category theory - Is unique morphism `m` which maps "best" product type to "suboptimal" product type truly unique?

I'm working through Bartosz Milewski's awesome blogs about category theory. I'm stuck on the one on products and coproducts.Bartosz says that a product of two objects a and b is the object c equipped with two projections such that for any other object c' equipped with two projections there is a unique morphism m from c' to c that factorizes those projections.Of course we can find a suitable example in the category of sets and functions. The product of two types Int and Bool is the pair (Int, Bool). The two projections are p (int, _) = int and q...Read more

category theory - What does it mean for a value to "have a functor"?

I'm new to this, and I may be missing something important. I've read part one of Category Theory for Programmers, but the most abstract math I did in university was Group Theory so I'm having to read very slowly.I would ultimately like to understand the theory to ground my use of the techniques, so whenever I feel like I've made some progress I return to the fantasy-land spec to test myself. This time I felt like I knew where to start: I started with Functor.From the fantasy-land spec: Functor => a) is equivalent to u (identity...Read more

set theory - In the category of sets, why are singleton sets terminal?

I'm trying to understand why the category of sets is defined the way it is, with singleton sets as terminal objects. If the "Set" category contains all of the possible sets, and all of the possible morphisms between those sets, why wouldn't there be injective, non-surjective morphisms from the singleton sets to all other sets with infinite cardinality? In this case, there wouldn't be any terminal objects.So what is the rule that leads it to being defined the way it is defined, rather than being defined with infinite sets and morphisms. I guess ...Read more

category theory - Amalgam of trees

Definition A tree is a partially ordered set $(T, <)$ such that for each $t \in T$, the set $\{s \in T : s < t\}$ is well-ordered by the relation $<$.For trees $(T,<_T)$, $(S,<_S)$, $T\prec S$, if there is an embedding of $T$ into $S$, i.e. there exists an injection $f:T\rightarrow S$ such that for all $x,y\in T$, $x<_T y$ iff $f(x)<_S f(y)$.Let $f_1,f_2$ witness that $T\prec S_1,S_2$ respectively. The "amalgam of $S_1,S_2$ over $T$" is a tree $R$ and embeddings $g_1,g_2$ such that $g_1,g_2$ witness that $S_1,S_2\prec R$ a...Read more

category theory - Exercise on categories of $G$-set

The following is exercise 6 from chapter I of Mac Lane and Moerdijk's Sheaves in Geometry and Logic:I'm currently having trouble with point $(b)$, probably because of my little ability with group actions and topological groups in general. We are asked to define a functor $r_G$ sending a set $X$ with an action $X\times G\rightarrow X$ to the set $r_G(X)$: the restriction of the action is now continuous. The trouble is: how can I define, for an equivariant map $f:X\rightarrow Y$, the corresponding $r_G(f)$? The only obvious choice is the restrict...Read more

Formalising Statements in Category Theory with Regards to Universes (with an example of the end of a functor)

I am currently in the process of changing the way I think about category theory, by adopting the notion of Grothendiek universe and trying not to think of proper classes. Think of a statement such as If $\mathcal C$ and $\mathcal D$ are categories, and $F,G \colon \mathcal C \to \mathcal D$ are functors, then $$ \int_{c \colon \mathcal C} \mathcal D(Fc, Gc) = \mathrm{Nat}(F,G). $$This doesn't make much sense as written, since depending on the foundations you use there are potentially size problems and $\mathsf{Set}$ might not recive the funct...Read more

category theory - Y combinator as an application of Lawvere's fixed point theorem

Lawvere's fixed point theorem states that that in a cartesian closed category, if there is a morphism $ϕ: A \to B^A$ which is point-surjective (i.e., for every point $q : 1 \to B^A$ there exists a point $p : 1 \to A$ such that $ϕp = q$), then every endomorphism $f : B \to B$ has a fixed point.The fixed point of $f$ is obtained as $ϕ(p)(p)$ where $p : 1 \to A$ is the point of $A$ such that $ϕp = q$ for $q : 1 \to B^A$ defined in lambda calculus notation as $q = λa:A.fϕ(a)(a)$.I've seen it remarked several times that the existence of Y combinator...Read more

category theory - coequalizer of simplicial sets

This is the statement whose first line of proof I am confused. This is on page 10, of Goerss, Jardine's Simplicial Homotopy Theory.(i) How does one prove the "presentation" of $\partial \Delta^n$? (ii) How does one obtain the first coequalizer? In general, I am just confused in how to prove coequalizing diagram for simplicial sets. Can ignore: I have typed up my attempt for (i). I would be happy to see for a better presentation (hand drawn is great too)....Read more

What is an equationally presentable category?

I encountered this term in [1]. I can’t find a formal definition. Here is the definition from [1, p. 23]: We shall call a category algebraic if it is monadic over Set, and equationally presentable if its objects can be described by (a proper class of) operations and equations…The term “described” has no formal definition. From [2, p. 334]: Recall that a category is called algebraic if it is monadic over Set, and equationally presentable if its objects can be prescribed by (a proper class of) operations and equations.Again, no definition o...Read more

category theory - Natural transformations and small categories

If $\mathcal{A}$ and $\mathcal{B}$ are small categories (i.e. objects are sets, not proper classes) and $F,G:\mathcal{A} \to \mathcal{B}$ are both (contra/co)variant functors, then the the 'collection' of natural transformations $Nat(F,G)$ is also a set.Say we now have arbitary categories $\mathcal{C}$ and $\mathcal{D}$, with functors $S,T$ such that $S,T:\mathcal{C} \to \mathcal{D}$. Then it seems logical that $Nat(S,T)$ may be a proper class, and not a set.Can anyone provide any examples?Moreover (this is a bit more philosophical) - are such ...Read more

category theory - Difference between large and small categories

In my book the following definition is given:A category $C$ is called small both the collection of objects and arrows are sets. Otherwise the category is called large.A set is delfined to be a collection of distinct objects. Now I am a little confused since the category of all finte sets is said to be small, which is ok for me since the objects and functions can be considered as sets. But why isnt the category of groups a small category. All groups may be considered as distinct objects and hence a set, same argument goes for group homomorphisms...Read more

category theory - How to prove a set with at least one element is a separator?

I'm reading Lawvere's Sets for Mathematics and got stuck at Exercise 1.15 In the category of abstract sets S, any set A with at least one element $1 \xrightarrow{x} A$ is a separator.I can see that with the axiom that "the terminal object $1$ is a separator" and the following diagram, I ought to deduce the statement, but I cannot seem to write it down formally:)Any hint please?...Read more

Ordinals in category theory?

I am just learning category theory and I am wondering if there is some way to define ordinals, and ordinal arithmetic purely through categorical means. I have a notion of taking a category with 0, 1, 2, etc elements but I have no idea how you could define addition on these categories with the usual nice properties (associativity, commutativity in the finite case, etc). I also don't know how this would generalize to categories representing ordinals $\geq \omega$. Through the basic notions of category theory, what is an ordinal? And what does ord...Read more