Aside from some spelling, punctuation and font errors please note the following corrections to CATEGORIES, ALLEGORIES by Peter Freyd and Andre Scedrov. 1.725 (p 120) 5'th line down (3'rd equation) should be: z ^ (x <--> y) = z ^ [(z ^ x) <--> (z ^ y)]. using <--> for the double-arrow operation and ^ for the meet operation. 1.82(10) (p 143) 13'th line down (first in italics) should be: A functor that preserves pre-limits preserves limits. 1.947 (p 172) 4'th line down. Formula should read R(^F) < ^F using ^ for the intersection and < for containment. 2.11 (p 196) Please note that we begin by saying that an allegory is a category. All equations of 1.1 are to be considered part of the definition of allegory. 2.158 (p 207-210) small print [thanks to Roger Maddox] The sentence about graphs on page 209, 8 lines down, is false: "If one identifies any one or any two of the pairs of vertices, the resulting graph is not in G-bar." The trouble is that if the vertices labeled s and t are identified the result is in G-bar. Worse, the displayed formula _is_ a consequence of the allegory equations (in which ' is used for reciprication): 1 ^ (R ^ R')(R ^ R')(S ^ S')(S ^ S ) < [distributivity] 1 4 2 3 2 3 1 4 1 ^ (R R ^ R'R')(S'S' ^ S S ) = [2.124] 1 2 4 3 2 1 3 4 Dom((R R ^ R'R') ^ (S'S'^ S S )') = 1 2 4 3 2 1 3 4 Dom((R R ^ S S) ^ (R R ^ S S )') = [2.124] 1 2 1 2 3 4 3 4 1 ^ (R R ^ S S )(R R ^ S S ). 1 2 1 2 3 4 3 4 The subscripts in the complicated formula in the middle of page 210 are remarkably wrong. They should be: n-1 1 ^ (R ^ R' )[ INTERSECT (R ^ R' )(S' ^ S )](S' ^ S ) 0 2n-1 i = 1 2i-1 2i 2i-1 2i 0 2n-1 n-1 < PRODUCT (R R ^ S S ) i = 0 2i 2i+1 2i 2i+1 The argument that these formulas are not consequences of the allegory equations is OK for n > 2. 2.357 (p 234) 9'th line down. Of the two equations on this line the first is, of course, just a restatement of the definition of domain of simplicity. The second, however, should be referenced with [2.124]. 2.4 (p 235) 5'th line up. Of the four R's two should be F's. The subscripts should remain R's. The "numerators" should become F's. 2.412 (p 236) 12'th line up. The reference [2.357] is wrong. It should be [2.124] 2.418 (p 238) small print 16'th line up. The formula [X]/E should be X/I. 2.444 (p 248) small print Not a correction but an improvement. A nicer example of the failure of the law of metonymy is the full sub-allegory of the allegory of Z-sets (sets with automorphisms) of all those Z-sets in which no orbit has more than 3 elements. For power-objects start with the usual construction and remove all orbits with more than 3 elements. If A is a 5-element Z-set consisting of two orbits (one with 2 and one with 3 elements) then the epsiloff relation from [A] to A is not tabular. B.211 (p 272) Last sentence. There are two rules for existential quantification and the second is needed for equality. The best correction seems to be simply to remove this last sentence (and, of course, the index entry for Horn logic). B.229 (p 274) Not a correction but an addition. The rules for the commu- tivity and idempotence of existential quantifiers are given. The same rules for universal quantifiers should also be given. B.3 (p 275-6) The definition of DERIVED PREDICATE is correct but too terse. Be warned. corrections for SUBJECT INDEX page Entries to be added: 287 ASSEMBLIES 2.153 287 CARRIER 2.153 288 CAUCUSES 2.153 289 *effective topos 2.418 292 MODULUS 2.153 Finally, there should not be an asterisk on the index entry for SUBTERMINATOR on page 295. =========================================================================