Definition df-termc 50131

Description: Definition of the proper class (termcnex 50234) of terminal categories, or final categories, i.e., categories with exactly one object and exactly one morphism, the latter of which is an identity morphism (termcid 50144). These are exactly the thin categories with a singleton base set. Example 3.3(4.c) of [Adamek] p. 24.

As the name indicates, TermCat is the class of all terminal objects in the category of small categories (termcterm3 50173). TermCat is also the class of categories to which all categories have exactly one functor (dftermc2 50178). See also dftermc3 50189 where TermCat is defined as categories with exactly one disjointified arrow.

Unlike https://ncatlab.org/nlab/show/terminal+category 50189, we reserve the term "trivial category" for (SetCat‘1_o), justified by setc1oterm 50149.

Followed directly from the definition, terminal categories are thin (termcthin 50135). The opposite category of a terminal category is "almost" itself (oppctermco 50163). Any category 𝐶 is isomorphic to the category of functors from a terminal category to the category 𝐶 (diagcic 50198).

Having defined the terminal category, we can then use it to define the universal property of initial (dfinito4 50159) and terminal objects (dftermo4 50160). The universal properties provide an alternate proof of initoeu1 18064, termoeu1 18071, initoeu2 18069, and termoeu2 49896. Since terminal categories are terminal objects, all terminal categories are mutually isomorphic (termcciso 50174).

The dual concept is the initial category, or the empty category (Example 7.2(3) of [Adamek] p. 101). See 0catg 17740, 0thincg 50116, func0g 49747, 0funcg 49743, and initc 49749.

(Contributed by Zhi Wang, 16-Oct-2025.)

Assertion

Ref	Expression
df-termc	⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

Distinct variable group:   𝑥,𝑐

Detailed syntax breakdown of Definition df-termc

Step	Hyp	Ref	Expression
1		ctermc 50130	. 2 class TermCat
2		vc	. . . . . . 7 setvar 𝑐
3	2	cv 1566	. . . . . 6 class 𝑐
4		cbs 17265	. . . . . 6 class Base
5	3, 4	cfv 6534	. . . . 5 class (Base‘𝑐)
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1566	. . . . . 6 class 𝑥
8	7	csn 4591	. . . . 5 class {𝑥}
9	5, 8	wceq 1567	. . . 4 wff (Base‘𝑐) = {𝑥}
10	9, 6	wex 1806	. . 3 wff ∃𝑥(Base‘𝑐) = {𝑥}
11		cthinc 50075	. . 3 class ThinCat
12	10, 2, 11	crab 3423	. 2 class {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
13	1, 12	wceq 1567	1 wff TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

This definition is referenced by:  istermc  50132