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Fix Typos in categories.tex #1152

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Jan 11, 2024
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Fix Typos in categories.tex #1152

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48626d2
Fix Typos in categories.tex
niyoushanajmaei Jan 2, 2024
fd00fe5
more consistent punctuation
niyoushanajmaei Jan 4, 2024
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8 changes: 4 additions & 4 deletions categories.tex
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Expand Up @@ -339,7 +339,7 @@ \section{Functors and transformations}
\end{lem}
\begin{proof}
If $\gamma$ is an isomorphism, then we have $\delta:G\to F$ that is its inverse.
By definition of composition in $B^A$, $(\delta\gamma)_a\jdeq \delta_a\gamma_a$ and similarly.
By definition of composition in $B^A$, $(\delta\gamma)_a\jdeq \delta_a\gamma_a$ and similarly $(\gamma\delta)_a\jdeq \gamma_a\delta_a$.
Thus, $\id{\delta\gamma}{1_F}$ and $\id{\gamma\delta}{1_G}$ imply $\id{\delta_a\gamma_a}{1_{Fa}}$ and $\id{\gamma_a\delta_a}{1_{Ga}}$, so $\gamma_a$ is an isomorphism.

Conversely, suppose each $\gamma_a$ is an isomorphism, with inverse called $\delta_a$, say.
Expand Down Expand Up @@ -453,7 +453,7 @@ \section{Functors and transformations}
\cref{ct:functor-assoc} is coherent, i.e.\ the following pentagon\index{pentagon, Mac Lane} of equalities commutes:
\[ \xymatrix{ & K(H(GF)) \ar@{=}[dl] \ar@{=}[dr]\\
(KH)(GF) \ar@{=}[d] && K((HG)F) \ar@{=}[d]\\
((KH)G)F && (K(HG))F \ar@{=}[ll] }
((KH)G)F && (K(HG))F \ar@{=}[ll].}
\]
\end{lem}
\begin{proof}
Expand All @@ -464,7 +464,7 @@ \section{Functors and transformations}
We have a similar coherence result for units.

\begin{lem}\label{ct:units}
For a functor $F:A\to B$, we have equalities $\id{(1_B\circ F)}{F}$ and $\id{(F\circ 1_A)}{F}$, such that given also $G:B\to C$, the following triangle of equalities commutes.
For a functor $F:A\to B$, we have equalities $\id{(1_B\circ F)}{F}$ and $\id{(F\circ 1_A)}{F}$, such that given also $G:B\to C$, the following triangle of equalities commutes:
\[ \xymatrix{
G\circ (1_B \circ F) \ar@{=}[rr] \ar@{=}[dr] &&
(G\circ 1_B)\circ F \ar@{=}[dl] \\
Expand Down Expand Up @@ -551,7 +551,7 @@ \section{Equivalences}
\indexdef{precategory!equivalence of}%
\index{functor!equivalence}%
if it is a left adjoint for which $\eta$ and $\epsilon$ are isomorphisms.
We write $\cteqv A B$ for the type of equivalences of categories from $A$ to $B$.
We write $\cteqv A B$ for the type of equivalences of (pre)categories from $A$ to $B$.
\end{defn}

By \cref{ct:adjprop,ct:isoprop}, if $A$ is a category, then the type ``$F$ is an equivalence of precategories'' is a mere proposition.
Expand Down