Eric Auld November 25, 2019 Enriched Category Theory

The notion of a locally small category is so skeletal as to be mostly taxonomic. More structure is needed to prove nonvacuous theorems: for instance, the existence of limits, and knowing that certain functors preserve limits, or structures on the hom sets, which this note deals with. The presence of an identity morphism in each EndA paq, which we can think of as kind of a choice of basis for each element a P A , already gives us some vestige of an algebraic structure: EndA paq is a monoid. Namely, it is a set S with an associative multiplication map S ˆ S Ñ S and distinguished element e P S which satisfies s ¨ e “ e ¨ s “ s for all s.

The set of endomorphisms of an abelian group A are again a monoid, in the same way. We would somehow like to express the extra structure that function composition is bilinear with respect to addition of functions. This means that composition is a map EndAb A b End A Ñ End A. This clues us into the fact that defining a monoid object in a category shouldn’t always necessarily be using the cartesian product to model the multiplication.

What would be the analog in Cat, if I do use the cartesian product? I’d need a category C , an associative multiplication map C ˆ C Ñ C , and an identity object in C . It turns out that this is pretty much the kind of C I need to create a monoid object inside of C : I just need to allow the unitality and the associativity to be a natural isomorphism, instead of on the nose. (Mostly for philosophical reasons...how objects are defined in Set, for instance.)

Monoidal Categories

A monoidal category consists of a category V, a bifunctor ´ b ´ : V ˆ V Ñ V, a special object I P V, and three natural isomorphisms α, λ, ρ.

pV,W,Xq V bpW bXq V IbV V V bI α – λ – ρ – pV,W,Xq pV bW qbX V V V V

These must be coherent in the following ways

α V b pW b pX b Y qq pV b W q b pX b Y q V b pI b W q pV b Iq b W

1bλ ρb1 V b ppW b Xq b Y q ppV b W q b Xq b Y V b W

λ pV b pW b Xqq b Y I b I I ρ

A strict monoidal category is when all the isomorphisms are required to be identity mor- phisms.

1 Example: From Mac Lane’s text, examples of (generally weak) monoidal categories:

Example: Any category with finite products can be given a cartesian monoidal structure with unit as terminal element. Examples are pSet, ˆ, t˚uq, pAb, ‘, 0q, pSet˚, ˆ, t˚uq.

Example: Any category with finite coproducts can be given a cocartesian monoidal structure with unit as

initial element. Examples are pSet, \, ∅q, pCRing, bZ, Zq (or more generally pRCAlg, bR,Rq), pSet˚, _, t˚uq

Example: The category 2 is monoidal (strict, in fact). We think of it as false Ñ true, and the product ^ being conjunction (“and”), and the unit being true.

Example: pSet˚, ^, t˚, ˚uq is monoidal. Recall S ^ t˚, ˚u – S.

I would say the genesis of a monoidal category is defining some operation, and picking out how we would like it to interact with whatever structure is there already (e.g. I want an operation which is bilinear with respect to the addition that is already there). Of course, the whole thing is kind of circular: a strict monoidal category is a monoid object in the monoidal category pCat, ˆ, 1q.

Monoidal categories allow us to define monoid objects inside them...

A (lax) monoidal functor pV, b,Iq ÝÑpF U, d,Jq is a normal functor V Ñ U, along with data

pV,W q F pV qdF pW q J ÝÑ F pIq pV,W q F pV bW q

Note that these are not required to be isomorphisms, and note that they kind of seem to go the wrong way at first. (They sort of “undistribute” F , or take me not from “where I landed” F pIq to “where I should be” J, but the other way.) This is the direction that gives “image of monoid object is monoid object”. They are required to satisfy unitality and associativity

F pW q d J F pW q d F pIq

ρ pV,W,Xq F pV qdF pW qdF pXq (two ways of “unrolling” U – “ F pV qdF pW qdF pXqq ρV F pV,W,Xq F pV bW bXq F pW q – F pW b Iq (sim left unital)

2 Remark: A lax monoidal functor takes a monoid object to a monoid object, as follows. If F : pV, b,Iq Ñ pU, d,Jq is lax monoidal, and pV, µ, eq is a monoid object in V, then we get F µ a map F pV q d F pV q Ñ F pV b V q ÝÝÑ F pV q.

