Basic Category Theory

We are currently reading “Basic Category Theory” by Tom Leinster.Tom Leinster, Basic Category Theory, Cambridge Studies in Advanced Mathematics 143 (Cambridge University Press, 2014, 2016), http://arxiv.org/abs/1612.09375v1.

Ch. 0: Introduction

Exercise 0.10: Find a universal property of the indiscrete topological space $I(S)$.

While all functions out of the discrete topology are continuous, all functions into the indiscrete topology are continuous. Thus, we can reverse the $X$ arrows of Example 0.5:

$$\xymatrix{
S \ar[r]^i & I(S) \\
& \forall X
    \ar[ul]^{\forall \text{\ functions\ } f}
    \ar@{.>}[u]_{\exists! \text{\ continuous\ } \bar{f}}
}$$

Exercise 0.11: Find a universal property satisfied by the pair $(\text{ker}(\theta),\iota)$ of diagram 0.2.

Diagram 0.2 was:

$$\xymatrix{
\text{ker}(\theta) \lhook\mkern-7mu \ar[r]^\iota & G \ar@<.5ex>[r]^\theta \ar@<-.5ex>[r]_\varepsilon & H
}$$

Having $\theta \circ \iota = \varepsilon \circ \iota$, with any object $F$ and $m : F \to G$, if $\theta \circ m = \varepsilon \circ m$, there exists a unique $u : F \to E$ with $\iota \circ u = m$.

$$\xymatrix{
\text{ker}(\theta) \lhook\mkern-7mu \ar[r]^\iota & G \ar@<.5ex>[r]^\theta \ar@<-.5ex>[r]_\varepsilon & H \\
\text{F} \ar@{.>}[u] \ar[ru]^m &
}$$

Exercise 0.12: $X$ is a pushout of $(i,j)$.

Exercise 0.13: (a) $\phi$ is uniquely defined by $\phi(1) \mapsto 1$, $\phi(x) \mapsto r$. All polynomial terms of one variable over $\ZZ$ can be expressed as sums and products of these primitives. (Thus $\phi(x^n) \mapsto r^n$.) (b) For the isomorphism we just need the inverse function of $\iota$, which is uniquely given by $\iota^{-1} : A \to \ZZ[x], \iota^{-1}(a) = x$.

Exercise 0.14: TBD.

Ch. 1: Categories, functors and natural transformations

Exercise 1.1.12:

Exercise 1.1.13: Assume a different $h : B \to A$. Then $h = h \circ 1_B  = h \circ (f \circ g)  = (h \circ f) \circ g  = 1_A \circ g  = g$.

Exercise 1.1.14: $$((f \times g) \circ (f' \times g'))(A, B) =
f(f'(A)) \times g(g'(B))$$

$$\text{id}_{\mathcal{A \times B}}(A, B) =
\text{id}_{\mathcal{A}}(A) \times \text{id}_{\mathcal{B}}(B)$$

Exercise 1.1.15: We need to know composition of continuous maps is closed, is associative and has an identity. Two objects are isomorphic if there are two continuous maps between them such that their composition is homotopic to the identity.