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Yoneda Embedding

Let \( \mathcal C \) be a (small) category. Then the Yoneda embedding is the functor \( \mathcal C \rightarrow [\mathcal C ^{op}, \Set] \). Note that the “contravariant Yoneda embedding” \( \mathcal C ^{op} \rightarrow [\mathcal C, \Set]\) is the Yoneda embedding of \( \mathcal C^{op}\). The “co-Yoneda embedding” (or dual Yoneda embedding) of \( \mathcal C \) is \( \mathcal C \rightarrow [\mathcal C, \Set]^{op}\), the dual functor of the Yoneda embedding of \( \mathcal C ^{op}\).

Let \( \mathcal C \) be a locally small category. The Yoneda embedding of \( \mathcal C \) is

\begin{equation*} \gamma \colon \mathcal C \rightarrow [\mathcal C ^{op}, \Set], X \mapsto \mathcal C(-,X). \end{equation*}

Let \( \mathcal C \) be a locally small category. The co-Yoneda embedding of \( \mathcal C \) is

\begin{equation*} \gamma ' \colon \mathcal C \rightarrow [\mathcal C, \Set]^{op}, X \mapsto \mathcal C(X,-). \end{equation*}

Let \( \mathcal C \) be a locally small category. Then:

  1. The Yoneda embedding \( \gamma \) is fully faithful (Yoneda lemma).
  2. \( \gamma \) preserves limits, ie \( \gamma \) is continuous.
  3. \( \gamma \) is the free cocompletion of \( \mathcal C \).

Let \( \mathcal C \) be a locally small category. Then:

  1. The co-Yoneda embedding \( \gamma ' \) is fully faithful.
  2. \( \gamma ' \) preserves colimits, ie \( \gamma ' \) is cocontinuous.
  3. \( \gamma ' \) is the free completion of \( \mathcal C \).

Author: Frederik Gebert

Created: 2025-02-10 Mon 21:45