Finish

Example: The forgetful functor Ab Ñ Set is lax monoidal pAb, b, Zq Ñ pSet, ˆ, t˚uq. To give the data of a lax monoidal functor, I need to choose a map t˚u Ñ |Z|, and a natural transformation

pA,Bq |A|ˆ|B|

pA,Bq |AbB|

1 We choose to take pa, bq ÞÑ a b b, and t˚u ÝÑ Z.

|A| ˆ t˚u |A| ˆ |Z|

– Note that it takes a ring (monoid object in Ab to its multiplicative monoid.

|A| – |A bZ Z|

Notice that this sort of tells us why we have the maps going the way it does...how would we come up with a natural map |A b B| Ñ |A| ˆ |B|?

Example: More generally, it turns out that the “forgetful” functor VpI, ´q : V Ñ Set is lax monoidal from pV, b,Iq Ñ pSet, ˆ, t˚uq. To show this I need natural maps

VpI, vq ˆ VpI, wq Ñ VpI, v b wq and t˚u Ñ VpI,Iq.

For the former, I can do

fbg pf, gq ÞÑ I ÝÑ„ I b I ÝÝÝÑ v b w.

Example: Let R be a commutative ring. Then the free functor pSet, \, ∅q Ñ pRMod, ‘, 0q is strong monoidal. (Flesh this out)

Why does R have to be commutative here?

Example: Similarly, the free functor pSet, ˆ, t˚uq Ñ pRMod, b,Rq is strong monoidal when R is commutative. (I guess it takes a monoid to an R algebra?)

Example: In homological algebra, we get a natural map H‚pC‚q b H‚pD‚q Ñ H‚pC‚ b D‚q, the algebraic homology cross product. Note on the left hand side we have a tensor product of graded modules (not chain complexes), so just HjC‚ b HkD‚. This map makes homology into a lax monoidal i j`k“i functor pCh‚pRModq, b,Rq Ñ pGràpRModà , b,Rq.

(The Kunneth theorem says that when R is a PID, the homology cross product is injective, and has cokernel some stuff involving Tor.)

A strong monoidal functor is one where the maps are required to be isomorphisms. A strict monoidal functor is where they are required to be identities.

3 Closed Monoidal Categories

Let V be monoidal such that for each w P V, the functor v ÞÑ v b w has a right adjoint Vpw, ´q.

´bw Vpv b w, xq – Vpv, Vpw, xqq V K V Vpw,´q

Claim: Vp´, ´q is the action on objects of a bifunctor VˆV Ñ V, and there is an isomorphism of trifunctors Vp´ b ´, ´q – Vp´, Vp´, ´qq.

f f ˚“Vpf,xq Proof: Given w ÝÑ w1, I want to define Vpw, xq ÐÝÝÝÝÝÝ Vpw1, xq by

Vpv b w, xq – Vpv, Vpw, wqq

p1bfq˚

1 1 Vpv b w , xq – Vpv, Vpw , xqq

Let A be any category.

ϕ 1 1 Lemma: To define a map a ÝÑ a , it is sufficient to define set maps A px, aq Ñ A px, a q for all x, natural in x. (Yoneda embedding full...just take x “ a.)

Remark: Of course it is necessary also...choosing ϕ gives us those natural set maps.

Remark: In particular the map of hom objects Vpw1, xq ÝÑ Vpw, xq is defined by its effect on generalized points Vpv, Vpw1, xqq ÝÑ Vpv, Vpw, xqq.

This assignment of arrows is functorial, because p1 b fq˚ is. (I can first precompose with f, then g, or precompose with gf.)

Therefore I have defined the isomorphism Vp´ b ´, ´q – Vp´, Vp´, ´qq in a way that is natural in w. I have assumed naturality in v and x (adjunction), so I’m done.

The unit for this adjunction looks like v ÞÑ Vpw, v b wq, and the counit (“evaluation”) looks like Vpw, vq b w Ñ v.

As an example of the unit, think about the identity map A b B Ñ A b B. It should in- duce a map A Ñ rB,A b Bs which just says “tensor with whichever element you chose”.

Enriched Categories

Given a monoidal category V, a category C enriched over V (AKA a V-category) is

4 • A collection of objects c P C

• For each c, c1 P C , a “hom object” C pc, c1q in V

• Composition morphisms C pa, bq b C pb, cq ÝÑ C pa, cq

• “Identities” I ÝÑ C pc, cq (generalized point in C pc, cq) These are supposed to satisfy the following coherence axioms:

´1 I b C pb, cq λ C pb, cq C pa, bq b C pb, cq b C pc, dq ˝b1 1b˝

i b1 a C pa, cq b C pc, dq C pa, bq b C pb, dq

˝ ˝ C pa, aq b C pb, cq ˝ C pb, cq C pa, dq (and similarly for right unit) (unitality) (associativity)

Note that any V category C has a easily formed opposite V category C op.

