Category Theory - Definition of Hom(X,X) in Awodey

[Math Amateur]

I am reading Steve Awodey's book, "Category Theory" (Second Edition).

In Chapter 1 within a small section on monoids, Awodey defines \(\displaystyle Hom_{Sets} (X,X) \) as follows:

" ... ... for any set X, the set of functions from X to X, written as \(\displaystyle Hom_{Sets} (X,X) \) is a monoid under the operation of composition."

Despite the notation "Hom" suggesting that we are dealing with a set of homomorphisms, Awodey just defines \(\displaystyle Hom_{Sets} (X,X) \) as a set of functions ...

Are these functions actually homomorphisms for some reason, or is the suggestive notation simply misleading in this sense ...?

Hope someone can help.

The relevant text from Awodey is as follows:

View attachment 2519

I just note in passing that D&F first define Hom in the context of R-modules on page 345 as follows:

"Let M and N be R-modules and define \(\displaystyle Hom_R (M,N) \) to be the set of R-module homomorphisms from M into N"

I understand that the context of the two definitions are different, one pertaining to modules and one to a set X, but the notation "Hom" is used in both cases, suggesting homomorphisms.

Hoping someone can clarify this issue.

Peter

 

[Deveno]

One of the impetuses for the creation of category theory was to consolidation common notions that occur ACROSS different structures. This led to the notion of categories at first being "structured things" and morphisms being "structure-preserving maps".

It turned out that this idea is "too restrictive": many mathematical objects are not necessarily "sets with structure". However, many of them ARE, and for these objects "structure-preserving maps" are often called "homomorphisms".

Sets can be viewed as "sets with null structure", that is, a set is a set, together with a null binary operation (no operation at all!). Since a set "homomorphism" has no operation to preserve, it is just a plain old function.

Categories DO have a structure: their arrows form a monoid-like structure (or rather, the analogue for monoids that groupoids are for groups). "Morphisms" of categories are called functors.

There is a "natural" functor for any category $\mathcal{C}$ from $\mathcal{C}^{\text{op}}\times \mathcal{C} \to \mathbf{Set}$ called the "hom-functor", and often written like so:

$\text{Hom}_{\mathcal{C}}(-,-)$

which assigns to any pair of objects $A,B \in \mathcal{C}$ the SET of morphisms $f:A \to B$, and assigns to any pair of morphisms:

$f:A' \to A$ (the "prime" comes first because we are in $\mathcal{C}^{\text{op}}$)
$g:B \to B'$

a map $\text{Hom}_{\mathcal{C}}(A,B) \to \text{Hom}_{\mathcal{C}}(A',B')$ that sends: $h: A \to B$ to $g \circ h \circ f$.

It is often easier to understand this if one "fixes" an object $A$ of $\mathcal{C}$ and considers the two functors:

$\text{Hom}_{\mathcal{C}}(A,-): \mathcal{C}^{\text{op}} \to \mathbf{Set}$

$\text{Hom}_{\mathcal{C}}(-,A): \mathcal{C} \to \mathbf{Set}$

individually.

In some kinds of categories, we have a related functor that takes values in $\mathcal{C}$ itself, instead of $\mathbf{Set}$, that is, that sets of morphisms of a structure have the SAME kind of structure as the objects they are morphisms OF. This is the case with $R$-modules (and thus for abelian groups, and vector spaces, the two "extreme cases" of $R$-modules).

The notation, then "$\text{Hom}_{\mathcal{C}}$" should be read as "morphisms of category $\mathcal{C}$", which for:

$\mathcal{C} = \mathbf{Set}$

means: functions.

 

[Math Amateur]

One of the impetuses for the creation of category theory was to consolidation common notions that occur ACROSS different structures. This led to the notion of categories at first being "structured things" and morphisms being "structure-preserving maps".

It turned out that this idea is "too restrictive": many mathematical objects are not necessarily "sets with structure". However, many of them ARE, and for these objects "structure-preserving maps" are often called "homomorphisms".

Sets can be viewed as "sets with null structure", that is, a set is a set, together with a null binary operation (no operation at all!). Since a set "homomorphism" has no operation to preserve, it is just a plain old function.

Categories DO have a structure: their arrows form a monoid-like structure (or rather, the analogue for monoids that groupoids are for groups). "Morphisms" of categories are called functors.

There is a "natural" functor for any category $\mathcal{C}$ from $\mathcal{C}^{\text{op}}\times \mathcal{C} \to \mathbf{Set}$ called the "hom-functor", and often written like so:

$\text{Hom}_{\mathcal{C}}(-,-)$

which assigns to any pair of objects $A,B \in \mathcal{C}$ the SET of morphisms $f:A \to B$, and assigns to any pair of morphisms:

$f:A' \to A$ (the "prime" comes first because we are in $\mathcal{C}^{\text{op}}$)
$g:B \to B'$

a map $\text{Hom}_{\mathcal{C}}(A,B) \to \text{Hom}_{\mathcal{C}}(A',B')$ that sends: $h: A \to B$ to $g \circ h \circ f$.

