Example 4.27.
The following are some examples of presentations of logoi:
As discussed, the initial logos (no generators and no relations) is \(\An\):
\[\lra{\emptyset \mid \emptyset} \quad = \quad \An.\]The free logos on a single generator \(X\) is \(\An[X]\):
\[\lra{\{X\} \mid \emptyset} \quad = \quad \An[X] \quad = \quad \Fun(\An^{\fin}, \An).\]More generally, the logos presented by \((C,\emptyset)\) is simply the free logos generated by \(C\):
\[\lra{C \mid \emptyset} \quad = \quad \An[C] \quad = \quad \PSh(C^{\lex}).\]The free logos generated by some initial object \(X\) is simply \(\An\):
\[\lra{\{X\} \mid \emptyset \to X} \quad \simeq \quad \An.\]Indeed, logos morphisms \(F\colon \lra{\{X\} \mid \emptyset \to X} \to T\) correspond to logos morphisms \(\An[X] = \lra{\{X\} \mid \emptyset} \to T\) sending the generator \(X\) to the initial object of \(T\), but by definition of \(\An[X]\) there is a unique such morphism.
The free logos in which the initial object is terminal is the one-object logos:
\[\lra{\{X\} \mid \emptyset \to *} \quad \simeq \quad *.\]Indeed, any logos in which the initial and terminal objects agree is trivial: for all \(Y\) we have \(\emptyset \simeq \emptyset \times Y \simeq * \times Y \simeq Y\).
The free logos generated by an \(n\)-truncated object is
\[\lra{\{X\} \mid X \to X^{S^{n+1}}} \quad \simeq \quad \Fun((\An_{\leq n})^{\fin},\An),\]where \((\An_{\leq n})^{\fin} \subseteq \An_{\leq n}\) is the subcategory generated under finite colimits by the point. The left-hand side classifies the assignment \(T \mapsto T_{\leq n}\) of \(n\)-truncated objects. Indeed, an object of \(T\) corresponds to a left exact functor \((\An^{\fin})\catop\to T\). This object is \(n\)-truncated precisely when the corresponding functor factors through the opposite of the truncation functor
\[\tau_n\colon \An^{\fin}\longrightarrow (\An_{\leq n})^{\fin}.\]The free logos generated by an \(n\)-connected object \(X\) is
\[\lra{\{X\} \mid \tau_n(X) \to *}.\]It arises as a localization of \(\An[X] = \lra{* \mid \emptyset}\).
The pointed object classifier is the logos
\[\lra{[1] \mid \mathrm{source} = *} \quad \simeq \quad \PSh((\An_*^{\fin})\catop).\]Note that a logos morphism \(\lra{[1] \mid \mathrm{source} = *} \to T\) is just a morphism in \(T\) whose source is terminal, i.e. a pointed object. But the right-hand side also classifies pointed objects: we have
\[\Fun_{\bbLog}(\PSh((\An_*^{\fin})\catop),T) \, \simeq \, \Fun^{\lex}((\An_*^{\fin})\catop, T) \, \simeq \, \Fun^{\lex}((\An_*^{\fin})\catop, T_*) \, \simeq \, T_*.\]Given a Lawvere theory \(L\), there is the \(L\)-algebra classifier \(\lra{L \mid \text{finite products}}\): for every logos \(T\) we have
\[\Fun_{\bbLog}(\lra{L \mid \text{finite products}}, T) \quad \simeq \quad \Alg_L(T) \quad := \quad \Fun^{\times}(L,T).\]This logos may be explicitly described as
\[\lra{L \mid \text{finite products}} \quad \simeq \quad \PSh((\PSh_{\Sigma}(L)^{\fin})\catop).\]Here \(\PSh_{\Sigma}(L)\subseteq\Fun(L\catop,\An)\) denotes the full subcategory of finite-product-preserving presheaves.