diff --git a/errata.md b/errata.md index 856913d..e2f3aff 100644 --- a/errata.md +++ b/errata.md @@ -8,6 +8,14 @@ * [#157](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/157) - Adding paragraph indent +### 12. Limits and Colimits + +* [#162](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/162) - Fix grammatical error + +### 14. Representable Functors + +* [#162](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/162) - Fix grammatical error + ### 18. Adjunctions * [#160](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/160) - Fix spelling of "counit" @@ -20,6 +28,11 @@ ### 20. Monads - Programmer's Definition * [#160](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/160) - Fix grammatical error +* [#162](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/162) - Fix grammatical error + +### 22. Monads Categorically + +* [#162](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/162) - Fix grammatical error ### 23. Comonads @@ -30,6 +43,7 @@ * [#158](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/158) - fixed incorrect typesetting of `set` * [#159](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/159) - fixed incorrect typesetting of category terms * [#160](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/160) - Fix spelling of "counit" and "morphisms", fix subscript spacing +* [#162](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/162) - Fix grammatical errors ### 26. Ends and Coends @@ -43,7 +57,13 @@ ### 28. Enriched Categories * [#160](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/160) - Fix subscript spacing +* [#162](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/162) - Fix grammatical error + +### 29. Topoi + +* [#162](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/162) - Fix grammatical error ### 30. Lawvere Theories * [#160](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/160) - Fix spelling of "coequalizer" +* [#162](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/162) - Fix grammatical errors and a typesetting error diff --git a/src/content/2.2/Limits and Colimits.tex b/src/content/2.2/Limits and Colimits.tex index 34bd013..c97f5e0 100644 --- a/src/content/2.2/Limits and Colimits.tex +++ b/src/content/2.2/Limits and Colimits.tex @@ -83,7 +83,7 @@ identities) in $\cat{2}$. A generalization of this construction to categories other than $\cat{2}$ --- ones that, for instance, contain non-trivial morphisms --- will impose naturality conditions on the transformation between -$\Delta_c$ and $D$. We call such transformation a \emph{cone}, +$\Delta_c$ and $D$. We call such a transformation a \emph{cone}, because the image of $\Delta$ is the apex of a cone/pyramid whose sides are formed by the components of the natural transformation. The image of $D$ forms the base of the cone. diff --git a/src/content/2.4/Representable Functors.tex b/src/content/2.4/Representable Functors.tex index b3e7ed4..bc6a155 100644 --- a/src/content/2.4/Representable Functors.tex +++ b/src/content/2.4/Representable Functors.tex @@ -139,7 +139,7 @@ $\cat{C}$ as sets and functions in $\Set$. The functor $\cat{C}(a, -)$ itself is sometimes called representable. More generally, any functor $F$ that is naturally isomorphic to the hom-functor, for some choice of $a$, is called -\newterm{representable}. Such functor must necessarily be +\newterm{representable}. Such a functor must necessarily be $\Set$-valued, since $\cat{C}(a, -)$ is. I said before that we often think of isomorphic sets as identical. More diff --git a/src/content/3.12/Enriched Categories.tex b/src/content/3.12/Enriched Categories.tex index 24ef391..922a93a 100644 --- a/src/content/3.12/Enriched Categories.tex +++ b/src/content/3.12/Enriched Categories.tex @@ -299,7 +299,7 @@ preorder). The monoidal structure is given by addition, with zero serving as the unit object. In other words, the tensor product of two numbers is their sum. -A metric space is a category enriched over such monoidal category. A +A metric space is a category enriched over such a monoidal category. A hom-object $\cat{C}(a, b)$ from object $a$ to $b$ is a non-negative (possibly infinite) number that we will call the distance from $a$ to $b$. Let's see what we get for identity and diff --git a/src/content/3.13/Topoi.tex b/src/content/3.13/Topoi.tex index 0a04af8..4a3bb82 100644 --- a/src/content/3.13/Topoi.tex +++ b/src/content/3.13/Topoi.tex @@ -9,7 +9,7 @@ of Haskell's type system with dependent types, or the exploration on homotopy type theory in programming. So far I've been casually identifying types with \emph{sets} of values. -This is not strictly correct, because such approach doesn't take into +This is not strictly correct, because such an approach doesn't take into account the fact that, in programming, we \emph{compute} values, and the computation is a process that takes time and, in extreme cases, might not terminate. Divergent computations are part of every Turing-complete diff --git a/src/content/3.14/Lawvere Theories.tex b/src/content/3.14/Lawvere Theories.tex index 9ba3645..602f13d 100644 --- a/src/content/3.14/Lawvere Theories.tex +++ b/src/content/3.14/Lawvere Theories.tex @@ -139,7 +139,7 @@ $1 \times 1$ (or $1^2$) in $\cat{L}$. In this sense, the category $\cat{F}$ behaves like the logarithm of $\cat{L}$. Among morphisms in $\cat{L}$ we have those transferred by the functor -$I_{\cat{L}}$ from $\cat{F}$. They play structural role in $\cat{L}$. In +$I_{\cat{L}}$ from $\cat{F}$. They play a structural role in $\cat{L}$. In particular coproduct injections $i_k$ become product projections $p_k$. A useful intuition is to imagine the projection: \[p_k \Colon 1^n \to 1\] @@ -219,7 +219,7 @@ The preservation of products by models means that the image of $M$ in $\Set$ is a sequence of sets generated by powers of the set $M\ 1$ --- the image of the object $1$ from $\cat{L}$. Let's call this set $a$. (This set is sometimes -called a \emph{sort}, and such algebra is called \newterm{single-sorted}. There +called a \emph{sort}, and such an algebra is called \newterm{single-sorted}. There exist generalizations of Lawvere theories to multi-sorted algebras.) In particular, binary operations from $\cat{L}$ are mapped to functions: \[a \times a \to a\] @@ -252,7 +252,7 @@ The functors that define models form a category of models, $\cat{Mod}(\cat{L}, \Set)$, with natural transformations as morphisms. Consider a model for the trivial Lawvere category -$\Fop$. Such model is completely determined by +$\Fop$. Such a model is completely determined by its value at $1$, $M\ 1$. Since $M\ 1$ can be any set, there are as many of these models as there are sets in $\Set$. Moreover, every morphism in $\cat{Mod}(\Fop, \Set)$ (a @@ -279,7 +279,7 @@ elegant construction. Every monoid must have a unit, so we have to have a special morphism $\eta$ in $\cat{L}_{\cat{Mon}}$ that goes from $0$ to $1$. Notice that there can be no corresponding morphism in -$\cat{F}$. Such morphism would go in the opposite direction, from +$\cat{F}$. Such a morphism would go in the opposite direction, from $1$ to $0$ which, in $\cat{FinSet}$, would be a function from the singleton set to the empty set. No such function exists. @@ -334,7 +334,7 @@ $\cat{L}$. Another way of deriving $U$ is by exploiting the fact that $\Fop$ is the initial object in $\cat{Law}$. It means that, for any Lawvere theory $\cat{L}$, there is a unique -functor $\Fop \to L$. This functor induces the +functor $\Fop \to \cat{L}$. This functor induces the opposite functor on models (since models are functors \emph{from} theories to sets): \[\cat{Mod}(\cat{L}, \Set) \to \cat{Mod}(\Fop, \Set)\] diff --git a/src/content/3.4/Monads - Programmer's Definition.tex b/src/content/3.4/Monads - Programmer's Definition.tex index 5d89e48..0439a83 100644 --- a/src/content/3.4/Monads - Programmer's Definition.tex +++ b/src/content/3.4/Monads - Programmer's Definition.tex @@ -85,7 +85,7 @@ We have previously arrived at the \hyperref[kleisli-categories]{writer monad} by embellishing regular functions. The particular embellishment was done by pairing their return values with strings or, more generally, -with elements of a monoid. We can now recognize that such embellishment +with elements of a monoid. We can now recognize that such an embellishment is a functor: \src{code/haskell/snippet02.hs} diff --git a/src/content/3.6/Monads Categorically.tex b/src/content/3.6/Monads Categorically.tex index 112ef31..d649af4 100644 --- a/src/content/3.6/Monads Categorically.tex +++ b/src/content/3.6/Monads Categorically.tex @@ -524,7 +524,7 @@ This is indeed the implementation of \code{join} for the It turns out that not only every adjunction gives rise to a monad, but the converse is also true: every monad can be factorized into a -composition of two adjoint functors. Such factorization is not unique +composition of two adjoint functors. Such a factorization is not unique though. We'll talk about the other endofunctor $L \circ R$ in the next diff --git a/src/content/3.9/Algebras for Monads.tex b/src/content/3.9/Algebras for Monads.tex index c2239a7..6ad9938 100644 --- a/src/content/3.9/Algebras for Monads.tex +++ b/src/content/3.9/Algebras for Monads.tex @@ -15,7 +15,7 @@ every adjunction \hyperref[monads-categorically]{defines a monad} (and a comonad). The question is: Can every monad (comonad) be derived from an adjunction? The answer is positive. There is a whole family of adjunctions that generate a given monad. I'll show you two -such adjunction. +such adjunctions. Let's review the definitions. A monad is an endofunctor $m$ equipped with two natural transformations that satisfy some coherence conditions. The components of these transformations at $a$ are: @@ -165,7 +165,7 @@ the pair $(T\ a, \mu_a)$. So in order to define the component of the counit $\varepsilon$ at $(a, f)$, we need the right morphism in the Eilenberg-Moore category, or a homomorphism of T-algebras: \[(T\ a, \mu_a) \to (a, f)\] -Such homomorphism should map the carrier $T\ a$ to $a$. +Such a homomorphism should map the carrier $T\ a$ to $a$. Let's just resurrect the forgotten evaluator $f$. This time we'll use it as a homomorphism of T-algebras. Indeed, the same commuting diagram that makes $f$ a T-algebra may be re-interpreted to show