It is often easier to understand this if one "fixes" an object $A$ of $\mathcal{C}$ and considers the two functors:

$\text{Hom}_{\mathcal{C}}(A,-): \mathcal{C}^{\text{op}} \to \mathbf{Set}$

$\text{Hom}_{\mathcal{C}}(-,A): \mathcal{C} \to \mathbf{Set}$

individually.

In some kinds of categories, we have a related functor that takes values in $\mathcal{C}$ itself, instead of $\mathbf{Set}$, that is, that sets of morphisms of a structure have the SAME kind of structure as the objects they are morphisms OF. This is the case with $R$-modules (and thus for abelian groups, and vector spaces, the two "extreme cases" of $R$-modules).

The notation, then "$\text{Hom}_{\mathcal{C}}$" should be read as "morphisms of category $\mathcal{C}$", which for:

$\mathcal{C} = \mathbf{Set}$

means: functions.

Thanks Deveno ... most helpful ... definitely taken my study of category theory forward!

Just working through your post in detail now.

Peter

 

[Math Amateur]

One of the impetuses for the creation of category theory was to consolidation common notions that occur ACROSS different structures. This led to the notion of categories at first being "structured things" and morphisms being "structure-preserving maps".

It turned out that this idea is "too restrictive": many mathematical objects are not necessarily "sets with structure". However, many of them ARE, and for these objects "structure-preserving maps" are often called "homomorphisms".

Sets can be viewed as "sets with null structure", that is, a set is a set, together with a null binary operation (no operation at all!). Since a set "homomorphism" has no operation to preserve, it is just a plain old function.

Categories DO have a structure: their arrows form a monoid-like structure (or rather, the analogue for monoids that groupoids are for groups). "Morphisms" of categories are called functors.

There is a "natural" functor for any category $\mathcal{C}$ from $\mathcal{C}^{\text{op}}\times \mathcal{C} \to \mathbf{Set}$ called the "hom-functor", and often written like so:

$\text{Hom}_{\mathcal{C}}(-,-)$

which assigns to any pair of objects $A,B \in \mathcal{C}$ the SET of morphisms $f:A \to B$, and assigns to any pair of morphisms:

$f:A' \to A$ (the "prime" comes first because we are in $\mathcal{C}^{\text{op}}$)
$g:B \to B'$

a map $\text{Hom}_{\mathcal{C}}(A,B) \to \text{Hom}_{\mathcal{C}}(A',B')$ that sends: $h: A \to B$ to $g \circ h \circ f$.

It is often easier to understand this if one "fixes" an object $A$ of $\mathcal{C}$ and considers the two functors:

$\text{Hom}_{\mathcal{C}}(A,-): \mathcal{C}^{\text{op}} \to \mathbf{Set}$

$\text{Hom}_{\mathcal{C}}(-,A): \mathcal{C} \to \mathbf{Set}$

individually.

In some kinds of categories, we have a related functor that takes values in $\mathcal{C}$ itself, instead of $\mathbf{Set}$, that is, that sets of morphisms of a structure have the SAME kind of structure as the objects they are morphisms OF. This is the case with $R$-modules (and thus for abelian groups, and vector spaces, the two "extreme cases" of $R$-modules).

The notation, then "$\text{Hom}_{\mathcal{C}}$" should be read as "morphisms of category $\mathcal{C}$", which for:

$\mathcal{C} = \mathbf{Set}$

means: functions.

Hi Deveno,

Just a couple of basic questions.

You write:"There is a "natural" functor for any category $\mathcal{C}$ from $\mathcal{C}^{\text{op}}\times \mathcal{C} \to \mathbf{Set}$ called the "hom-functor", and often written like so:

$\text{Hom}_{\mathcal{C}}(-,-)$"Can you explain what is meant by \(\displaystyle \mathcal{C}^{\text{op}}\)?Also, you write:"$\text{Hom}_{\mathcal{C}}(-,-)$

which assigns to any pair of objects $A,B \in \mathcal{C}$ the SET of morphisms $f:A \to B$, and assigns to any pair of morphisms:

$f:A' \to A$ (the "prime" comes first because we are in $\mathcal{C}^{\text{op}}$)
$g:B \to B'$"Why is the prime - ' - first in one case and second in the other? Is this a typo?Hope you can help.Peter

 

[Deveno]

$\mathcal{C}^{\text{op}}$ is the "opposite category of $\mathcal{C}$" which has the same OBJECTS, but an arrow $f:A \to B$ in $\mathcal{C}$ is an arrow $g:B \to A$ in $\mathcal{C}^{\text{op}}$, in other words the arrows go "the opposite direction".

The image of an arrow $f:B \to C$ under the hom-functor:

$H = \text{Hom}_{\mathcal{C}}(A,-)$ is often written: $\text{Hom}_{\mathcal{C}}(A,f)$

or, perhaps more intelligibly: $H(f) = f'$ where for any $g \in \text{Hom}_{\mathcal{C}}(A,B)$

$f'(g) = f \circ g \in \text{Hom}_{\mathcal{C}}(A,C)$ (you may recognize this idea from Dummit and Foote).

