Definition df-termc 49634

Description: Definition of the proper class (termcnex 49737) 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 49647). 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 49676). TermCat is also the class of categories to which all categories have exactly one functor (dftermc2 49681). See also dftermc3 49692 where TermCat is defined as categories with exactly one disjointified arrow.

Unlike https://ncatlab.org/nlab/show/terminal+category 49692, we reserve the term "trivial category" for (SetCat‘1_o), justified by setc1oterm 49652.

Followed directly from the definition, terminal categories are thin (termcthin 49638). The opposite category of a terminal category is "almost" itself (oppctermco 49666). Any category 𝐶 is isomorphic to the category of functors from a terminal category to the category 𝐶 (diagcic 49701).

Having defined the terminal category, we can then use it to define the universal property of initial (dfinito4 49662) and terminal objects (dftermo4 49663). The universal properties provide an alternate proof of initoeu1 17926, termoeu1 17933, initoeu2 17931, and termoeu2 49399. Since terminal categories are terminal objects, all terminal categories are mutually isomorphic (termcciso 49677).

The dual concept is the initial category, or the empty category (Example 7.2(3) of [Adamek] p. 101). See 0catg 17602, 0thincg 49619, func0g 49250, 0funcg 49246, and initc 49252.

(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 49633	. 2 class TermCat
2		vc	. . . . . . 7 setvar 𝑐
3	2	cv 1540	. . . . . 6 class 𝑐
4		cbs 17127	. . . . . 6 class Base
5	3, 4	cfv 6489	. . . . 5 class (Base‘𝑐)
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1540	. . . . . 6 class 𝑥
8	7	csn 4577	. . . . 5 class {𝑥}
9	5, 8	wceq 1541	. . . 4 wff (Base‘𝑐) = {𝑥}
10	9, 6	wex 1780	. . 3 wff ∃𝑥(Base‘𝑐) = {𝑥}
11		cthinc 49578	. . 3 class ThinCat
12	10, 2, 11	crab 3396	. 2 class {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
13	1, 12	wceq 1541	1 wff TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

This definition is referenced by:  istermc  49635