1,C, and C. The comma category $(S \downarrow Id_C)$ is most often written as X/ C, and is called the slice category and can be seen as the category of “objects of C ‘under’ X.” Specifically:

The objects of X/C are triples $(*, \beta, f)$ , with $f: S(*) \to T(\beta)$ , with $\beta$ an object of T. The objects are usually simplified to $(\beta,f)$ , since $*$ is the only object in 1.
The morphisms are $F: (\beta, f) \to (\gamma, g)$ , with $F: \beta \to \gamma$ a morphism in C such that $F \circ f = g$ .
Composition is defined in the only natural way (it’s a trivial exercise to check).

One can, of course, define the co-slice category which is the same as the slice category, except the directions of all the arrows are reversed. These are the “objects ‘over’ X.”

Remember the category of “pointed topological spaces” from before? It turns out that this is actually a comma category! Let $S : \textbf{1} \to \textbf{Top}$ be the functor with value $S(*) = \{pt\}=p$ any singleton set $p$ . Then the category $p/ \textbf{Top}$ has objects $(X, f)$ with $X$ a topological space and $f: p \to X$ an inclusion of a point into $X$ . We can then make the obvious identification $(X,f) \cong (X, f(p))$ . The morphisms here are precisely the basepoint preserving ones.

Here’s another cool example: Let C be a category with an initial object $x$ . Then I want to show that $x/ \textbf{C}$ is “isomorphic as a category” to C. I haven’t yet defined what that means, sorry. It just means that there are functors $U: x/ \textbf{C} \to \textbf{C}$ and $T: \textbf{C} \to x/ \textbf{C}$ such that $T \circ U = Id_{x/\textbf{C}}$ and $U \circ T= Id_{\textbf{C}}$ . Anyway, let $U: x/\textbf{C} \to \textbf{C}$ be the functor that sends each pair $(\beta, f)$ to the C-object $\beta$ , and each morphism $h: (\beta, f) \to (\beta',f')$ to the map $h: \beta \to \beta'$ . This is another instance of a “forgetful functor,” by the way.

Since $x$ is an initial object, for any other C-object $\beta$ there is one and only one morphism $f: x \to \beta$ . With this in mind, we define $T: \textbf{C} \to x/\textbf{C}$ via $T(\beta) = (\beta,f: x \to \beta)$ , and for any morphism $g: \beta \to \beta'$ , $T(g) = g : (\beta, f) \to (\beta',f')$ . It’s then trivial to check that these functors compose to get the identity functors on both sides. Therefore they are isomorphic.

Obviously the dual statement holds for categories C with a terminal object $y$ and the co-slice category $\textbf{C}/y$. (note: I owe these above cool examples to this fantastic post: http://drexel28.wordpress.com/2012/01/10/comma-categories-pt-i/. You should really visit this guy’s blog.)

Almost Slice Categories

Let’s step up the abstraction a bit. Let C and D be two categories, and let $U: \textbf{D} \to \textbf{C}$ be a functor. Let $S: \textbf{1} \to \textbf{C}$ be the functor that picks out a C-object $X$ . Then the comma category $(S \downarrow U)$ , written most often as $(X \downarrow U)$ , is the category of “morphisms from $X$ to $U$ ” (so sayeth the wiki page). You can think of these as (almost) slice categories, in that $X$ is now “over” objects of the form $U(\beta)$ for $\beta$ an object in D instead of just all C-objects.

Remember the example of the kernel of a group homomorphism $\varphi: G \to H$ ? We can now almost talk about that whole business of “the largest group that is killed off by $\varphi$ .” let D be the subcategory of Grp whose objects are groups $K$ such that for any group homomorphism $f: K \to G$ , the composition $\varphi \circ f$ is the zero map to $H$ . The morphisms are simply those induced by the parent category Grp.

Then if we let $S: \textbf{1} \to \textbf{Grp}$ pick out $G$ , and $U : \textbf{D} \to \textbf{Grp}$ be the functor that sends each object and morphism of D to itself, then $(G \downarrow U)$ is the category that simply “pairs off” groups $K$ and morphisms $i_K : K \to G$ such that $\varphi \circ i_K = 0_H$ .

What would a terminal object be in $(G \downarrow U)$ ? 🙂 Try to find it!

Universal Properties: a Prelude

So I want to take some time to talk about universal properties. I personally think they’re awesome because if you look hard enough, you start to see them everywhere in mathematics. Especially in abstract algebra and algebraic geometry. They admit a fairly intuitive explanation, but the actual details of their definition require a lot of work. A lot.

Universal properties are used to define certain “special” objects in a category. That is, a universal property, in a sense, picks out the “best possible” object in a category that satisfies a certain property. Any other object that satisfies this property then has a morphism to/from (depending on the type of property) this “best possible” object. The nicest thing is then that an object satisfying a universal property is unique up to unique isomorphism. Yes, there are two “uniques” there.

For example (still working intuitively), let’s look at the kernel of a group homomorphism. If $\varphi: G \to H$ is a group homomorphism, then we usually define the kernel to be the normal subgroup $\text{Ker}\varphi = \{ g \in G | \varphi(g) = 0\}$ of $G$ . Easy enough. This also relies on the fact that groups are also sets. Another way of looking at the kernel: It is the largest group $K$ that is “killed off” by $\varphi$ , i.e. if $i_K : K \to G$ is a group homomorphism, then $\varphi \circ i_k = 0_H$ , the map that sends everything in $K$ to 0 in $H$ . The same idea holds for the cokernel of $\varphi$ , and tons of other special objects that you’ve undoubtedly run into before.

The technical details of the universal property are in the “largest group such that…” part. Here, this means that for any other group $K'$ and homomorphism $i_{K'} : K' \to G$ such that $\varphi \circ i_{K'} = 0_H$ , there is a unique group $K$ , homomorphism $i_K : K \to G$ , and homomorphism $g: K' \to K$ such that $i_{K'} = i_K \circ g$ .

Yes, that is a bit wordy, it’s not just you. The point of this post (and probably the next as well), is to unravel that mess of words into something tractable. The first step to understanding these properties is to understand initial and terminal objects.

Initial and Terminal Objects

These aren’t too bad. Let C be some category. An initial object in C is an object $a$ such that for any other object $b$ , there is a unique morphism $a \to b$ . A simple example to keep in mind is the empty set in the category Set. Since the empty set is a subset of every set, it admits a unique map into any set (there’s only one way to send nothing to nothing!).

A terminal object in C is an object $a$ such that for any other object $b$ , there is a unique morphism $b \to a$ . Again, a simple example is any singleton set in the category of sets. To see this, if $X$ is any set, there is a unique map that sends everything everything in $X$ to the singleton set. I’m lying a tiny bit here (in what the “universal ” object is), and I’ll tell you later what it was. Is “the” singleton set unique? Of course not. It is, however, unique up to a unique isomorphism.

The lesson here is that initial objects capture the idea of “the most efficient” or “the smallest such that” and that terminal objects capture the idea of “the largest such that.” The key, however, is to define them in the right category.