This is called a "co-variant" functor, since the direction of the arrows is preserved.

The other hom-functor does the same thing, but instead of post-composition with $f$, we do pre-composition with $f$, which reverses the directions of the arrows:

for $F =\text{Hom}_{\mathcal{C}}(-,A)$, we still get a set $\text{Hom}_{\mathcal{C}}(X,A)$ for any $X \in \text{obj}(\mathcal{C})$, but now, for any arrow $f:X \to Y$, we have:

$F(f) = f_{\ast}$ where for any $g \in \text{Hom}_{\mathcal{C}}(Y,A)$, we have $f_{\ast}(g) = g \circ f$ so that:

$f_{\ast}:\text{Hom}_{\mathcal{C}}(Y,A) \to \text{Hom}_{\mathcal{C}}(X,A)$

(we've "reversed the flow" of FROM $X$ TO $Y$).

Such a functor is called "contra-variant", and is akin to "anti-homomorphisms" one often encounters in rings:

$h(xy) = h(y)h(x)$ (one example of such an "anti-homomorphism" is the transpose map on the ring of $n \times n$ matrices over a commutative ring),

and these are often characterized as (regular) homomorphisms from $R \to R^{\text{op}}$.

With functions (indeed with many categories, not just $\mathbf{Set}$) this is not a "perfect mirror symmetry", the domain and co-domain play asymmetrical roles in a function. As you saw in studying modules, just because we have a surjection:

$M \to M/N$

there is no reason to believe we have an injection $M/N \to M$ (for a concrete example of how this might happen, consider the canonical surjection:

$\Bbb Z \to \Bbb Z/n\Bbb Z$

where it is clear we can have several homomorphisms going left-to-right (just send 1 to a divisor of $n$), but only one possible morphism going right-to-left (the 0-map, which is very much NOT injective)).

So, in general, $\mathcal{C}$ and $\mathcal{C}^{\text{op}}$ behave rather differently. Nevertheless, if something is true for any category, then it can be formulated as a statement about the opposite category of any given category, which often tells us something new and unexpected. This principle is called "categorical duality".

 

[Math Amateur]

$\mathcal{C}^{\text{op}}$ is the "opposite category of $\mathcal{C}$" which has the same OBJECTS, but an arrow $f:A \to B$ in $\mathcal{C}$ is an arrow $g:B \to A$ in $\mathcal{C}^{\text{op}}$, in other words the arrows go "the opposite direction".

The image of an arrow $f:B \to C$ under the hom-functor:

$H = \text{Hom}_{\mathcal{C}}(A,-)$ is often written: $\text{Hom}_{\mathcal{C}}(A,f)$

or, perhaps more intelligibly: $H(f) = f'$ where for any $g \in \text{Hom}_{\mathcal{C}}(A,B)$

$f'(g) = f \circ g \in \text{Hom}_{\mathcal{C}}(A,C)$ (you may recognize this idea from Dummit and Foote).

This is called a "co-variant" functor, since the direction of the arrows is preserved.

The other hom-functor does the same thing, but instead of post-composition with $f$, we do pre-composition with $f$, which reverses the directions of the arrows:

for $F =\text{Hom}_{\mathcal{C}}(-,A)$, we still get a set $\text{Hom}_{\mathcal{C}}(X,A)$ for any $X \in \text{obj}(\mathcal{C})$, but now, for any arrow $f:X \to Y$, we have:

$F(f) = f_{\ast}$ where for any $g \in \text{Hom}_{\mathcal{C}}(Y,A)$, we have $f_{\ast}(g) = g \circ f$ so that:

$f_{\ast}:\text{Hom}_{\mathcal{C}}(Y,A) \to \text{Hom}_{\mathcal{C}}(X,A)$

(we've "reversed the flow" of FROM $X$ TO $Y$).

Such a functor is called "contra-variant", and is akin to "anti-homomorphisms" one often encounters in rings:

$h(xy) = h(y)h(x)$ (one example of such an "anti-homomorphism" is the transpose map on the ring of $n \times n$ matrices over a commutative ring),

and these are often characterized as (regular) homomorphisms from $R \to R^{\text{op}}$.

With functions (indeed with many categories, not just $\mathbf{Set}$) this is not a "perfect mirror symmetry", the domain and co-domain play asymmetrical roles in a function. As you saw in studying modules, just because we have a surjection:

$M \to M/N$

there is no reason to believe we have an injection $M/N \to M$ (for a concrete example of how this might happen, consider the canonical surjection:

$\Bbb Z \to \Bbb Z/n\Bbb Z$

where it is clear we can have several homomorphisms going left-to-right (just send 1 to a divisor of $n$), but only one possible morphism going right-to-left (the 0-map, which is very much NOT injective)).

So, in general, $\mathcal{C}$ and $\mathcal{C}^{\text{op}}$ behave rather differently. Nevertheless, if something is true for any category, then it can be formulated as a statement about the opposite category of any given category, which often tells us something new and unexpected. This principle is called "categorical duality".

Thanks Deveno … appreciate the help ...

Just working through the post in detail now ...

Peter