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

By Lawvere F.W.

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We denote these functors co-adjoint and adjoint to A respectively, when they exist, or by lim A , ←D ←D A →D lim if there is any danger of confusion. ) If D f ✲ A, then we also sometimes write ← f = limA ({f }) ←D → f = limA ({f }) →D if these exist, where {f } is the object in AD corresponding to f . Note that the latter notation is unambiguous since f determines its domain D and codomain A. ) f and f are objects in A if they exist. In particular, if D is a set we write Π = lim D D ←D = lim →D in any A for which the latter exist, and we call these operations product and coproduct, respectively.

F ✲ B, with A, B left complete has an adjoint if and only if Theorem 4. A functor A f commutes with equalizers and all small products and for every B ∈ |B|, there exists a vA ✲ Af , A ∈ SB , such that for every A ∈ |A| small set SB of objects in A and maps B x y and for every B ✲ A f in B there is some A ∈ SB and a map A ✲ A in A such that x = vA (yf ). Proof. 3. 2. 4. Now construct a small left pacing CB the second condition of the theorem may be phrased thus: there is a small set SB and a v ✲ (B, f ) such that the property (P) holds for u = v (by taking x = A, functor SB y = yf ).

H ❆ €€ .. ❆ € .. €€ f .. ❆ €€ €€❘. ❆❯ € q S0 is commutative in C1 . e. S0 S0 C h ✲ C in ({S0 }, C1 ) such B ✲ A S0 B is the coproduct in ({S0 }, C1 ) of A, B. ) In view of the nature of comeets in C1 , x this means that in the ({S0 }, C1 )-coproduct any map n ✲ m is represented by a string n x0 ✲ n0 ✲ n1 → · · · → n −2 x −1 ✲m x1 where each xi is a map ni−1 → ni in either A or B (n−1 being n and n −1 being m); the only relations imposed on strings are that σA = σB for all σ ∈ S0 and that x0 x1 = y0 if x0 x1 = y0 in A or in B (and consequences of these relations).

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