The Clowder Project

Let $(X,x_{0})$ and $(Y,y_{0})$ be pointed sets.

1. 1.

   Functoriality. The assignments $X,Y,(X,Y)\mapsto \boldsymbol {\mathsf{Sets}}_{*}(X,Y)$ define functors

   \[ \begin{array}{ccc} \boldsymbol {\mathsf{Sets}}_{*}(X,-)\colon \mkern -15mu & \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ \boldsymbol {\mathsf{Sets}}_{*}(-,Y)\colon \mkern -15mu & \mathsf{Sets}^{\mathrlap {\mathsf{op}}}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ \boldsymbol {\mathsf{Sets}}_{*}(-_{1},-_{2})\colon \mkern -15mu & \mathsf{Sets}^{\mathsf{op}}_{*}\times \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*}. \end{array} \]

   In particular, given pointed maps

   \begin{align*} f & \colon (X,x_{0}) \to (A,a_{0}),\\ g & \colon (Y,y_{0}) \to (B,b_{0}), \end{align*}

   the induced map

   \[ \boldsymbol {\mathsf{Sets}}_{*}(f,g)\colon \boldsymbol {\mathsf{Sets}}_{*}(A,Y)\to \boldsymbol {\mathsf{Sets}}_{*}(X,B) \]

   is given by

   \[ [\boldsymbol {\mathsf{Sets}}_{*}(f,g)](\phi )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}g\circ \phi \circ f \]

   for each $\phi \in \boldsymbol {\mathsf{Sets}}_{*}(A,Y)$.

2. 2.

   Adjointness. We have adjunctions

   
   witnessed by bijections
   \begin{align*} \operatorname {\mathrm{Hom}}_{\mathsf{Sets}_{*}}(X\wedge Y,Z) & \cong \operatorname {\mathrm{Hom}}_{\mathsf{Sets}_{*}}(X,\boldsymbol {\mathsf{Sets}}_{*}(Y,Z)),\\ \operatorname {\mathrm{Hom}}_{\mathsf{Sets}_{*}}(X\wedge Y,Z) & \cong \operatorname {\mathrm{Hom}}_{\mathsf{Sets}_{*}}(X,\boldsymbol {\mathsf{Sets}}_{*}(A,Z)), \end{align*}

   natural in $(X,x_{0}),(Y,y_{0}),(Z,z_{0})\in \operatorname {\mathrm{Obj}}(\mathsf{Sets}_{*})$.