Enriched Categories Self-Enriched Categories References

Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References

Enriched Categories

Alexander Rogovskyy

Proseminar “Category Theory”, Summer 2020

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Table of Contents

Motivation and Intuition

Constructing Enriched Categories

Self-Enriched Categories

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Example: Vec

5 f 3 R R

I The morphisms between two objects are a Set I Hom(R5, ): Vec → Set I Set is special. Can we try to get away from that?

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Special structure of morphisms

I We often have special structure in morphisms I A notion of “combining two morphisms” into one I Concatenation: ◦ : Hom(A, B) × Hom(B, C) → Hom(A, C) I Tensoring ⊗ : Hom(A, B) ⊗ Hom(B, C) → Hom(A, C) I Can we generalize the notion of morphisms to reﬂect that?

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Motivation for Enriched Categories

Two central questions: 1. Can we generalize over Set? 2. Can we employ special structure of morphisms (⇒ relationships) between objects

Alexander Rogovskyy Enriched Categories 5/ 33 Classical categories can be seen as enriched over Set.

Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References

Answer: Instead of sets of morphisms (hom-sets)

f g A B h

we use hom-objects from another category V to represent the relationship between A and B!

A C(A, B) B

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Answer: Instead of sets of morphisms (hom-sets)

f g A B h

we use hom-objects from another category V to represent the relationship between A and B!

A C(A, B) B

Classical categories can be seen as enriched over Set.

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An example: R-Vec as a Set-Category

3 Hom(R , C)

5 3 3 ◦ 5 Hom(R , R ) × Hom(R , C) Hom(R , C)

5 3 Hom(R , R )

5 Hom(R , C)

5 5 3 3 3 R Hom(R , R ) R Hom(R , C) C

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An example: R-Vec as a Set-Category

3 Hom(R , C)

5 3 3 ◦ 5 Hom(R , R ) × Hom(R , C) Hom(R , C)

5 3 Hom(R , R )

5 Hom(R , C)

5 5 3 3 3 R Hom(R , R ) R Hom(R , C) C

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Another point of view

I Essentially, enriched categories can be thought of as directed graphs with edge labels I The edge labels come from another (regular, non-enriched) category V I We have a hom-object between any two objects =⇒ the graph is complete

A C

Important! We do not have individual morphisms anymore!

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Step 1: Monoidal Categories

I What structure should our category of hom-objects V have? I We saw that something like a cartesian product × could be useful

Alexander Rogovskyy Enriched Categories 9/ 33 =⇒ essentially a categorical version of a monoid

Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References

Deﬁnition (Monoidal Category) A monoidal category consists of I A category V I A functor ⊗ : V × V → V (“tensor product”) I An element 1 ∈ Ob(V) (“unit”) such that I ⊗ is associative (up to isomorphism) I 1 is the left and right unit w.r.t ⊗ (up to isomorphism) I certain coherence diagrams commute (see next slide)

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=⇒ essentially a categorical version of a monoid

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Some coherence Axioms

((W ⊗ X ) ⊗ Y ) ⊗ Z (W ⊗ X ) ⊗ (Y ⊗ Z) W ⊗ (X ⊗ (Y ⊗ Z))

(W ⊗ (X ⊗ Y )) ⊗ Z W ⊗ ((X ⊗ Y ) ⊗ Z)

(X ⊗ 1) ⊗ Y X ⊗ (1 ⊗ Y )

X ⊗ Y

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Examples

I Set with × and {·} I k-Vec with × (=⊕) and {0} I k-Vec with ⊗ and k I Grp with × and {e} I Graph with the tensor product of graphs and I Cat with × and I R+ (with ≥ for morphisms), + and 0.

Alexander Rogovskyy Enriched Categories 12/ 33 I We call this adjoint functor the internal hom

I cartesian if ⊗ is the category-theoretic product and 1 is the terminal object I closed if for every X , the functor ⊗ X has a right adjoint

Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References

Special cases and variations

A monoidal category is called... I symmetric if ⊗ is commutative up to isomorphism (and some additional coherence diagramms commute)

Alexander Rogovskyy Enriched Categories 13/ 33 I We call this adjoint functor the internal hom

I closed if for every X , the functor ⊗ X has a right adjoint

Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References

Special cases and variations

Alexander Rogovskyy Enriched Categories 13/ 33 I We call this adjoint functor the internal hom

Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References

Special cases and variations

A monoidal category is called... I symmetric if ⊗ is commutative up to isomorphism (and some additional coherence diagramms commute) I cartesian if ⊗ is the category-theoretic product and 1 is the terminal object I closed if for every X , the functor ⊗ X has a right adjoint

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Special cases and variations

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Step 2: Enriched Categories

Remember that we are replacing hom-sets with hom-objects. We need to have a composition morphism ◦ := ◦ABC

C(A, B) ⊗ C(B, C) −→◦ C(A, C)

In V = Set, this is just the usual composition of functions.

Alexander Rogovskyy Enriched Categories 14/ 33 Solution Require a morphism jA : 1 → C(A, A) that represents “picking” an identity

C(A, A)

jA j in V 1 B C(B, B)

...

Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References

I But what about the identity morphism idA? I We don’t have individual morphisms anymore, only hom-objects! I We need a way to “pick” an identity

Alexander Rogovskyy Enriched Categories 15/ 33 C(A, A)

jA j in V 1 B C(B, B)

...

Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References

I But what about the identity morphism idA? I We don’t have individual morphisms anymore, only hom-objects! I We need a way to “pick” an identity

Solution Require a morphism jA : 1 → C(A, A) that represents “picking” an identity

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I But what about the identity morphism idA? I We don’t have individual morphisms anymore, only hom-objects! I We need a way to “pick” an identity

Solution Require a morphism jA : 1 → C(A, A) that represents “picking” an identity

C(A, A)

jA j in V 1 B C(B, B)

...

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Deﬁnition (Enriched Category) Let (V, ⊗, 1) be a monoidal category. A V-Category C consists of I A collection of objects Ob(C) I For A, B ∈ Ob(C) a hom-object C(A, B) ∈ V I For A, B, C ∈ Ob(C) a composition morphism

◦ABC : C(A, B) ⊗ C(B, C) → C(A, C)

I For A ∈ Ob(C) an identity selector jA : 1 → C(A, A) such that the associativity and unit laws (next slides) hold.

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Associativity of ◦

(C(A, B) ⊗ C(B, C)) ⊗ C(C, D) C(A, B) ⊗ (C(B, C) ⊗ C(C, D))

◦⊗id id⊗◦ C(A, C) ⊗ C(C, D) C(A, B) ⊗ C(B, D)

◦ ◦ C(A, D)

Alexander Rogovskyy Enriched Categories 17/ 33 “If we pick the right identity, it will act as a unit w.r.t ◦!”

Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References

Unit law for jA

id⊗j C(A, B) ⊗ 1 A C(A, B) ⊗ C(A, A) =∼ ◦ C(A, B)

(symmetric law omitted)

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Unit law for jA

id⊗j C(A, B) ⊗ 1 A C(A, B) ⊗ C(A, A) =∼ ◦ C(A, B)

(symmetric law omitted)

“If we pick the right identity, it will act as a unit w.r.t ◦!”

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The Triangle of Truth B

C(A, B) C(B, C) ⊗

C(A, B) ⊗ C(B, C)

◦

A C(A, C) C

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Examples of Enriched Categories

I Any category C can be seen as enriched over Set I k − Vec as a (k − Vec)-Category with ⊗ and k I k − Vec as a (k − Vec)-Category with ⊕ and {0} I Preorders as categories enriched over 0 1 I (Generalized) metric spaces as categories over R+

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A closer look: Lawvere Metric Spaces

Let V := R+ with x → y ⇐⇒ x ≥ y, ⊗ := + and 1 := 0:

1 = 0 1 2 ...

Let C be a V-Category. What can we say about its structure?

Alexander Rogovskyy Enriched Categories 21/ 33 This is exactly the 4-inequality!

Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References

Example diagram in a R+-Category

2 3 +

2 + 3 = 5 ≥ A 4 C

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Example diagram in a R+-Category

2 3 +

2 + 3 = 5 ≥ A 4 C

This is exactly the 4-inequality!

Alexander Rogovskyy Enriched Categories 22/ 33 A R+-category is thus a generalized metric space: I The hom-objects represent the metric (“distance” between objects) I The triangle inequality holds I C(A, A) = 0

Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References

What about the identity selector?

≥ 1 = 0 C(A, A) A

This implies C(A, A) = 0!

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What about the identity selector?

≥ 1 = 0 C(A, A) A

This implies C(A, A) = 0! A R+-category is thus a generalized metric space: I The hom-objects represent the metric (“distance” between objects) I The triangle inequality holds I C(A, A) = 0

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Enriched Functors

A regular functor is a mapping between objects and hom-sets. An enriched functor is a mapping between objects and hom-objects:

A C(A, B) B

F FAB F V-Functor F : C → D FA D(FA, FB) FB

Alexander Rogovskyy Enriched Categories 24/ 33 ...and identity selections

jA jFA C(A, A) F D(FA, FA)

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A V-Functor has to respect composition...

C(A, B) ⊗ C(B, C) ◦ C(A, C)

F F D(FA, FB) ⊗ D(FB, FC) ◦ D(FA, FC)

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A V-Functor has to respect composition...

C(A, B) ⊗ C(B, C) ◦ C(A, C)

F F D(FA, FB) ⊗ D(FB, FC) ◦ D(FA, FC)

...and identity selections

jA jFA C(A, A) F D(FA, FA)

Alexander Rogovskyy Enriched Categories 25/ 33 But we already saw how to pick items of the hom-object: We use morphisms from 1:

αA FA D(FA, GA) GA

Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References

Enriched natural transformations An ordinary natural transformation α : F → G (F , G : C → D) assigns to each object A a morphism

α FA A GA

In other words: We select an object of Hom(FA, GA).

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Enriched natural transformations An ordinary natural transformation α : F → G (F , G : C → D) assigns to each object A a morphism

α FA A GA

In other words: We select an object of Hom(FA, GA). But we already saw how to pick items of the hom-object: We use morphisms from 1:

αA FA D(FA, GA) GA

Alexander Rogovskyy Enriched Categories 26/ 33 =⇒ We need to express hom-sets in V as objects within V!.

Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References

Self-Enriched Categories

Set can be thought as being enriched over itself. Can we generalize this idea?

Ordinary Category V V-Category C

I Objects I Objects I Hom-Sets I Hom-Objects in V

Alexander Rogovskyy Enriched Categories 27/ 33 =⇒ We need to express hom-sets in V as objects within V!.

Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References

Self-Enriched Categories

Set can be thought as being enriched over itself. Can we generalize this idea?

Ordinary Category V V-Category V

I Objects I Objects I Hom-Sets I Hom-Objects in V

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Self-Enriched Categories

Set can be thought as being enriched over itself. Can we generalize this idea?

Ordinary Category V V-Category V

I Objects I Objects I Hom-Sets I Hom-Objects in V

=⇒ We need to express hom-sets in V as objects within V!.

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Theorem Any closed symmetrical monoidal category V can be self-enriched, i.e. it is equivalent to a V-Category.

Reminder (V, ⊗, 1) is closed if every functor ⊗ A has a right adjoint [A, ].

Hom(A ⊗ B, C) =∼ Hom(A, [B, C])

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Theorem Any closed symmetrical monoidal category V can be self-enriched, i.e. it is equivalent to a V-Category.

Reminder (V, ⊗, 1) is closed if every functor ⊗ A has a right adjoint [A, ].

Hom(A ⊗ B, C) =∼ Hom(A, [B, C]) =⇒ Hom(B, C) =∼ Hom(1, [B, C])

Alexander Rogovskyy Enriched Categories 28/ 33 Thus we deﬁne the V-category V as:

Ob(V) := Ob(V) V(A, B) := [A, B] ∈ Ob(V )

Now we need to provide a concatenation and a unit selector

◦ :[A, B] ⊗ [B, C] → [A, C] (1)

jA : 1 → [A, A] (2) (2) is easy: Just set X := 1, Y , Z := A in the adjunction property

Hom(X ⊗ Y , Z) =∼ Hom(X , [Y , Z])

Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References

How to turn V into a V-Category This gives us an ideal candidate for the hom-object:

[A, B] ∈ Ob(V)

Alexander Rogovskyy Enriched Categories 29/ 33 Now we need to provide a concatenation and a unit selector

◦ :[A, B] ⊗ [B, C] → [A, C] (1)

jA : 1 → [A, A] (2) (2) is easy: Just set X := 1, Y , Z := A in the adjunction property

Hom(X ⊗ Y , Z) =∼ Hom(X , [Y , Z])

Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References

How to turn V into a V-Category This gives us an ideal candidate for the hom-object:

[A, B] ∈ Ob(V)

Thus we deﬁne the V-category V as:

Ob(V) := Ob(V) V(A, B) := [A, B] ∈ Ob(V )

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How to turn V into a V-Category This gives us an ideal candidate for the hom-object:

[A, B] ∈ Ob(V)

Thus we deﬁne the V-category V as:

Ob(V) := Ob(V) V(A, B) := [A, B] ∈ Ob(V )

Now we need to provide a concatenation and a unit selector

◦ :[A, B] ⊗ [B, C] → [A, C] (1)

jA : 1 → [A, A] (2) (2) is easy: Just set X := 1, Y , Z := A in the adjunction property

Hom(X ⊗ Y , Z) =∼ Hom(X , [Y , Z])

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Similarly, for ◦, we use

Hom([A, B] ⊗ [B, C], [A, C]) =∼ Hom(([A, B] ⊗ [B, C]) ⊗ A, C)

and deﬁne

([A, B] ⊗ [B, C]) ⊗ A

By associativity and commutativity [B, C] ⊗ ([A, B] ⊗ A)

By “almost invertability” of adjoints [B, C] ⊗ B

By “almost invertability” of adjoints C

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Examples of Self-Enrichment

The categories I Set with [A, B] = {f : A → B} I k − Vec with the usual tensor product ⊗ and [A, B] = {f : A → B linear } I Cat with the categorical product where [A, B] is the functor category BA. I Graph with the tensor product of graphs and a kind of “exponential graph” as the internal hom I SimplyTyped−λ are all cartesian (and thus symmetrically) closed and hence are enriched over themselves.

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Thank you for listening!

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G. M. Kelly. “Basic concepts of enriched category theory”. In: Repr. Theory Appl. Categ. 10 (2005). url: http://emis.dsd.sztaki.hu/journals/TAC/reprints/ articles/10/tr10abs.html. Bartosz Milewski. Category Theory for Programmers. url: https://github.com/hmemcpy/milewski-ctfp-pdf. nLab authors. enriched category. http://ncatlab.org/nlab/show/enriched%20category. Revision 94. Sept. 2020. Emily Riehl. Categorical homotopy theory. Vol. 24. New Mathematical Monographs. Cambridge University Press, Cambridge, 2014. isbn: 978-1-107-04845-4. url: http://www.math.jhu.edu/~eriehl/cathtpy.pdf.

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