By Lawvere F.W.

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F is faithful iﬀ the induced map is a monomorphism of sets for every pair of objects. f is dense iﬀ for every object b ∈ |B| there is an object a ∈ |A| such that ✲ C2 , Mi ✲ Ci , and Si ✲ Ci are full af ∼ = b in B. For example, the inclusions C1 and faithful, but not dense. A proof of the following proposition will be found, for example, in Freyd’s dissertation [Freyd, 1960]. Proposition. A functor A f ✲ B is an equivalence iﬀ it is full, faithful, and dense. 2. Adjoint functors As pointed out in Section 1, for any two objects 1 A ✲ ✲ A in a category A, the category A (A, A ) is a set; however it need not be a small set (or even a ‘large’ set in our sense) so ✲ S1 (although the latter is of that in general ( , ) does not deﬁne a functor A∗ × A course true for many categories of interest).

Hence K ∼ = A. k 3 Regular epimorphisms and monomorphisms 59 For the next two propositions assume that our category has ﬁnite limits. Proposition 2. A map k is a regular monomap iﬀ k = (j1 q)E(j2 q) where q = (kj1 )E ∗ (kj2 ). K k ✲ A j1 ✲ ✲ A A q ✲Q j2 Proof. Suppose k = f Eg. Deﬁne t by A j1 ✲ A A✛ j2 A ❅ ❅ t ❅ f ❅ g ❘ ❅ ❄✠ B and let h = (j1 q)E(j2 q). Then obviously k ≤ h. To show h ≤ k, note that kj1 t = kj2 t u ✲ B such that since k = (j1 t)E(j2 t). That is, t ‘coequalizes’ kj1 , kj2 . Q t = qu.

A functor C t ✲ A with A left complete, D limA (ut) ∼ = limA (t) in A ←C ←D (and in particular, the latter exists). The following two theorems are also due in essence to [Freyd, 1960]. f ✲ B be a functor with A, B left complete. Then there exists g adTheorem 3. Let A joint to f iﬀ f is left continuous and for every B ∈ |B|, there exists a small category CB u ✲ (B, f ). and a left pacing functor CB t Proof. Suppose f has an adjoint g, and suppose D is any small category and D ✲ A is any functor.

### Functorial semantics of algebraic theories(free web version) by Lawvere F.W.

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