Example: Ab can be promoted to Ab, enriched over pAb, b, Zq. Notice that map composition is bilinear and so forms a map AbpA, Bq b AbpB,Cq ÝÑ˝ AbpA, Cq.

pf ` gq; h “ fh ` gh.

Categories enriched over Ab, b, Zq are called “preadditive” categories. (They become “additive” once they admit all finite biproducts.)

Lemma: In a preadditive category A , the following conditions are equivalent: i. A has an initial object. ii. A has a terminal object. iii. A has a zero object.

Proof: If a P A is either terminal or initial, that implies the identity map Z Ñ A pa, aq is the zero map. Then we can show that a is both terminal and initial. Supposing we know it is initial, then we just need to show that A pb, aq “ x0y for all b. The square

λ´1 A pb, aq b Z A pb, aq

1b0“0

A pb, aq b A pa, aq ˝ C pb, aq

is supposed to commute, which tells us that the identity map on A pb, aq is the zero map.

5 Lemma: In a preadditive category A , the following conditions are equivalent: i. x, y P A admit a product. ii. x, y P A admit a coproduct. iii. There is an object P P A and maps

x ix px x P y iy py y

so that pxix ` pyiy “ 1P .

Proof: Suppose x ˆ y exists. Take px and py to be the projection maps, and take ix to be the map into P induced by p1, 0q, and similarly iy. Now it suffices to show that ppxix ` pyiyqpx “ px and ppxix ` pyiyqpy “ py. But ixpy “ 0 and iypy “ 1 and similarly for x.

Now do the same thing in A op for ii ùñ iii. The direction iii ùñ i is pretty easy.

Example: More generally, left R modules over can be given an enriched structure over pAb, bZ, Zq. If we try to enrich them over RMod, we immediately run into the problem that “left multiplication by r is not a left R-linear map.” (i.e. left multiplication by r does not in general commute with left multiplication by s P R.)

(This seems similar, but really is not, to “addition of a does not preserve the additive structure.”)

If R is commutative, of course it works.

The general way to put it is that if B is the category of RS bimodules, I can create a category B, with the same objects, BpM,Nq being maps between M and N which we only require to be left R-linear, and we can give BpM,Nq the structure of a left S module (yes, strangely, the right S module structure on M gives a left S module structure to the hom set) via

psϕqprmq :“ ϕprmsq. Note prsqϕ “ rpsϕq, because

pprsqϕqpmq “ ϕpmrsq and prpsϕqpmq “ psϕqpmrq “ ϕpmrsq.

If I try to require the maps to be maps of RS bimodules, I lose it again because I can’t expect actions of S on the right to commute with each other. (Multiplication by s inside RMod maps might not preserve RModS as a subset. I could also look at right S module maps, and give them a right R module structure.

The RS bimodule case subsumes all the previous ones, since a left R module is an RZ bimodule, and when R is a commutative ring, an R module is an RR bimodule (with R acting the same on the left as on the right).

The general statement is that if M is a left R module, N is an SR bimodule, and P is a left S module, then

S ModpN bR M,P q – RModpM, S ModpN,P qq and this is an isomorphism of trifunctors to Ab. I think after I learn about enriched adjunctions, I want to express this as as an Ab enriched adjunction

Nb´

RMod K SMod [N,´]

6 Example: If V is monoidal closed, then V can be promoted to V, a V category, as follows. Let the...(finish)

Example: Ordinary (locally small) categories can be seen as enriched over Set, ˆ, t˚u. A map t˚u Ñ A pa, bq is exactly an ordinary morphism, and composition is a map A pa, bq ˆ A pb, cq Ñ A pa, cq.

Example: Let S be a preordered set, i.e. let it have a reflexive and transitive relation on it. (A poset would require antisymmetry as well.) One way to see this as a category is Sps, tq is a singleton if s ď t and null otherwise. We can also view S as enriched over p2, ^, true), in the sense that it returns true iff s ď t. This way happens to encode the reflexivity and transitivity as well. To see this, recall that we need a map Sps, tq ^ Spt, rq Ñ Sps, rq for all s, t, r. Any two objects in 2 have at most one map between them, and a map exists iff we’re not asking for true Ñ false. Transitivity then becomes clear. For reflexivity, note that I need a map true Ñ Sps, sq giving an identity, which means Sps, sq must be true.

How could I require antisymmetry?

Example: Suppose A is a category with a zero object. Then I can promote A to A , enriched over pSet˚, ^, t˚, ˚uq. To see this, I need

t˚, ˚u Ñ A pa, aq A pa, bq ^ A pb, cq Ñ A pa, cq

´1 t˚, ˚u ^ A pa, bq λ A pa, bq

A pa, aq ^ A pa, bq ˝ C pa, bq

to be pointed set maps. When I unravel this, it means that I need the zero map to be sort of magnetic, consuming all it touches. (Remember too that t˚, ˚u ^ S – S.)

Really I only need zero maps, not a zero object.

Claim: If V Ñ U is a lax monoidal functor, and C is a V category, we can V can be demoted to a U category.

Example: The lax monoidal functor VpI, ´q : V Ñ Set gives us a Set enriched category out of a V enriched category. This is referred to as the “underlying” category C0. It has the same objects as C , and has morphism sets C0pa, bq :“ VpI, C pa, bqq.

f g To compose arrows I ÝÑ C pa, bq and I ÝÑ C pb, cq, take

„ fbg ˝ I ÝÑ I b I ÝÝÝÑ C pa, bq b C pb, cq ÝÑ C pa, cq.

f Function composition is associative here because it is in C . Note that arrows I ÝÑ C pc, dq can act on hom g objects C pb, cq (think of it like post-composition) and arrows I ÝÑ C pa, bq can act on hom objects C pb, cq (think of it like precomposition) as follows:

ρ´1 1bf ˝ C pb, cq ÝÝÑ C pb, cq b I ÝÝÝÑ C pb, cq b C pc, dq ÝÑ C pb, dq

˝ C pb, cq ÝÑ I b C pb, cq ÝÑ C pa, bq b C pb, cq ÝÑ C pa, cq

One way to express this action is that I can view C pc, ´q as a functor from C0 to V, and C p´, cq as a op functor C0 Ñ V.

7 „ Example: An example of the above: Ab... the map Z ÝÑ Z b Z can be seen as “factor if you want”...

Example: The underlying category of a preorder enriched over 2 is just the locally small cate- gory where hom sets are t˚u or ∅.

Remark: The “underlying category” functor V Cat Ñ Cat has a left adjoint called the “free V category on C ”, where C is Set enriched.

If C and D are V-categories, a V-enriched functor F : C Ñ D is a map on objects |C | Ñ |D|, along with maps of hom objects C pa, bq Ñ DpF a, F bq. The functoriality requirement (which normally says F pIdq “ Id and F pfgq “ pF fq; pF gq is manifested as

i ˝C I a C pa, aq C pa, bq b C pb, cq C pa, cq

F F F iF a DpF a, F aq DpF a, F bq b DpF b, F cq DpF a, F cq ˝D

To define a natural transformation, we’d normally like to find an arrow between F a and Ga for each a P C , such that

αc c F c Gc For each f F f Gf c1 F c1 Gc1 αc1 I can think of a slick way to express this without mentioning particular arrows (mostly):

HompF c, F c1q DpF c, F c1q F ´;αc1 F ?

Hompc, c1q HompF c, Gc1q C pc, c1q DpF c, Gc1q

˚ G pαcq G ? HompGc, Gc1q DpGc, Gc2q

But for the notion of “choosing an arrow” and “post composing/precomposing” I’m going to have to go to the underlying category, and define a V enriched natural transformation F

αa C α D between V functors to be a choice of map I ÝÑ DpF a, Gaq for all a P C so that

G

C pb, cq F DpF b, F cq

G αc˚

DpGb, Gcq ˚ DpF b, Gcq αb

8 My first thought about what naturality should mean was: the action of αa on the hom objects commutes with composition, like so:

DpF a, F bq b DpF b, F cq ˝ DpF a, F cq

1bαc˚ ´;αc

DpF a, F bq b DpF b, Gcq ˝ DpF a, Gcq and similarly for precomposition.

The problem with this is that it doesn’t say anything about naturality as such; notice that only the covariant αc˚ is involved, defeating what we mean by naturality. Rather, this says that “fake” composition associates with real composition. This is true for any “fake” arrow, as follows:

C pa, bq b C pb, cq C pa, bq b C pb, cq b I

C pa, bq b C pb, cq b C pc, dq

The last one is associative with respect to composition. Now composing left terms first is the same as . . . going in the above diagram, and composing right terms first is the same as going . . .

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